#
Chapter 1
Linear and Nonlinear Partial Integro-Differential Equations arising from
Finance

José M. T. S. Cruz1, Maria do R. L. Grossinho1, Daniel Ševčovič2, and Cyril
Izuchukwu Udeani2
1 ISEG, University of Lisbon, Portugal,
2 Comenius University in Bratislava, Slovakia Corresponding author: D.
Ševčovič; email<EMAIL_ADDRESS>
Acknowledgments: Support of the Slovak Research and Development Agency under
the project APVV-20-0311 (C.U.) is kindly acknowledged. The research was also
supported by the VEGA 1/0611/21 grant (D.Š.).

###### Abstract

The purpose of this review paper is to present our recent results on nonlinear
and nonlocal mathematical models arising from modern financial mathematics. It
is based on our four papers written jointly by J. Cruz, M. Grossinho, D.
Ševčovič, and C. Udeani [20], [19], [58], [66], as well as parts of PhD thesis
by J. Cruz [18]. We investigated linear and nonlinear partial integro-
differential equations (PIDEs) arising from option pricing and portfolio
selection problems and studied the systematic relationships between the PIDEs
with option pricing theory and Black–Scholes models. First, we relax the
liquid and complete market assumptions and extend the models that study the
market’s illiquidity to the case where the underlying asset price follows a
Lévy stochastic process with jumps. Then, we establish the corresponding PIDE
for option pricing under suitable assumptions. The qualitative properties of
solutions to nonlocal linear and nonlinear PIDE are presented using the theory
of abstract semilinear parabolic equation in the scale of Bessel potential
spaces. The existence and uniqueness of solutions to the PIDE for a general
class of the so-called admissible Lévy measures satisfying suitable growth
conditions at infinity and origin are also established in the multidimensional
space. Additionally, the qualitative properties of solutions to the
generalized PIDE are investigated by considering a general shift function
arising from nonlinear option pricing models, which takes into account a large
trader stock-trading strategy with the underlying asset price following the
Lévy process. For the portfolio management problem, we present the existence
and uniqueness results of the fully nonlinear HBJ equation arising from the
stochastic dynamic optimization problem in Sobolev spaces using the theory of
monotone operator technique, which can also be viewed as PIDE in some sense.
Furthermore, a stable, convergent, and consistent numerical scheme that can
give an approximate solution to such PIDE is presented, and various numerical
experiments are conducted to illustrate the influence of a large trader and
the intensity of jumps on the option price.

Keywords: Lévy measure, Option pricing, Partial integro-differential equation,
Hamilton-Jacobi-Bellman equation, Maximal monotone operator, Dynamic
stochastic portfolio optimization

###### Contents

  1. 1\. Introduction
  2. 2\. Background and Motivation
  3. 3\. Preliminaries and Definitions
    1. 3.1. Exponential Lévy models
    2. 3.2. Examples of Lévy processes in finance
    3. 3.3. Admissible activity Lévy measures
  4. 4\. Multidimensional Linear and Nonlinear PIDE
    1. 4.1. Existence and uniqueness results of PIDE
    2. 4.2. Maximal monotone operator technique for solving nonlinear parabolic equations
  5. 5\. Applications to Option Pricing
    1. 5.1. Linearization of PIDE
  6. 6\. Feedback effects under jump-diffusion asset price dynamics
    1. 6.1. Numerical simulation for the underlying PIDE
      1. 7. Hamilton-Jacobi-Bellman equation
        1. 7.1. Static Markowitz model for portfolio optimization
          1. 7.2. Riccati transformation of the HJB equation and application to optimal portfolio selection problem
            1. 8\. Conclusions

##  1\. Introduction

This review paper contain our recent advances in the research focused on
nonlinear and nonlocal mathematical models arising from modern financial
mathematics. The major parts of this chapter are based on our four papers
jointly written by J. Cruz, M. Grossinho, D. Ševčovič, and C. Udeani [20],
[19], [58], [66], as well as parts of PhD thesis by J. Cruz.

The classical Black–Scholes model has been widely used in financial industry
because of its simplicity and existence of analytical formula for pricing
derivative securities. This model relies on the restrictive assumptions, such
as completeness, frictionless of the market, and the assumption that the
underlying asset price follows a geometric Brownian motion. However, the
assumption that an investor can trade large amount of assets without affecting
the underlying asset price is usually not satisfied, especially in illiquid
markets. It is also known that the fully nonlinear Hamilton–Jacobi–Bellman
equation plays an essential role in finance. For instance, it gives the
necessary and sufficient condition of a control with respect to the value
function. Therefore, this chapter investigates linear and nonlinear partial
integro-differential equations (PIDEs) arising from option pricing and
portfolio selection problem. We investigate the systematic relationships of
the PIDEs with option pricing theory and Black–Scholes models. First, we relax
the liquid and complete market assumptions and extend the models that study
market’s illiquidity to the case where the underlying asset price follows a
Lévy stochastic process with jumps. Then, we establish the corresponding PIDE
for option pricing under suitable assumptions. The qualitative properties of
solutions to nonlocal linear and nonlinear PIDE are presented using the theory
of abstract semilinear parabolic equation in the scale of Bessel potential
spaces. The existence and uniqueness of solutions to the PIDE for a general
class of the so-called admissible Lévy measures satisfying suitable growth
conditions at infinity and origin are also established, and the solution to
the corresponding PIDE are presented in the multidimensional space.
Additionally, the qualitative properties of solutions to the generalized PIDE
are investigated by considering a general shift function arising from
nonlinear option pricing models, which takes into account a large trader
stock-trading strategy with the underlying asset price following the Lévy
process. For the portfolio management problem, we present the existence and
uniqueness results to the fully nonlinear HBJ equation arising from stochastic
dynamic optimization problem in Sobolev spaces using the theory of monotone
operator technique, which can also be viewed as PIDE in some sense.
Furthermore, a stable, convergent, and consistent numerical scheme that can
give an approximate solution of such PIDE is presented, and various numerical
experiments are conducted to illustrate the influence of a large trader and
intensity of jumps on the option price.

The Black–Scholes model and Hamilton–Jacobi–Bellman (HJB) equation have been
widely used in financial markets. However, evidence from the stock market
observation shows that the Black-Scholes model is not the most realistic one
because it depends on some restrictive assumptions, such as the liquidity,
completeness, and frictionless of the market. Additionally, the linear
Black–Scholes equation provides a solution that corrsponds to a perfectly
replicated portfolio, which is not a desirable property. For this reason,
several attempts have been made to generalize and relax some of these
assumptions. Some authors relaxed these assumptions by considering the
presence of transaction costs (see Kwok [41] and Avellaneda and Paras [6]),
feedback and illiquid market effects due to large traders choosing given
stock-trading strategies (Schönbucher and Willmott [55], Frey and Patie [25],
Frey and Stremme [26]), and risk from the unprotected portfolio (Jandačka and
Ševčovič [32]). In these generalizations, the constant volatility was replaced
by a nonlinear function depending on the second derivative of the option
price. Frey and Stremme derived a nonlinear Black–Scholes model that plays an
essential role in the class of the generalized Black–Scholes equation with
such a nonlinear diffusion function [25, 32, 27]). In this model, the asset
dynamics considers the presence of feedback effects due to a large trader
choosing his/her stock-trading strategy [55]. Another important direction in
generalizing the original Black–Scholes equation arises from the fact that the
sample paths of a Brownian motion are continuous; however, the realized stock
price of a typical company exhibits random jumps over the intraday scale,
making the price trajectories discontinuous. In the classical Black–Scholes
model, the logarithm of the price process has normal distribution. However,
the empirical distribution of stock returns exhibits fat tails. Meanwhile,
when calibrating the theoretical prices to the market prices, the implied
volatility is not constant as a function of strike price nor a function of
time to maturity, contradicting the prediction of the Black–Scholes model.
However, the models with jumps and diffusion can solve the problems inherent
to the Black–Scholes model. The jump models also plays an essential role in
the option market. In the Black–Scholes model, the market is complete,
implying that every payoff can exactly be replicated; meanwhile, there is no
perfect hedge in jump models, making the way of options not redundant.

Market Illiquidity has been widely studied in the literature [33, 52, 64, 56,
28]. The first major contribution was made by Robert Jarrow, in 1994, who
studied the market manipulation strategies that may arise in illiquid markets.
The author also studied the option pricing theory in discrete time when there
is a large trader. The pricing argument used was a condition to ensure that no
market manipulation strategy is used by the large trader and large trader’s
optimality conditions; thus, replacing the usual free-arbitrage argument.
Then, Frey (1998) extended Jarrow’s analysis to the continuous time case and
established the existence and uniqueness of solution of a nonlinear partial
differential equation satisfied by the large trader’s hedging strategy.
Additionally, Platten and Schweizer (1998) proposed an explanation for the
smile and skewness for the implied volatilities and showed that hedging
strategies followed by large traders can lead to option price bias. Sircar and
Papanicolaou (1998) also proposed a model where the derivative security price
is characterized by a nonlinear partial differential equation that becomes the
Black–Scholes equation when there is no feedback. When the programme traders
are a small fraction of the economy, numerical and analytical methods can be
used to analyze the nonlinear partial differential equation through
perturbation. This equation is derived using an argument similar to the one
used in deriving classical Black–Scholes equation. Consequently, they obtained
that this model also predict increased implied volatilities as in Platten and
Schweizer. Furthermore, Schonbucher and Willmott (2000) analyzed the feedback
effects from the presence of hedging strategies. They also derived a nonlinear
partial differential equation for an option replication strategy and studied
these effects for a put option. The effects are more pronounced in markets
with low liquidity, which can induce discontinuities in the price process.
However, none of these studies that investigated jump models [45, 44, 10, 14,
49, 15, 22] considered in the market’s illiquidity. Meanwhile, investors and
risk managers have realized that financial models based on the assumption that
an investor can trade large amounts of an asset without affecting its price is
no longer true in markets that are not liquid. Therefore, in this chapter, we
relax the liquid and complete market hypothesis and extend the models that
study market’s illiquidity to the case where the underlying asset price
follows a Lévy stochastic process with jumps to obtain a model for pricing
European and American call and put options on an underlying asset
characterized by a Lévy measure. In this way, it is assumed that trading
strategies affects the stock price, and the possibility to account for sudden
jumps that might occur when the market is under stress .

Recently, the relationships between more general nonlocal operators and jump
processes have been widely investigated. For instance, there is an actual
connection between the solution to PIDE and properties of the corresponding
Markov jump process (c.f., Abels and Kassmann [2]; Florescu and Mariani [24]).
In the past decades, the role of PIDEs has been investigated in various
fields, such as pure mathematical areas, biological sciences, and economics
[3, 4, 69]. PIDE problems arising from financial mathematics, especially from
option pricing models, have been of great interest to many researchers. In
most cases, standard methods for solving these problems lead to the study of
parabolic equations. Mikulevičius and Pragaraustas [47] investigated solutions
of the Cauchy problem to the parabolic PIDE with variable coefficients in
Sobolev spaces. They employed their results to obtain solutions of the
corresponding martingale problem. Crandal _et al._ [31] employed the notion of
a viscosity solution to investigate the qualitative results. Soner _et al._
[12] and Barles _et al._ [8] extended and generalized their results for the
first and second order operators, respectively. Florescu and Mariani [24]
employed the Schaefer fixed point argument to establish existence of a weak
solution of the generalized PIDE. Amster _et al._ [57] used the notion of
upper and lower solutions to obtain the solution to such PIDEs. They proved
the existence of solutions in a general domain for multiple assets and the
regime switching jump-diffusion model. Cont _et al._ [16] investigated the
actual connection between option pricing in exponential Lévy models and the
corresponding PIDEs for European options and those with single or double
barriers. They discussed and established the conditions for which prices of
option are classical solution of the corresponding PIDE. In this chapter, we
obtain a certain partial integro-differential equation (PIDE) for option
pricing in illiquid market by assuming a certain dynamics for the stock’s
price. The existence of solution and localization results of the associated
PIDE are also established. We investigated and established the qualitative
properties of solutions to the nonlocal linear and nonlinear PIDE in the scale
of Bessel potential spaces using the theory of abstract semilinear parabolic
equation. Furthermore, we present the existence and uniqueness results for
nonlinear parabolic equations using monotone operator technique, Fourier
transform, and Banach fixed point argument. We considered the fully nonlinear
HJB equation arising portfolio selection problem, where the goal of an
investor is to optimize the condition expected value of the terminal utility
of the portfolio. Such nonlinear parabolic equation is presented in an
abstract setting, which can also be viewed as a nonlinear PIDE. Many previous
studies have developed numerical methods for PIDEs, such as finite difference
and finite element methods. However, the equation corresponding to the case of
illiquid markets is more difficult. Therefore, this chapter also presents a
stable, convergent, and consistent numerical scheme that can give an
approximate solution of such PIDEs. Various numerical experiments are
presented to illustrate the influence of a large trader and intensity of jumps
on the option price.

##  2\. Background and Motivation

Based on the classical theory developed by Black, Scholes, and Merton, the
price $V(t,S)$ of an option in a stylized market at time $t\in[0,T]$ and the
underlying asset price $S$ can be calculated as a solution to the following
linear Black–Scholes parabolic equation:

$\frac{\partial V}{\partial
t}(t,S)+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}V}{\partial
S^{2}}(t,S)+rS\frac{\partial V}{\partial S}(t,S)-rV(t,S)=0,\mbox{
}t\in[0,T),S>0.$ (1)

Here, $\sigma>0$ is the historical volatility of the underlying asset driven
by the geometric Brownian motion, and $r>0$ is the risk-free interest rate of
zero-coupon bond. The solution to above equation is subject to the terminal
payoff condition $V(T,S)=\Phi(S)$ at maturity $t=T$. Meanwhile, evidence from
stock markets observations indicates that the model is not the most realistic
one because it assumes that the market is liquid, complete, frictionless and
without transaction costs. It is also known that the linear Black–Scholes
equation provides a solution corresponding to a perfectly replicated
portfolio, which need not be a desirable property. To solve this issues,
several attempts have been made to generalized the linear Black–Scholes
equation (1) by replacing the constant volatility $\sigma$ with a nonlinear
function $\tilde{\sigma}(S\partial_{S}^{2}V)$ depending on the second
derivative $\partial_{S}^{2}V$ of the option price. In this regard, Frey and
Stremme derived a nonlinear Black–Scholes model, which plays an essential role
in the class of generalized Black–Scholes equation with such a nonlinear
diffusion function [25, 27]). They considered that case in which the asset
dynamics takes into account the presence of feedback effects due to a large
trader choosing his/her stock-trading strategy (see also [55]). The diffusion
coefficient is non-constant, and it is given by

$\tilde{\sigma}(S\partial^{2}_{S}V)^{2}=\sigma^{2}\left(1-\varrho
S\partial^{2}_{S}V\right)^{-2},$ (2)

where $\sigma$ and $\varrho>0$ are constants.

Furthermore, several researchers have attempted to generalized the original
Black–Scholes equation arises from the fact that the sample paths of a
Brownian motion are continuous. However, the realized stock price of a typical
company exhibits random jumps over the intraday scale, making the price
trajectories discontinuous. The underlying asset price process is usually
assumed to follow a geometric Brownian motion in the classical Black–Scholes
model. However, the empirical distribution of stock returns exhibits fat
tails. The models with jumps and diffusion can solve the problems inherent to
the linear Black–Scholes model and play an essential role in options pricing.
It is well-known that the market is complete in the Black–Scholes model,
illustrating that each payoff can be perfectly replicated; however, there is
no perfect hedge in jump–diffusion models, making the options not redundant.
It turns out that the option price can be computed from the solution $V(t,S)$
to the following PIDE Black–Scholes equation [20]:

$\displaystyle\frac{\partial V}{\partial
t}(t,S)+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}V}{\partial
S^{2}}(t,S)+rS\frac{\partial V}{\partial S}(t,S)-rV(t,S)$
$\displaystyle+\int_{\mathbb{R}}V(t,S+H(z,S))-V(t,S)-H(z,S)\frac{\partial
V}{\partial S}(t,S)\nu(\mathrm{d}z)=0,$ (3)

where $H(z,S)=S(e^{z}-1)$, and $\nu$ is the so-called Lévy measure
characterizing the underlying asset process with random jumps in time and
space. It is worth noting that (3) reduces to the classical linear
Black–Scholes equation (1) if $\nu=0$.

In this chapter, we consider both directions of generalizations of the
Black–Scholes equation. First, we relaxed the assumption of liquid market
following the Frey–Stremme model by assuming that the underlying asset price
follows a Lévy stochastic process with jumps and established the corresponding
PIDEs. Then, we present the existence and uniqueness results to the linear and
nonlinear nonlocal PIDE in the framework of Bessel potential spaces for the
multidimensional case. A more generalized nonlinear nonlocal PIDE is also
presented by considering a shift function $\xi=\xi(\tau,x,z)$ depending on the
variables $x,z\in\mathbb{R}^{n}$. Furthermore, we derive, analyze, and perform
numerical computation of the model. We also show that the corresponding PIDE
nonlinear equation has the following form:

$\displaystyle\frac{\partial V}{\partial
t}+\frac{1}{2}\frac{\sigma^{2}}{\left(1-\varrho
S\partial_{S}\phi\right)^{2}}S^{2}\frac{\partial^{2}V}{\partial
S^{2}}+rS\frac{\partial V}{\partial S}-rV$
$\displaystyle+\int_{\mathbb{R}}V(t,S+H(t,z,S))-V(t,S)-H(t,z,S)\frac{\partial
V}{\partial S}\nu(\mathrm{d}z)=0.$ (4)

It is worth noting that the function $H(t,z,S)$ may depend on the large trader
strategy function $\phi=\phi(t,S)$ and the delta $\partial_{S}V$ of the price
$V$ if $\varrho>0$.

We consider a stylized economy with two traded assets: a riskless asset (a
bond with a price $B_{t}$ taken as numeraire) and a risky asset (stock with a
price $S_{t}$). Here, we assume that the bond market is perfectly elastic
since it is more liquid than stocks. Here, we consider two type of traders:
reference and program traders. The program traders are also known as portfolio
insurers because they use dynamic hedging strategies to hedge portfolio
against jumps in stock prices. They are classified as single traders or group
of traders acting together. It is assumed that their trades influence the
stock price equilibrium. In contrast, the reference traders can be considered
as representative traders of many small agents. We assume that they act as
price takers. Generally, $\tilde{D}(t,Y_{t},S_{t})$ represents the reference
trader demand function that depends on the income process $Y_{t}$ or some
other fundamental state variable influencing the reference trader demand. The
aggregate demand of program traders is denoted by
$\varphi(t,S_{t})=\xi\phi(t,S_{t})$, where $\xi$ is the number of written
identical securities that the program traders are trying to hedge, and
$\phi(t,S_{t})$ is the demand per unit of the security being hedged. For
simplicity, we assume that $\xi$ is the same for every program trader. For a
more general case where different securities are considered, see [64]. Suppose
the supply of a stock with the price $\tilde{S_{0}}$ is constant, and let
$D(t,Y,S)=\frac{\tilde{D}(t,Y,S)}{\tilde{S_{0}}}$ be the quantity demanded by
a reference trader per unit supply. Then, the total demand relative to the
supply at time $t$ is given by $G(t,Y,S)=D(t,Y,S)+\rho\phi(t,S)$, where
$\rho=\frac{\xi}{\tilde{S_{0}}}$, and $\rho\phi(t,S)$ is the proportion of the
total supply of the stock traded by program traders. Therefore, to obtain the
market equilibrium, the variables $Y$ and $S$ should satisfy $G(t,Y,S)=1$.
Assume that the function $G$ is monotone with respect to the variables $Y$ and
$S$, and it is sufficiently smooth. Then, we can solve the implicit equation
$G(t,Y_{t},S_{t})=1$ to obtain $S_{t}=\psi(t,Y_{t})$, where $\psi$ is a
sufficiently smooth function. By employing the approach in [64], we assume
that the stochastic process $Y_{t}$ has the following dynamics:

$\mathrm{d}Y_{t}=\mu(t,Y_{t})\mathrm{d}t+\eta(t,Y_{t})\mathrm{d}W_{t}.$

Then, using Itô’s lemma for the process $S_{t}=\psi(t,Y_{t})$, we have

$\mathrm{d}S_{t}=\left(\partial_{t}\psi+\mu\partial_{y}\psi+\frac{\eta^{2}}{2}\partial^{2}_{y}\psi\right)\mathrm{d}t+\eta\partial_{y}\psi\mathrm{d}W_{t}\equiv
b(t,S_{t})S_{t}\mathrm{d}t+v(t,S_{t})S_{t}\mathrm{d}W_{t}.$ (5)

It means that $S_{t}$ follows a geometric Brownian motion with a nonconstant
volatility function $v(t,S)=\eta(t,Y)\partial_{Y}\psi(t,Y)/\psi(t,Y)$, where
$Y=\psi^{-1}(t,S)$. Thus, we follow the argument used in the derivation of the
original Black–Scholes equation to obtain a generalization of the
Black–Scholes partial differential equation with a nonconstant volatility
function $\sigma=v(t,S)$. We employ the Frey and Stremme’s approach (cf. [25,
27]) to prescribe a dynamics for the underlying stock price instead of
deriving it using the market equilibrium and dynamics for the income process
$Y_{t}$ as it is done in [64]. In this way, Frey and Stremme derived the same
stock price dynamics as in [64] corresponding to a situation where the demand
function is of logarithmic type, $D(Y,S)=\ln(\frac{Y^{\gamma}}{S})$, where
$\gamma=\frac{\sigma}{\eta_{0}}$, and the income process $Y_{t}$ follows a
geometric Brownian motion, i.e.,

$\displaystyle\partial_{Y}D(Y,S)=\gamma\frac{1}{Y},\
\partial_{S}D(Y,S)=-\frac{1}{S},\ \
\mathrm{d}Y_{t}=\mu_{0}Y_{t}\mathrm{d}t+\eta_{0}Y_{t}\mathrm{d}W_{t},$ (6)
$\displaystyle
v(t,S)=\eta(t,Y)\frac{\partial_{Y}\psi(t,Y)}{\psi(t,Y)}=-\frac{\eta_{0}Y}{S}\frac{\gamma\frac{1}{Y}}{-\frac{1}{S}+\rho\frac{\partial\phi}{\partial
S}}=\frac{\sigma}{1-\rho S\frac{\partial\phi}{\partial S}}.$

Assuming the delta hedging strategy with $\phi(t,S)=\partial_{S}V(t,S)$ and
substituting the volatility function $v(t,S)$ in (5), we obtain the
generalized Black–Scholes equation with the nonlinear diffusion function of
the form (2).

In this chapter, we first generalized the Frey–Stremme model by considering an
underlying asset following a Lévy process with jumps. Then, we establish the
corresponding PIDE for option pricing. Furthermore, we investigated the
existence and uniqueness of solutions to such PIDE in the multidimensional
spaces.

##  3\. Preliminaries and Definitions

This section presents some basic definitions and properties of Lévy measures
and notion of admissible activity Lévy measures. Here,$|\cdot|$ and
$\|\cdot\|$ represent the Euclidean norm in $\mathbb{R}^{n}$ and the norm in
an infinite dimensional function space (e.g.,
$L^{p}(\mathbb{R}^{n}),X^{\gamma}$). HIn what follows, $a\cdot b$ stands for
the usual Euclidean product in $\mathbb{R}^{n}$ with the norm
$|z|=\sqrt{z\cdot z}$.

######  Definition 1.

[58] A Lévy process on $\mathbb{R}^{n}$ is a stochastic (right continuous)
process $X=\\{X_{t},t\geq 0\\}$ having the left limit with independent
stationary increments. It is uniquely characterized by its Lévy exponent
$\phi$:

$\mathbb{E}_{x}(e^{iy\cdot X_{t}})=e^{-t\phi(y)},~{}y\in\mathbb{R}^{n}.$

The subscript $x$ in the expectation operator $\mathbb{E}_{x}$ indicate that
the process $X_{t}$ starts from a given value $x$ at the origin $t=0$. The
Lévy exponent $\phi$ has the following unique decomposition:

$\phi(y)=ib\cdot
y+\sum_{i,j=1}^{n}a_{ij}y_{i}y_{j}+\int_{\mathbb{R}^{n}}\left(1-e^{iy\cdot
z}+iy\cdot z1_{|z|\leq 1}\right)\nu(\mathrm{d}z),$

where $b\in\mathbb{R}^{n}$ is a constant vector; $(a_{ij})$ is a constant
matrix, which is positive semidefinite; $\nu(\mathrm{d}z)$ is a nonnegative
measure on $\mathbb{R}^{n}\setminus\\{0\\}$ such that
$\int_{\mathbb{R}^{n}}\min(1,|z|^{2})\nu(\mathrm{d}z)<\infty$ (c.f., [50]).

###  3.1. Exponential Lévy models

Let $X_{t},t\geq 0,$ be a stochastic process. The Poisson random measure
$\nu(A)$ of a Borel set $A\in\mathcal{B}(\mathbb{R})$ is defined by
$\nu(A)=\mathbb{E}\left[J_{X}([0,1]\times A)\right]$, where $J_{X}([0,t]\times
A)=\\#\left\\{s\in[0,t]:\Delta X_{s}\in A\right\\}$. This measure gives the
mean number per unit time of jumps whose amplitude belongs to the set $A$. It
is worth noting that the Lévy–Itô decomposition provides a representation of
$X_{t}$ interpreted as a combination of a Brownian motion with a drift
$\omega$ and an infinite sum of independent compensated Poisson processes with
variable jump sizes (see [19]), i.e.,

$\mathrm{d}X_{t}=\omega\mathrm{d}t+\sigma\mathrm{d}W_{t}+\int_{\left|x\right|\geq
1}xJ_{X}\left(\mathrm{d}t,\mathrm{d}x\right)+\int_{\left|x\right|<1}x\widetilde{J}_{X}\left(\mathrm{d}t,\mathrm{d}x\right),$

where $\widetilde{J}_{X}\left(\left[0,t\right]\times
A\right)=J_{X}\left(\left[0,t\right]\times A\right)-t\nu\left(A\right)$ is the
compensation of $J_{X}$.

######  Remark 1.

[19, Remark 1] Note that any Lévy process is a strong Markov process, and the
associated semigroup is a convolution semigroup. Its infinitesimal generator
$L:u\mapsto L[u]$ is a nonlocal partial integro-differential operator given by

$\displaystyle L[u](x)$ $\displaystyle=$ $\displaystyle\lim_{h\to
0^{+}}\frac{\mathbb{E}\left[u\left(x+X_{h}\right)\right]-u\left(x\right)}{h}$
(7) $\displaystyle=$
$\displaystyle\frac{\sigma^{2}}{2}\frac{\partial^{2}u}{\partial
x^{2}}+\gamma\frac{\partial u}{\partial
x}+\int_{\mathbb{R}}\left[u\left(x+y\right)-u\left(x\right)-y1_{\left|y\right|\leq
1}\frac{\partial u}{\partial x}\left(x\right)\right]\nu(\mathrm{d}y),$

which is well-defined for any compactly supported function $u\in
C^{2}_{0}\left(\mathbb{R}\right)$.

Let $S_{t},t\geq 0,$ be a stochastic process representing an underlying asset
process under a filtered probability space
$\left(\Omega,\mathcal{F},\left\\{\mathcal{F}_{t}\right\\},\mathbb{P}\right)$.
The filtration $\left\\{\mathcal{F}_{t}\right\\}$ represents the price history
up to the time $t$. In an arbitrage-free market, there is an equivalent
measure $\mathbb{Q}$ under which discounted prices of all traded assets are
$\mathbb{Q}-$ martingales, which is called the fundamental theorem of asset
pricing (see, [17]). The measure $\mathbb{Q}$ is also known as the risk-
neutral measure. We consider the exponential Lévy model in which the risk-
neutral price process $S_{t}$ under $\mathbb{Q}$ is given by
$S_{t}=e^{rt+X_{t}}$, where $X_{t}$ is a Lévy process under $\mathbb{Q}$ with
the characteristic triplet $\left(\sigma,\gamma,\nu\right)$. Then, the
arbitrage-free market hypothesis states that
$\widehat{S}_{t}=S_{t}e^{-rt}=e^{X_{t}}$ is a martingale, which is equivalent
to the following conditions imposed on the triplet
$\left(\sigma,\gamma,\nu\right)$:

$\int_{\left|y\right|\geq 1}e^{y}\nu(\mathrm{d}y)<\infty,\ \
\gamma\in\mathbb{R},\ \
\gamma=-\frac{\sigma^{2}}{2}-\int_{-\infty}^{+\infty}\left(e^{y}-1-y1_{\left|y\right|\leq
1}\right)\nu(\mathrm{d}y).$ (8)

The risk-neutral dynamics of $S_{t}$ under $\mathbb{Q}$ is given by

$\mathrm{d}S_{t}=rS_{t}\mathrm{d}t+\sigma
S_{t}\mathrm{d}W_{t}+\int_{\mathbb{R}}\left(e^{y}-1\right)S_{t}\widetilde{J}_{X}\left(dt,dy\right).$
(9)

The exponential price process, $e^{X_{t}},t\geq 0$, is also a Markov process
with the state space $\left(0,\infty\right)$. It has the following
infinitesimal generator:

$\displaystyle L^{S}[V](S)$ $\displaystyle=$ $\displaystyle\lim_{h\rightarrow
0}\frac{\mathbb{E}[V(Se^{X_{h}})]-V(S)}{h}=rS\frac{\partial V}{\partial
S}+\frac{\sigma^{2}}{2}S^{2}\frac{\partial^{2}V}{\partial S^{2}}$ (11)
$\displaystyle+\int_{\mathbb{R}}\left[V(Se^{y})-V(S)-S(e^{y}-1)\frac{\partial
V}{\partial S}\right]\nu(\mathrm{d}y)$

(see [17]). A Lévy process with the following representation:

$\mathrm{d}X_{t}=\omega\mathrm{d}t+\sigma\mathrm{d}W_{t}+\int_{|x|\geq
1}K(t,x)J_{X}(\mathrm{d}t,\mathrm{d}x)+\int_{|x|<1}H(t,x)\tilde{J}_{X}(\mathrm{d}t,\mathrm{d}x).$

is called the Lévy type stochastic integral. The following variant of Itô’s
lemma is an essential results, which will be needed later in this study.

######  Theorem 1.

[20, Therorem 2.1] Let $f\in C^{1,2}([0,T]\times\mathbb{R})$ and $H,K\in
C([0,T]\times\mathbb{R})$. Suppose $X_{t},t\geq 0,$ is a Lévy stochastic
process. Then,

$\displaystyle\mathrm{d}f(t,X_{t})$ $\displaystyle=$
$\displaystyle\frac{\partial f}{\partial t}\mathrm{d}t+\frac{\partial
f}{\partial x}\mathrm{d}X_{t}+\frac{1}{2}\frac{\partial^{2}f}{\partial
x^{2}}\mathrm{d}[X_{t},X_{t}]$ $\displaystyle+\int_{|x|\geq
1}f(t,X_{t}+K(t,x))-f(t,X_{t})J_{X}(\mathrm{d}t,\mathrm{d}x)$
$\displaystyle+\int_{|x|<1}f(t,X_{t}+H(t,x))-f(t,X_{t})\tilde{J}_{X}(\mathrm{d}t,\mathrm{d}x)$
$\displaystyle+\int_{|x|<1}f(t,X_{t}+H(t,x))-f(t,X_{t})-H(t,x)\frac{\partial
f}{\partial x}(t,X_{t})\nu(\mathrm{d}x)\mathrm{d}t.$

###  3.2. Examples of Lévy processes in finance

There are two types of exponential Lévy models considered in the literature.
The first types are jump-diffusion models, where the log-price is represented
as a Lévy process with a nonzero diffusion part $(\sigma>0)$ and a jump
process with finite activity $($i.e., $\nu(\mathbb{R})<\infty)$. The second
types of models are infinite activity pure jump models, where there is no
diffusion part and only a jump process with infinite activity $($i.e.,
$\nu(\mathbb{R})=\infty)$. This section presents different types of
exponential Lévy models that differ in the choice of the Lévy measure.

####  3.2.1. Jump-Diffusion models

A Lévy process with jump-diffusion has the following general form:

$X_{t}=\gamma t+\sigma W_{t}+\sum_{i=1}^{N_{t}}Y_{i},$

where $\sigma>0$, and $N_{t}$ is a Poisson process with intensity $\lambda$
that counts the jumps of $X_{t}$, and $Y_{i},i=1,2,3...$ are independent and
identically distributed random variables with distribution $\mu$. The Lévy
measure $\nu$ is $\lambda\mu$, and the drift $\gamma$ is given by

$-\frac{\sigma^{2}}{2}-\int_{\mathbb{R}}\left(e^{y}-1-y1_{\left|y\right|\leq
1}\right)\nu\left(\mathrm{d}y\right).$

#### Merton’s model

This is the first jump-diffusion model proposed by Merton [45] in the
financial application. The random variables $Y_{i},i=1,2,3...$, are normally
distributed with mean $m$ and variance $\delta^{2}$. It has the following Lévy
density:

$\nu(\mathrm{d}z)=\lambda\frac{1}{(2\pi\delta^{2})^{n/2}}e^{-\frac{|z-m|^{2}}{2\delta^{2}}}\mathrm{d}z\,,$
(13)

where the parameters $m\in\mathbb{R}^{n},\lambda,\delta>0,$ are given.
Merton’s measure is a finite activity Lévy measure, i.e.,
$\nu(\mathbb{R}^{n})=\int_{\mathbb{R}^{n}}\nu(\mathrm{d}z)<\infty$, with
finite variation $\int_{|z|\leq 1}|z|\nu(\mathrm{d}z)<\infty$. Therefore, the
probability density of $X_{t}$ can be obtain as a series that converges
rapidly (see [17]): rCl p_t(x)&=∑_j=0^∞e^-λt(λt)^je-—z-γt-jm—22(σ2t+jδ2)j!2
π(σ2t+jδ2). Thus, the price of an European call option can be expressed as a
weighted sum of Black–Scholes prices:

$C_{Merton}(S_{0},K,T,\sigma,r)=e^{-rT}\sum_{j=0}^{\infty}e^{-\lambda
t}\frac{(\lambda
t)^{j}}{j!}e^{r_{j}T}C_{BS}(S_{0}e^{\frac{j\delta^{2}}{T}},K,T,\sigma_{j},r_{j}),$
(14)

where $r_{j}=r-\lambda(e^{m+\frac{\delta^{2}}{2}}-1)+\frac{jm}{T}$,
$\sigma_{j}=\sqrt{\sigma^{2}+\frac{j\delta^{2}}{T}}$, and
$C_{BS}(S,K,T,\sigma,r)$ is the well-known Black–Scholes formula.

####  3.2.2. Infinite activity pure jump models

The variance Gammma and normal inverse Gaussian (NIG) processes are obtained
by a subordination of a Brownian motion and a tempered $\alpha$-stable
process; variance Gamma and NIG processes correspond to $\alpha=0$ and
$\alpha=1/2$, respectively. These models are widely used in finance because of
the existence of probability density of the subordinator in a closed form for
these values of $\alpha$ (see [17]).

#### Variance Gamma process

This is a pure discontinuous process of infinite activity and finite variation
($\int_{|x|\leq 1}|x|\nu(\mathrm{d}x)<\infty$) that is widely used in the
financial modeling. It has the following Lévy measure:

$\nu\left(x\right)=\frac{1}{\kappa\left|x\right|}e^{Ax-B\left|x\right|}\text{
with }A=\frac{\theta}{\sigma^{2}}\text{ and
}B=\frac{\sqrt{\theta^{2}+2\frac{\sigma^{2}}{\kappa}}}{\sigma^{2}}.$

Here, $\sigma$ and $\theta$ are parameters related to the volatility and drift
of the Brownian motion, respectively; $\kappa$ is the parameter related to the
variance of the subordinator, which is a the Gamma process (see [17]). The
probability density is given by

$p_{t}(x)=Ce^{Ax}|x|^{\frac{t}{k}}K_{\frac{t}{k}-\frac{1}{2}}(|x|),$

where $K$ is the modified Bessel function of second kind. The characteristic
function of $X_{t}+\gamma t$ is given by

$\Phi_{t}\left(u\right)=e^{itu\gamma}\phi_{t}\left(u\right)=e^{itu\gamma}\left(1+\frac{\sigma^{2}u^{2}\kappa}{2}-i\theta\kappa
u\right)^{-t/\kappa},$

where $\gamma$ is determined by the martingale condition, and
$\phi_{t}\left(u\right)$ is the characteristic function of $X_{t}$. Moreover,
we have

$\mathbb{E}[e^{-rT}S_{T}|\mathbb{F}_{t}]=e^{-rt}S_{t},$ (15)

where

$S_{t}=S_{0}e^{rt+\gamma t+X_{t}}$ (16)

is the risk-neutral process introduced in [44]. Therefore,
$\gamma=\frac{1}{\kappa}log(1-\frac{\sigma^{2}\kappa}{2}-\theta\kappa).$

#### Normal inverse Gaussian model

The NIG process is a process of infinite activity and infinite variation
without any Brownian component. It has the following Lévy measure [17]

$\nu\left(x\right)=\frac{C}{\left|x\right|}e^{Ax}K_{1}\left(B\left|x\right|\right)$

and

$C=\frac{\sqrt{\theta^{2}+\frac{\sigma^{2}}{\kappa}}}{2\pi\sigma\sqrt{\kappa}},A=\frac{\theta}{\sigma^{2}},B=\frac{\sqrt{\theta^{2}+\frac{\sigma^{2}}{\kappa}}}{\sigma^{2}},$

where $\theta$, $\sigma$, and $\kappa$ have the same meaning as in the
Variance Gamma process. The probability density is given by

$p_{t}(x)=Ce^{Ax}\frac{K_{1}(B\sqrt{x^{2}+\frac{t^{2}\sigma^{2}}{\kappa}})}{\sqrt{x^{2}+\frac{t^{2}\sigma^{2}}{\kappa}}}$

where $K$ is the modified Bessel function of second kind. The characteristic
function is given by

$\Phi_{t}\left(u\right)=e^{\frac{t}{\kappa}-\frac{t}{\kappa}\sqrt{1+u^{2}\sigma^{2}\kappa-2iu\theta\kappa}}.$
(17)

#### Generalized hyperbolic model

The generalized hyperbolic model is a process of infinite variation without
Gaussian part. It has the following characteristic function (see [17]):

$\phi_{t}(u)=e^{i\mu
u}(\frac{\alpha^{2}-\beta^{2}}{\alpha^{2}-(\beta+iu)^{2}})^{\frac{t}{2\kappa}}\frac{K_{\frac{t}{\kappa}}(\delta\sqrt{\lambda^{2}-(\beta+iu)^{2}})}{K_{\frac{t}{\kappa}}(\delta\sqrt{\alpha^{2}-\beta^{2}})},$
(18)

where $\delta$ is a scale parameter, $\mu$ is the shift parameter, and
$\kappa$ has the same meaning as in the variance Gamma process. The parameters
$\lambda$, $\alpha$, and $\beta$ determine the shape of the distribution. The
density function

$p_{t}(x)=C(\sqrt{\delta^{2}+(x-\mu)^{2}})^{\frac{t}{k}-\frac{1}{2}}K_{\frac{t}{\kappa}-\frac{1}{2}}(\alpha\sqrt{\delta^{2}-(x-\mu)^{2}})e^{\beta(x-\mu)},$

where $K$ is the modified Bessel function and

$C=\frac{(\sqrt{\alpha^{2}-\beta^{2}})^{\frac{t}{k}}}{\sqrt{2\pi}\alpha^{\frac{t}{\kappa}-\frac{1}{2}}\delta^{\frac{t}{\kappa}}K_{\frac{t}{\kappa}}(\delta\sqrt{\alpha^{2}-\beta^{2}})}.$

The variance Gamma process is obtained for $\mu=0$ and $\delta=0$. The NIG
process corresponds to $\lambda=-\frac{1}{2}$.

###  3.3. Admissible activity Lévy measures

This subsection presents the notion of an admissible activity Lévy measure
introduced by Cruz and Ševčovič [20, 19] for the one-dimensional case $n=1$,
which was later extended by Ševčovič and Udeani [58] for the multidimensional
case $n\geq 1$.

######  Definition 2.

[58, Definition 1] A measure $\nu$ in $\mathbb{R}^{n}$ is called an admissible
activity Lévy measure if there exists a nonnegative Lebesgue measurable
function $h:\mathbb{R}^{n}\to\mathbb{R}$ such that
$\nu(\mathrm{d}z)=h(z)\mathrm{d}z$ with

$0\leq h(z)\leq C_{0}|z|^{-\alpha}e^{-D|z|-\mu|z|^{2}},$ (19)

for all $z\in\mathbb{R}^{n}$ and the shape parameters $\alpha,\mu\geq
0,D\in\mathbb{R}$ ($D>0$ if $\mu=0)$, where $C_{0}>0$ is a positive constant.

######  Remark 2.

It is worth noting that the additional conditions
$\int_{\mathbb{R}}\min(|z|^{2},1)\nu(\mathrm{d}z)<\infty$ and
$\int_{\left|z\right|>1}e^{z}\nu(\mathrm{d}z)<\infty$ are satisfied provided
that $\nu$ is an admissible Lévy measure with shape parameters $\alpha<3$, and
either $\mu>0,D^{\pm}\in\mathbb{R}$, or $\mu=0$ and $D^{-}+1<0<D^{+}$. For the
Merton model, we have $\alpha=0,D^{\pm}=0$ and $\mu=1/(2\delta^{2})>0$.
Meanwhile, for the Kou model, we have
$\alpha=\mu=0,D^{+}=\lambda^{-},D^{-}=-\lambda^{+}$. For the variance Gamma
process, we have $\alpha=1,\mu=0,D^{\pm}=A\pm B$.

##  4\. Multidimensional Linear and Nonlinear PIDE

This section focuses on qualitative properties of solutions to the linear and
nonlinear nonlocal parabolic PIDE of the form:

$\displaystyle\frac{\partial u}{\partial\tau}$ $\displaystyle=$
$\displaystyle\frac{\sigma^{2}}{2}\Delta
u+\int_{\mathbb{R}^{n}}\left[u(\tau,x+z)-u(\tau,x)-z\cdot\nabla_{x}u(\tau,x)\right]\nu(\mathrm{d}z)+g(\tau,x,u,\nabla_{x}u),$
$\displaystyle u(0,x)=u_{0}(x),\quad x\in\mathbb{R}^{n},\tau\in(0,T),$

where $g$ is a given sufficiently smooth function; $\nu$ is a positive measure
on $\mathbb{R}^{n}$ such that its Radon derivative is a nonnegative Lebesgue
measurable function $h$ in $\mathbb{R}^{n}$, i.e.,
$\nu(\mathrm{d}z)=h(z)\mathrm{d}z$. Additionally, we will analyze the solution
of the following generalization of the above PIDE, in which the shift function
may depend on the variables $\tau>0,x,z\in\mathbb{R}$:

$\frac{\partial u}{\partial\tau}=\frac{\sigma^{2}}{2}\Delta
u+\int_{\mathbb{R}^{n}}\left[u(\tau,x+\xi)-u(\tau,x)-\xi\cdot\nabla_{x}u(\tau,x)\right]\nu(\mathrm{d}z)+g(\tau,x,u,\nabla_{x}u),$
(21)

where $\xi=\xi(\tau,x,z)$ is the shift function. An application of such a
general shift function $\xi$ can be found in nonlinear option pricing models
considering a large trader stock-trading strategy with the underlying asset
price dynamic following the Lévy process (c.f., Cruz and Ševčovič [20]). If
$\xi(x,z)\equiv z$, then (21) reduces to equation (4.). Fr example, the
nonlinearity $g$ often arises from applications occurring in pricing XVA
derivatives (c.f., Arregui _et al._ [4, 5]) or applications of the penalty
method for American option pricing under a PIDE model (c.f., Cruz and Ševčovič
[19]).

###  4.1. Existence and uniqueness results of PIDE

In this section, we present the existence and uniqueness results for the
general (21) for a class of Lévy measures using the theory of abstract
semilinear parabolic equation in the scale of Bessel potential spaces. First,
we rewrite the PIDE (21) in high-dimensional space as follows:

$\displaystyle\frac{\partial
u}{\partial\tau}+Au=f(u)+g(\tau,x,u,\nabla_{x}u),\;\;u(0,x)=u_{0}(x),\;x\in\mathbb{R}^{n},\tau\in(0,T),$
(22)

where $A=-(\sigma^{2}/2)\Delta$. The linear nonlocal operator $f$ is defined
by

$f(u)(\cdot)=\int_{\mathbb{R}^{n}}\left[u(\cdot+\xi)-u(\cdot)-\xi\cdot\nabla_{x}u(\cdot)\,\right]\nu(\mathrm{d}z),$
(23)

where $\xi=\xi(\tau,x,z)$ is a given shift function. The function $g$ is
assumed to be Hölder and Lipschitz continuous in the $\tau$ and other
variables, respectively. Then, we employ the theory of abstract semilinear
parabolic equations presented by Henry [29] to establish the existence,
continuation, and uniqueness of a solution. A solution to the PIDE (22) is
constructed in the scale of the Bessel potential spaces
${\mathscr{L}}^{p}_{2\gamma}(\mathbb{R}^{n}),\gamma\geq 0$ in high-dimensional
space, $n\geq 1$. These spaces can be viewed as a natural extension of the
classical Sobolev spaces $W^{k,p}(\mathbb{R}^{n})$ for non-integer values of
order $k$. It is worth noting that nested scale of Bessel potential spaces
allows for a finer formulation of existence and uniqueness results than the
classical Sobolev spaces.

######  Definition 3.

[29, Definition 1] An analytic semigroup is a family of bounded linear
operators $\left\\{S(t),t\geq 0\right\\}$ in a Banach space $X$ satisfying the
following conditions:

  * i)

$S(0)=I,S(t)S(s)=S(s)S(t)=S(t+s)$, for all $t,s\geq 0$;

  * ii)

$S(t)u\rightarrow u$ when $t\rightarrow 0^{+}$ for all $u\in X$;

  * iii)

$t\rightarrow S(t)u$ is a real analytic function on $0<t<\infty$ for each
$u\in X$.

The associated infinitesimal generator $A$ is defined as follows:
$Au=\lim_{t\rightarrow 0^{+}}\frac{1}{t}(S(t)u-u)$ and its domain
$D(A)\subseteq X$ consists of those elements $u\in X$ for which the limit
exists in the space $X$.

######  Definition 4.

[29] Let $S_{a,\phi}=\left\\{\lambda\in\mathbb{C}:\phi\leq\arg(\lambda-a)\leq
2\pi-\phi\right\\}$ be a sector of complex numbers. A closed densely defined
linear operator $A:D(A)\subset X\rightarrow X$ is called a sectorial operator
if there exists a constant $M\geq 0$ such that $\|(A-\lambda)^{-1}\|\leq
M/|\lambda-a|$ for all $\lambda\in
S_{a,\phi}\subset\mathbb{C}\setminus\sigma(A)$.

Next, we briefly recall the construction and basic properties of Bessel
potential spaces. It is worth noting if $A$ is a sectorial operator in a
Banach space $X$, then $-A$ is a generator of an analytic semigroup
$\left\\{e^{-At},t\geq 0\right\\}$ acting on $X$ (c.f., [29, Chapter I]). For
any $\gamma>0$, we can introduce the operator $A^{-\gamma}:X\to X$ as follows:
$A^{-\gamma}=\frac{1}{\Gamma(\gamma)}\int_{0}^{\infty}\xi^{\gamma-1}e^{-A\xi}\mathrm{d}\xi$.
Then, the fractional power space $X^{\gamma}=D(A^{\gamma})$ is the domain of
the operator $A^{\gamma}=(A^{-\gamma})^{-1}$, i.e., $X^{\gamma}=\left\\{u\in
X:\ \exists\varphi\in X,u=A^{-\gamma}\varphi\right\\}$. The norm is defined as
follows: $\|u\|_{X^{\gamma}}=\|A^{\gamma}u\|_{X}=\|\varphi\|_{X}$.
Furthermore, we have continuous embedding: $D(A)\equiv X^{1}\hookrightarrow
X^{\gamma_{1}}\hookrightarrow X^{\gamma_{2}}\hookrightarrow X^{0}\equiv X$,
for $0\leq\gamma_{2}\leq\gamma_{1}\leq 1$.

Let us recall the convolution operator
$(G*\varphi)(x)=\int_{\mathbb{R}^{n}}G(x-y)\varphi(y)\mathrm{d}{y}$. According
to [29, Section 1.6], $A=-(\sigma^{2}/2)\Delta$ is a sectorial operator in the
Lebesgue space $X=L^{p}(\mathbb{R}^{n})$ for any $p\geq 1,n\geq 1$, and
$D(A)\subset W^{2,p}(\mathbb{R}^{n})$. It follows from [65, Chapter 5] that
the space $X^{\gamma},\gamma>0,$ can be identified with the Bessel potential
space ${\mathscr{L}}^{p}_{2\gamma}(\mathbb{R}^{n})$, where

${\mathscr{L}}^{p}_{2\gamma}(\mathbb{R}^{n}):=\\{u\in X:\ \exists\varphi\in
X,u=G_{2\gamma}*\varphi\\}.$

Here, $G_{2\gamma}$ is the Bessel potential function,

$G_{2\gamma}(x)=\frac{1}{(4\pi)^{n/2}\Gamma(\gamma)}\int_{0}^{\infty}y^{-1+\gamma-n/2}e^{-(y+|x|^{2}/(4y))}\mathrm{d}y.$

The norm of $u=G_{2\gamma}*\varphi$ is given by
$\|u\|_{X^{\gamma}}=\|\varphi\|_{L^{p}}$. The space $X^{\gamma}$ is
continuously embedded in the fractional Sobolev–Slobodeckii space
$W^{2\gamma,p}(\mathbb{R}^{n})$ (c.f., [29, Section 1.6]).

In what follows, we denote $C_{0}>0$ as a generic constant, which is
independent of the solution $u$; however, it may depend on the model
parameters, e.g., $n\geq 1,p\geq 1,\gamma\in[0,1)$.

######  Proposition 1.

[58, Proposition 1] Let us define the mapping $Q(u,\xi)$ as follows:

$Q(u,\xi)=u(x+\xi(x))-\xi(x)\cdot\nabla_{x}u(x),\quad x\in\mathbb{R}^{n}.$

Then, there exists a constant $\hat{C}>0$ such that, for any vector valued
functions $\xi_{1},\xi_{2}\in(L^{\infty}(\mathbb{R}^{n}))^{n}$, and $u$ such
that $\nabla_{x}u\in(X^{\gamma-1/2})^{n}$, $1/2\leq\gamma<1$, the following
estimate holds:

$\|Q(u,\xi_{1})-Q(u,\xi_{2})\|_{L^{p}({\mathbb{R}^{n}})}\leq\hat{C}\|\xi_{1}-\xi_{2}\|_{\infty}^{2\gamma-1}(\|\xi_{1}\|_{\infty}+\|\xi_{2}\|_{\infty})\|\nabla_{x}u\|_{X^{\gamma-1/2}}.$

###### Proof.

Let $u\in X$ be such that $\nabla_{x}u\in(X^{\gamma-1/2})^{n}$, i.e.,
$\partial_{x_{i}}u\in X^{\gamma-1/2}~{}\text{for each}~{}i=1,\cdots,n$. Then,
$\nabla_{x}u=A^{-(2\gamma-1)/2}\varphi=G_{2\gamma-1}*\varphi$ for some
$\varphi\in(L^{p}(\mathbb{R}^{n}))^{n}$, and
$\|\nabla_{x}u\|_{X^{\gamma-1/2}}=\|A^{(2\gamma-1)/2}\nabla_{x}u\|_{X}=\|\varphi\|_{L^{p}}$.
Here, $\varphi=(\varphi_{1},\cdots,\varphi_{n})$ and
$\partial_{x_{i}}u=G_{2\gamma-1}*\varphi_{i}$. Let $x,\xi\in\mathbb{R}^{n}$.
Then,

$\nabla_{x}u(x+\xi)=G_{2\gamma-1}(x+\xi-\cdot)*\varphi(\cdot),\qquad\nabla_{x}u(x)=G_{2\gamma-1}(x-\cdot)*\varphi(\cdot).$

Recall that the following inequality holds for convolution operator:

$\|\psi*\varphi\|_{L^{p}(\mathbb{R}^{n})}\leq\|\psi\|_{L^{q}(\mathbb{R}^{n})}\|\varphi\|_{L^{r}(\mathbb{R}^{n})},$

where $p,q,r\geq 1$ and $1/p+1=1/q+1/r$ (see [29, Section 1.6]). In
particular, for $q=1$, we have
$\|\psi*\varphi\|_{L^{p}}\leq\|\psi\|_{L^{1}}\|\varphi\|_{L^{p}}$. The
following estimate holds for the modulus of continuity of the Bessel potential
function $G_{\alpha},\alpha\in(0,1)$:

$\|G_{\alpha}(\cdot+h)-G_{\alpha}(\cdot)\|_{L^{1}}\leq C_{0}|h|^{\alpha},$

for any $h\in\mathbb{R}^{n}$ (c.f., [65, Chapter 5.4, Proposition 7]). Let
$\xi_{1},\xi_{2}$ be bounded vector valued functions, i.e.,
$\xi_{1},\xi_{2}\in(L^{\infty}(\mathbb{R}^{n}))^{n}$. Then, for any
$x\in\mathbb{R}^{n}~{}\text{and}~{}\theta\in[0,1]$, we have

$\displaystyle
u(x+\xi_{1}(x))-u(x+\xi_{2}(x))-(\xi_{1}(x)-\xi_{2}(x))\cdot\nabla_{x}u(x)$
$\displaystyle=$ $\displaystyle
u(x+\xi_{1}(x))-\nabla_{x}u(x)-\xi_{1}(x)\cdot\nabla_{x}u(x)$
$\displaystyle-[u(x+\xi_{2}(x))-\nabla_{x}u(x)-\xi_{2}(x)\cdot\nabla_{x}u(x)]$
$\displaystyle=$
$\displaystyle(\xi_{1}(x)-\xi_{2}(x))\int_{0}^{1}\nabla_{x}u(x+\theta\xi_{1}(x))-\nabla_{x}u(x)\mathrm{d}{\theta}$
$\displaystyle+\int_{0}^{1}\nabla_{x}u(x+\theta\xi_{1}(x))-\nabla_{x}u(x+\theta\xi_{2}(x))\mathrm{d}{\theta}\
.$

Now,

$\displaystyle\|Q(u,\xi_{1})-Q(u,\xi_{2})\|^{p}_{L^{p}({\mathbb{R}^{n}})}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{n}}|u(x+\xi_{1}(x))-u(x+\xi_{2}(x))-(\xi_{1}(x)-\xi_{2}(x))\cdot\nabla_{x}u(x)|^{p}\mathrm{d}{x}$
$\displaystyle\leq$
$\displaystyle\int_{\mathbb{R}^{n}}\left|(\xi_{1}(x)-\xi_{2}(x))\int_{0}^{1}\nabla_{x}u(x+\theta\xi_{1}(x))-\nabla_{x}u(x)\mathrm{d}{\theta}\right|^{p}\mathrm{d}{x}$
$\displaystyle+\int_{\mathbb{R}^{n}}\left|\xi_{2}(x)\int_{0}^{1}\nabla_{x}u(x+\theta\xi_{1}(x))-\nabla_{x}u(x+\theta\xi_{2}(x))\mathrm{d}{\theta}\right|^{p}\mathrm{d}{x}$
$\displaystyle\leq$
$\displaystyle\|\xi_{1}-\xi_{2}\|_{\infty}^{p}\int_{0}^{1}\int_{\mathbb{R}^{n}}|\nabla_{x}u(x+\theta\xi_{1}(x))-\nabla_{x}u(x)|^{p}dxd\theta$
$\displaystyle+\|\xi_{2}\|_{\infty}^{p}\int_{0}^{1}\int_{\mathbb{R}^{n}}|\nabla_{x}u(x+\theta\xi_{1}(x))-\nabla_{x}u(x+\theta\xi_{2}(x)|^{p}dxd\theta$
$\displaystyle\leq$
$\displaystyle\|\xi_{1}-\xi_{2}\|_{\infty}^{p}\int_{0}^{1}\|\left(G_{2\gamma-1}(\cdot+\theta\xi_{1})-G_{2\gamma-1}(\cdot)\right)*\varphi\|_{L^{p}}^{p}d\theta$
$\displaystyle+\|\xi_{2}\|_{\infty}^{p}\int_{0}^{1}\|\left(G_{2\gamma-1}(\cdot+\theta\xi_{1})-G_{2\gamma-1}(\cdot+\theta\xi_{2})\right)*\varphi\|_{L^{p}}^{p}d\theta$
$\displaystyle\leq$
$\displaystyle\|\xi_{1}-\xi_{2}\|_{\infty}^{p}\int_{0}^{1}\|G_{2\gamma-1}(\cdot+\theta\xi_{1})-G_{2\gamma-1}(\cdot)\|^{p}_{L^{1}}d\theta\|\varphi\|_{L^{p}}^{p}$
$\displaystyle+\|\xi_{2}\|_{\infty}^{p}\int_{0}^{1}\|G_{2\gamma-1}(\cdot+\theta\xi_{1})-G_{2\gamma-1}(\cdot+\theta\xi_{2})\|^{p}_{L^{1}}d\theta\|\varphi\|_{L^{p}}^{p}$
$\displaystyle\leq$
$\displaystyle\left(\|\xi_{1}-\xi_{2}\|_{\infty}^{p}\|\xi_{1}\|^{(2\gamma-1)p}_{\infty}+\|\xi_{2}\|_{\infty}^{p}\|\xi_{1}-\xi_{2}\|^{(2\gamma-1)p}_{\infty}\right)C_{0}^{p}\|\nabla_{x}u\|^{p}_{X^{\gamma-1/2}}$
$\displaystyle\leq$
$\displaystyle\|\xi_{1}-\xi_{2}\|^{(2\gamma-1)p}_{\infty}\left(\|\xi_{1}\|_{\infty}^{p}+\|\xi_{2}\|^{(2-2\gamma)p}_{\infty}\|\xi_{1}\|^{(2\gamma-1)p}_{\infty}+\|\xi_{2}\|_{\infty}^{p}\right)C_{0}^{p}\|\nabla_{x}u\|^{p}_{X^{\gamma-1/2}}\
.$

By Young’s inequality, we have
$ab\leq\frac{a^{\alpha}}{\alpha}+\frac{b^{\beta}}{\beta}$ for any $a,b\geq 0$,
and $\alpha,\beta>1$ with $1/\alpha+1/\beta=1$ (c.f., [13]). Set
$\alpha=1/(2-2\gamma),\beta=1/(2\gamma-1)$. Then, $1/\alpha+1/\beta=1$, and we
obtain
$\|\xi_{2}\|^{(2-2\gamma)p}_{\infty}\|\xi_{1}\|^{(2\gamma-1)p}_{\infty}\leq(2-2\gamma)\|\xi_{2}\|_{\infty}^{p}+(2\gamma-1)\|\xi_{1}\|_{\infty}^{p}\leq
2\|\xi_{2}\|_{\infty}^{p}+\|\xi_{1}\|_{\infty}^{p}$. Therefore,

$\displaystyle\|Q(u,\xi_{1})-Q(u,\xi_{2})\|_{L^{p}({\mathbb{R}^{n}})}^{p}$
$\displaystyle\leq$ $\displaystyle
2\|\xi_{1}-\xi_{2}\|^{(2\gamma-1)p}_{\infty}\left(\|\xi_{1}\|_{\infty}^{p}+\|\xi_{2}\|_{\infty}^{p}\right)C_{0}^{p}\|\nabla_{x}u\|^{p}_{X^{\gamma-1/2}}$
$\displaystyle\leq$ $\displaystyle
2C_{0}^{p}\|\xi_{1}-\xi_{2}\|_{\infty}^{(2\gamma-1)p}(\|\xi_{1}\|_{\infty}+\|\xi_{2}\|_{\infty})^{p}\|\nabla_{x}u\|^{p}_{X^{\gamma-1/2}}.$

Hence, the pointwise estimate holds with the constant
$\hat{C}=2^{1/p}C_{0}>0$. ∎

Applying Proposition 1 with $\xi_{1}=\xi~{}\text{and}~{}\xi_{2}=0$, we obtain
the following corollary.

######  Corollary 1.

[58, Corollary 1] Let $u$ be such that $\nabla_{x}u\in(X^{\gamma-1/2})^{n}$
where $1>\gamma\geq 1/2$. Then, for any $\xi\in\mathbb{R}^{n}$, the following
pointwise estimate holds:

$\|Q(u,\xi)\|_{L^{p}({\mathbb{R}^{n}})}\leq
C_{0}|\xi|^{2\gamma}\|\nabla_{x}u\|_{X^{\gamma-1/2}}.$

Next, we consider the case when the nonlocal integral term depends on $x$ and
$z$ variables. It is a generalization of the result [20, Lemma 3.4] due to
Cruz and Ševčovič proven for the case where $\xi(x,z)\equiv z$.

######  Proposition 2.

[58, Proposition 2] Suppose that the shift mapping $\xi=\xi(x,z)$ satisfies
$\sup_{x\in\mathbb{R}}|\xi(x,z)|\leq C_{0}|z|^{\omega}(1+e^{D_{0}|z|})$ for
some constants $C_{0}>0,D_{0}\geq 0,\omega>0$ and any $z\in\mathbb{R}$. Assume
$\nu$ is a Lévy measure with the shape parameters $\alpha,D,$ and either
$\mu>0,D\in\mathbb{R}$, or $\mu=0$ and $D>D_{0}\geq 0$. Assume
$1/2\leq\gamma<1$, and $\gamma>(\alpha-n)/(2\omega)$. Then there exists a
constant $C_{0}>0$ such that

$\|f(u)\|_{L^{p}}\leq C_{0}\|\nabla_{x}u\|_{X^{\gamma-1/2}},$

provided that $\nabla_{x}u\in(X^{\gamma-1/2})^{n}$. If $u\in X^{\gamma}$ then
$\|f(u)\|_{L^{p}}\leq C\|u\|_{X^{\gamma}}$, i.e., $f:X^{\gamma}\to X$ is a
bounded linear operator.

###### Proof.

The Lévy measure $\nu(\mathrm{d}z)$ is given by
$\nu(\mathrm{d}z)=h(z)\mathrm{d}z$. Let us denote the auxiliary function
$\tilde{h}(z)=|z|^{\alpha}h(z)$. Then, $0\leq\tilde{h}(z)\leq
C_{0}e^{-D|z|-\mu|z|^{2}}$. Since
$h(z)=|z|^{-\alpha}\tilde{h}(z)=h_{1}(z)h_{2}(z)$, where
$h_{1}(z)=|z|^{-\beta}\tilde{h}(z)^{\frac{1}{2}}$ and
$h_{2}(z)=|z|^{\beta-\alpha}\tilde{h}(z)^{\frac{1}{2}}$. Applying Proposition
1 with $\xi_{1}=\xi,\xi_{2}=0$, and using the Hölder inequality, we obtain

$\displaystyle\|f(u)\|_{L^{p}}^{p}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{n}}\left|\int_{\mathbb{R}^{n}}(u(x+\xi(x,z))-u(x)-\xi(x,z)\cdot\nabla_{x}u(x))h(z)\mathrm{d}{z}\right|^{p}\mathrm{d}{x}$
$\displaystyle\leq$
$\displaystyle\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\left|u(x+\xi(x,z))-u(x)-\xi(x,z)\cdot\nabla_{x}u(x)\right|^{p}h_{1}(z)^{p}\mathrm{d}{z}$
$\displaystyle\quad\times\left(\int_{\mathbb{R}^{n}}h_{2}(z)^{q}\mathrm{d}{z}\right)^{p/q}\mathrm{d}{x}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{n}}\left(\int_{\mathbb{R}^{n}}\left|u(x+\xi(x,z))-u(x)-\xi(x,z)\cdot\nabla_{x}u(x)\right|^{p}\mathrm{d}{x}\right)h_{1}(z)^{p}\mathrm{d}{z}$
$\displaystyle\quad\times\left(\int_{\mathbb{R}^{n}}h_{2}(z)^{q}\mathrm{d}{z}\right)^{p/q}$
$\displaystyle\leq$ $\displaystyle
C_{0}^{p}\|\nabla_{x}u\|_{X^{\gamma-1/2}}^{p}\int_{\mathbb{R}^{n}}|\xi(x,z)|^{2\gamma
p}|z|^{-\beta
p}\tilde{h}(z)^{p/2}\mathrm{d}{z}\left(\int_{\mathbb{R}^{n}}h_{2}(z)^{q}\mathrm{d}{z}\right)^{p/q}$
$\displaystyle\leq$ $\displaystyle
C_{0}^{p}\|\nabla_{x}u\|_{X^{\gamma-1/2}}^{p}\int_{\mathbb{R}^{n}}|z|^{(2\gamma\omega-\beta)p}\tilde{h}(z)^{p/2}\mathrm{d}{z}\left(\int_{\mathbb{R}^{n}}h_{2}(z)^{q}\mathrm{d}{z}\right)^{p/q}.$

Assuming $p,q\geq 1,1/p+1/q=1$ are such that

$(2\gamma\omega-\beta)p>-n,\qquad(\beta-\alpha)q=(\beta-\alpha)\frac{p}{p-1}>-n,$

then, the integrals
$\int_{\mathbb{R}^{n}}|z|^{(2\gamma\omega-\beta)p}\tilde{h}(z)^{p/2}\mathrm{d}{z}$
and
$\int_{\mathbb{R}^{n}}h_{2}(z)^{q}\mathrm{d}{z}=\int_{\mathbb{R}^{n}}|z|^{(\beta-\alpha)q}\tilde{h}(z)^{q/2}\mathrm{d}{z}$
are finite, provided that the shape parameters satisfy: either
$\mu>0,D\in\mathbb{R}$, or $\mu=0,D>D_{0}\geq 0$. As
$\gamma>(\alpha-n)/(2\omega)$, there exists $\beta>1$ satisfying

$\alpha-n+n/p<\beta<2\gamma\omega+n/p.$

Therefore, there exists $C_{0}>0$ such that $\|f(u)\|_{L^{p}}\leq
C_{0}\|\nabla_{x}u\|_{X^{\gamma-1/2}}$. ∎

Let $C([0,T],X^{\gamma})$ be the Banach space consisting of continuous
functions from $[0,T]$ to $X^{\gamma}$ with the maximum norm. The following
proposition is due to Henry [29] (see also Cruz and Ševčovič [19]).

######  Proposition 3.

[29, Proposition 3.5] Suppose that the linear operator $-A$ is a generator of
an analytic semigroup $\left\\{e^{-At},t\geq 0\right\\}$ in a Banach space
$X$. Assume the initial condition $U_{0}$ belongs to the space $X^{\gamma}$
where $0\leq\gamma<1$. Suppose that the mappings $F:[0,T]\times X^{\gamma}\to
X$ and $h:(0,T]\to X$ are Hölder continuous in the $\tau$ variable,
$\int_{0}^{T}\|h(\tau)\|_{X}\mathrm{d}\tau<\infty$, and $F$ is Lipschitz
continuous in the $U$ variable. Then, for any $T>0$, there exists a unique
solution to the abstract semilinear evolution equation:
$\partial_{\tau}U+AU=F(\tau,U)+h(\tau)$ such that $U\in
C([0,T],X^{\gamma}),U(0)=U_{0},\partial_{\tau}U(\tau)\in X,U(\tau)\in D(A)$
for any $\tau\in(0,T)$. The function $U$ is a solution in the mild (integral)
sense, i.e.,
$U(\tau)=e^{-A\tau}U_{0}+\int_{0}^{\tau}e^{-A(\tau-s)}(F(s,U(s))+h(s))\mathrm{d}{s}$,
$\tau\in[0,T]$.

Applying Propositions 2 and 3, we can state the following result which is a
nontrivial generalization of the result shown by Ševčovič and Cruz [19] for
$n=1$.

######  Theorem 2.

[58, Theorem 1] Suppose that the shift mapping $\xi=\xi(x,z)$ satisfies
$\sup_{x\in\mathbb{R}}|\xi(x,z)|\leq
C_{0}|z|^{\omega}(1+e^{D_{0}|z|}),z\in\mathbb{R}^{n}$, for some constants
$C_{0}>0,D_{0}\geq 0,\omega>0$. Assume $\nu$ is an admissible activity Lévy
measure with the shape parameters $\alpha,D$, and, either
$\mu>0,D\in\mathbb{R}$, or $\mu=0,D>D_{0}\geq 0$. Assume $1/2\leq\gamma<1$ and
$\gamma>(\alpha-n)/(2\omega)$, $n\geq 1$. Suppose that
$g(\tau,x,u,\nabla_{x}u)$ is Hölder continuous in the $\tau$ variable and
Lipschitz continuous in the remaining variables, respectively. Assume
$u_{0}\in X^{\gamma}$, and $T>0$. Then, there exists a unique mild solution
$u$ to PIDE (21) satisfying $u\in C([0,T],X^{\gamma})$.

###  4.2. Maximal monotone operator technique for solving nonlinear parabolic
equations

This section presents the existence and uniqueness results of a fully
nonlinear parabolic equation using the monotone operator technique. We
consider the HJB equation arising from portfolio optimization selection, where
the goal is to maximize the conditional expected value of the terminal utility
of the portfolio. Such a fully nonlinear HJB equation presented in an abstract
setting can be viewed as a PIDE in some sense. First, we employ the so-called
Riccati transformation method to transform the fully nonlinear HJB equation
into a quasilinear parabolic equation, which can be viewed as the porous media
type of equation with source term. Then, we showed that the underlying
operator is maximally monotone in some Sobolev spaces. Next, we employed the
Banach’s fixed point theorem and Fourier transform technique to obtain the
existence and uniqueness of a solution to the general form of the transformed
parabolic equation in an abstract setting in high-dimensional spaces.
Furthermore, as a crucial requirement for solving the Cauchy problem, we
obtain that the diffusion function to the quasilinear parabolic equation is
globally Lipschitz continuous under some assumptions.

We consider the Cauchy problem for the nonlinear parabolic PDE of the
following form:

$\displaystyle\partial_{\tau}\varphi-\Delta\alpha(\tau,\varphi)=g_{0}(\tau,\varphi)+\nabla\cdot\bm{g}_{1}(\tau,\varphi),$
(24) $\displaystyle\varphi(\cdot,0)=\varphi_{0},$ (25)

where $\tau\in(0,T),x\in\mathbb{R}^{d},d\geq 1$. The solution
$\varphi=\varphi(x,\tau)$ to such a nonlinear parabolic equation is
established in some Sobolev spaces in high-dimensional spaces (see, [66]). To
achieve such results, we assumed that the diffusion function
$\alpha=\alpha(x,\tau,\varphi)$ is globally Lipschitz continuous and strictly
increasing in the $\varphi$-variable. An example of such a Lipschitz
continuous function $\alpha(x,\tau,\varphi)$ is the value function of the
following parametric optimization problem:

$\alpha(x,\tau,\varphi)=\min_{{\bm{\theta}}\in\triangle}\left(-\mu(x,t,{\bm{\theta}})+\frac{\varphi}{2}\sigma(x,t,{\bm{\theta}})^{2}\right),\quad\tau\in(0,T),x\in\mathbb{R}^{d},\varphi>\varphi_{min}\,,$
(26)

where $\mu$ and $\sigma^{2}$ are given $C^{1}$ functions, and
$\triangle\subset\mathbb{R}^{n}$ is a compact decision set. The properties of
the value function depend on the structure of the decision set $\triangle$. It
is smooth if $\triangle$ is a convex set; meanwhile, it can only be $C^{0,1}$
smooth if $\triangle$ is not connected.

####  4.2.1. Existence and uniqueness of a solution to the Cauchy problem

First, we define our underlying function spaces. Let $V\hookrightarrow
H\hookrightarrow V^{\prime}$ be a Gelfand triple, where

$H=L^{2}(\mathbb{R}^{d})=\\{f:\mathbb{R}^{d}\to\mathbb{R},\;\;\;\|f\|_{L^{2}}^{2}=\int_{\mathbb{R}^{d}}|f(x)|^{2}dx<\infty\\}$

is a Hilbert space endowed with the inner product
$(f,g)=\int_{\mathbb{R}^{d}}f(x)g(x)dx$. The Banach spaces $V$ and
$V^{\prime}$ are defined as follows:

$\quad V=H^{1}(\mathbb{R}^{d}),\quad V^{\prime}=H^{-1}(\mathbb{R}^{d}),$

where the Sobolev spaces $H^{s}(\mathbb{R}^{d})$ are defined by means of the
Fourier transform

$\hat{f}(\xi)=\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^{d}}e^{-ix\cdot\xi}f(x)dx,\quad\xi=(\xi_{1},\xi_{2},...,\xi_{d})^{T}\in\mathbb{R}^{d},$
$H^{s}(\mathbb{R}^{d})=\\{f:\mathbb{R}^{d}\to\mathbb{R},(1+|\xi|^{2})^{s/2}\hat{f}(\xi)\in
L^{2}(\mathbb{R}^{d})\\},\;s\in\mathbb{R}$

endowed with the norm
$\|f\|_{H^{s}}^{2}=\int_{\mathbb{R}^{d}}(1+|\xi|^{2})^{s}|\hat{f}(\xi)|^{2}d\xi$,
and $|\xi|=(\xi_{1}^{2}+\cdots+\xi_{d}^{2})^{1/2}$. Let the linear operator
$A:V\to V^{\prime}$ be defined as follows:

$A\psi=\psi-\Delta\psi.$

It is worth nothing that the operator $A$ is self-adjoint in the Hilbert space
$H=L^{2}(\mathbb{R}^{d})$ with the following Fourier transform representation:

$\widehat{A\psi}(\xi)=(1+|\xi|^{2})\hat{\psi}(\xi).$

The fractional powers of $A$ is defined by
$\widehat{A^{s}\psi}(\xi)=(1+|\xi|^{2})^{s}\hat{\psi}(\xi),\;s\in\mathbb{R}$.
In particular,

$\widehat{A^{\pm 1/2}\psi}(\xi)=(1+|\xi|^{2})^{\pm 1/2}\hat{\psi}(\xi),\quad$

and $A^{-1/2}$ is a self-adjoint operator in the Hilbert space
$H=L^{2}(\mathbb{R}^{d})$. Moreover, $A^{-1}=A^{-1/2}A^{-1/2}$.

In the sequel, we denote the duality pairing between the spaces $V$ and
$V^{\prime}$ by $\langle.,.\rangle$, i.e., the value of a functional $F\in
V^{\prime}$ at $u\in V$ is denoted by $\langle F,u\rangle$. We have the
following definitions.

######  Definition 5.

[66, 7] An operator (in general, nonlinear) $B:V\to V^{\prime}$ is said to be

  * (i)

monotone if

$\langle B(u)-B(v),u-v\rangle\geq 0,\;\forall\;u,v\in V,$

  * (ii)

strongly monotone if there exists a constant $C>0$ such that

$\langle B(u)-B(v),u-v\rangle\geq C\|u-v\|_{V}^{2},\;\forall\;u,v\in V,$

  * (iii)

hemicontinuous if for each $u,v\in V$, the real-valued function $t\mapsto
B(u+tv)(v)$ is continuous.

######  Theorem 3.

[7, 63] Let $V$ be a separable reflexive Banach space, dense, and continuous
in a Hilbert space $H$, which is identified with its dual, so
$V\hookrightarrow H\hookrightarrow V^{\prime}$. Let $p\geq 2$ and set
$\mathcal{V}=L^{p}((0,T);V).$ Assume a family of operators
$\mathcal{A}(\tau,.):V\to V^{\prime},0\leq\tau<T$, is given such that

  * (i)

for each $\varphi\in V$, the function $\mathcal{A}(.,\varphi):[0,T]\to
V^{\prime}$ is measurable,

  * (ii)

for a.e $\tau\in[0,T]$, the operator $\mathcal{A}(\tau,.):V\to V^{\prime}$ is
monotone, hemicontinuous, and bounded by $\|\mathcal{A}(\tau,\varphi)\|\leq
C(\|\varphi\|^{p-1}+k(\tau)),\varphi\in V,0\leq\tau<T,$ where $k\in
L^{p^{\prime}}(0,T)$,

  * (iii)

and there exists $\lambda>0$ such that
$\langle\mathcal{A}(\tau,\varphi),\varphi\rangle\geq\lambda\|\varphi\|^{p}-k(\tau),\varphi\in
V,0\leq\tau<T.$

Then, for each $\hat{f}\in\mathcal{V^{\prime}}$ and $\varphi_{0}\in H$, there
exists a unique solution $\varphi\in\mathcal{V}$ of the Cauchy problem

$\partial_{\tau}\varphi(\tau)+\mathcal{A}(\tau,\varphi(\tau))=\hat{f}(\tau)~{}\text{in}~{}\mathcal{V^{\prime}},\
\ \varphi(0)=\varphi_{0}.$

Consider the spaces $\mathcal{V}=L^{2}((0,T);V)$,
$\mathcal{H}=L^{2}((0,T);H)$, and
$\mathcal{V^{\prime}}=L^{2}((0,T);V^{\prime})$, i.e., $p=2$. Thus, we have
that these spaces satify the Gelfand triple, i.e.,
$\mathcal{V}\hookrightarrow\mathcal{H}\hookrightarrow\mathcal{V^{\prime}}$,
where $\mathcal{H}$ is a Hilbert space endowed with the norm

$\|\varphi\|^{2}_{\mathcal{H}}=\int_{0}^{T}\|\varphi(\tau)\|^{2}_{H}d\tau,\;\forall\varphi\in\mathcal{H}.$

For a given value $\varphi_{min}$, we denote
${\mathcal{D}}=\mathbb{R}^{d}\times(0,T)\times(\varphi_{min},\infty)$.

######  Theorem 4.

[66, Theorem 2] Assume that the above settings on $H$ and $V$ hold. Let
$g_{0},g_{1j}:[0,T]\times H\to H,j=1,\cdots,n,$ be globally Lipschitz
continuous functions. Suppose $\alpha\in C^{0,1}(\mathcal{D})$ is such that
there exist constants $\omega,L,L_{0}>0$ such that
$0<\omega\leq\alpha^{\prime}_{\varphi}(x,\tau,\varphi)\leq L$,
$|\nabla_{x}\alpha(x,\tau,\varphi)|\leq p(x,\tau)+L_{0}|\varphi|$,
$\alpha(x,\tau,0)=h(x,\tau)$ for a.e. $(x,\tau,\varphi)\in\mathcal{D}$ and
$p,h\in L^{\infty}((0,T);H)$. Then, for any $T>0\;\text{and}\;\varphi_{0}\in
H,$ there exists a unique solution $\varphi\in{\mathcal{V}}$ of the Cauchy
problem

$\partial_{\tau}\varphi+A\alpha(\cdot,\tau,\varphi)=g_{0}(\tau,\varphi)+\nabla\cdot\bm{g}_{1}(\tau,\varphi),\qquad\varphi(0)=\varphi_{0}.$
(27)

We remark here that the above result and its proof are contained in our recent
paper [66, Theorem 2].
P r o o f: Recall that $H=L^{2}(\mathbb{R}^{d})$ and
$V=H^{1}(\mathbb{R}^{d})$, its dual space being
$V^{\prime}=H^{-1}(\mathbb{R}^{d})$. Let the scalar products in $V$ and
$V^{{}^{\prime}}$ be respectively defined by

$(f,g)_{V}=(A^{1/2}f,A^{1/2}g)_{H}=(Af,g)_{H},\
(f,g)_{V^{\prime}}=(A^{-1/2}f,A^{-1/2}g)_{H}=(A^{-1}f,g)_{H}.$

Let us define the operator $\mathcal{A}(\tau,\cdot):V\to V^{\prime}$ by

$\langle\mathcal{A}(\tau,\varphi),\psi\rangle=(A^{-1}A\alpha(\cdot,\tau,\varphi),\psi)_{H}=(\alpha(\cdot,\tau,\varphi),\psi)_{H}.$

Therefore, we conclude that the mapping
$\varphi\mapsto\alpha(\cdot,\tau,\varphi)$ maps $V$ into $V$ under the
assumption made on the function $\alpha$. Indeed, if $\varphi\in V$ and
$\eta=\alpha(\cdot,\tau,\varphi)$, then
$\eta(x)=\alpha(x,\tau,\varphi(x))-\alpha(x,\tau,0)+\alpha(x,\tau,0)$, and so

$|\eta(x)|\leq(\max_{\varphi}\alpha^{\prime}_{\varphi}(x,\tau,\varphi))|\varphi(x)|+|h(x,\tau)|\leq
L|\varphi(x)|+|h(x,\tau)|.$

Thus, $\int_{\mathbb{R}^{d}}|\eta(x)|^{2}dx\leq
2\int_{\mathbb{R}^{d}}L^{2}|\varphi(x)|^{2}+|h(x,\tau)|^{2}dx\leq
2L^{2}\|\varphi\|^{2}_{H}+2\|h(\cdot,\tau)\|^{2}_{H}$. Since
$\nabla\eta(x)=\nabla_{x}\alpha(x,\tau,\varphi(x))+\alpha^{\prime}_{\varphi}(x,\tau,\varphi(x))\nabla\varphi(x)$,
we have

$\begin{split}\|\eta\|_{V}^{2}=&\int_{\mathbb{R}^{d}}|\eta(x)|^{2}+|\nabla\eta(x)|^{2}dx\\\
\leq&2\int_{\mathbb{R}^{d}}L^{2}|\varphi(x)|^{2}+|h(x,\tau)|^{2}dx+2\int_{\mathbb{R}^{d}}|p(x,\tau)|^{2}+L_{0}^{2}|\varphi(x)|^{2}dx+2\int_{\mathbb{R}^{d}}L^{2}|\nabla\varphi(x)|^{2}dx\\\
\leq&2(L^{2}\|\varphi\|_{V}^{2}+\|h(\cdot,\tau)\|_{H}^{2}+\|p(\cdot,\tau)\|_{H}^{2}+L_{0}^{2}\|\varphi\|_{H}^{2})<\infty,\end{split}$

because $p,h\in L^{\infty}((0,T);H)$. Consequently, $\eta\in V$, as claimed.

Next, we show that the operator $\mathcal{A}$ is monotone in the space
$V^{\prime}$. According to (76), we have
$(\alpha(x,\tau,\varphi_{1})-\alpha(x,\tau,\varphi_{2}))(\varphi_{1}-\varphi_{2})\geq\omega(\varphi_{1}-\varphi_{2})^{2}$,
for any
$\varphi_{1},\varphi_{2}\geq\varphi_{min},x\in\mathbb{R},\tau\in[0,T]$.

$\begin{split}\langle\mathcal{A}(\tau,\varphi_{1})-\mathcal{A}(\tau,\varphi_{2}),\varphi_{1}-\varphi_{2}\rangle&=(\alpha(\cdot,\tau,\varphi_{1})-\alpha(\cdot,\tau,\varphi_{2}),\varphi_{1}-\varphi_{2})\\\
&=\int_{\mathbb{R}^{d}}(\alpha(x,\tau,\varphi_{1}(x))-\alpha(x,\tau,\varphi_{2}(x)))(\varphi_{1}(x)-\varphi_{2}(x))dx\\\
&\geq\int_{\mathbb{R}^{d}}\omega|\varphi_{1}(x)-\varphi_{2}(x)|^{2}dx=\omega\|\varphi_{1}-\varphi_{2}\|^{2}_{H}.\end{split}$

This implies that the operator $\mathcal{A}(\tau,\cdot)$ is strongly monotone.

For a given $\tilde{\varphi}\in{\mathcal{H}}$, we have
$\hat{f}\in\mathcal{V}^{\prime}$, where
$\hat{f}(\tau)=g_{0}(\tau,\tilde{\varphi}(\cdot,\tau))+\nabla\cdot\bm{g}_{1}(\tau,\tilde{\varphi}(\cdot,\tau))$,
because $g_{0},g_{1j}:[0,T]\times H\to H$ are globally Lipschitz continuous,
$H\hookrightarrow V^{\prime}$, and the operator $\nabla$ maps $H$ into
$V^{\prime}$. The hemicontinuity, boundedness, and coercivity of the operator
$\mathcal{A}$ follow from the assumption that the function $\alpha$ is
globally Lipschitz continuous and strictly increasing.

Applying Theorem 3, we deduce the existence of a solution
$\varphi\in{\mathcal{V}}$ such that

$\partial_{\tau}\varphi+\mathcal{A}(\tau,\varphi)=\hat{f}(\tau),\qquad\varphi_{0}\in
H,$ (28)

where $\mathcal{A}(\tau,\varphi)=A\alpha(\cdot,\tau,\varphi)$. Next, we
multiply (28) by $A^{-1}$ to obtain

$\partial_{\tau}A^{-1}\varphi+\alpha(\cdot,\tau,\varphi)=f,$ (29)

where $f=f(\tau,\tilde{\varphi})=A^{-1}\hat{f}(\tau)$. For $\tau\in[0,T]$, we
denote
$\tilde{f}(\tilde{\varphi})=A^{-1/2}\hat{f}(\tau)=A^{-1/2}g_{0}(\tau,\tilde{\varphi})+A^{-1/2}\sum_{j=1}^{d}\partial_{x_{j}}g_{1j}(\tau,\tilde{\varphi})$.
The Fourier transform of $\tilde{f}$ is defined by

$\widehat{\tilde{f}(\tilde{\varphi})}(\xi)=\frac{1}{(1+|\xi|^{2})^{1/2}}\widehat{g_{0}(\tau,\tilde{\varphi}})(\xi)+\sum_{j=1}^{d}\frac{(-i\xi_{j})}{(1+|\xi|^{2})^{1/2}}\widehat{g_{1j}(\tau,\tilde{\varphi}})(\xi).$

Let $\beta>0$ be the Lipschitz constant of the mappings
$g_{0},g_{1j},j=1,\cdots,d$. Using Parseval’s identity and Lipschitz
continuity of $g_{0},g_{1j}$ in $H$, we obtain, for
$\tilde{\varphi}_{1},\tilde{\varphi}_{2}\in\mathcal{H}$,

$\begin{split}\|\tilde{f}(\tilde{\varphi}_{1})-\tilde{f}(\tilde{\varphi}_{2})\|^{2}_{H}&=\|\widehat{\tilde{f}(\tilde{\varphi}_{1})}-\widehat{\tilde{f}(\tilde{\varphi}_{2})}\|^{2}_{H}=\int_{\mathbb{R}^{d}}|\widehat{\tilde{f}(\tilde{\varphi}_{1})}(\xi)-\widehat{\tilde{f}(\tilde{\varphi}_{2})}(\xi)|^{2}d\xi\\\
&\leq
2\int_{\mathbb{R}^{d}}\frac{1}{1+|\xi|^{2}}|\widehat{g_{0}(\tau,\tilde{\varphi}_{1})}(\xi)-\widehat{g_{0}(\tau,\tilde{\varphi}_{2})}(\xi)|^{2}\\\
&\quad+\sum_{j=1}^{d}\frac{|\xi|^{2}}{1+|\xi|^{2}}|\widehat{g_{1j}(\tau,\tilde{\varphi}_{1})}(\xi)-\widehat{g_{1j}(\tau,\tilde{\varphi}_{2})}(\xi)|^{2}d\xi\\\
&\leq
2\|\widehat{g_{0}(\tau,\tilde{\varphi}_{1})}-\widehat{g_{0}(\tau,\varphi_{2})}\|^{2}_{H}+2\sum_{j=1}^{d}\|\widehat{g_{1j}(\tau,\tilde{\varphi}_{1})}-\widehat{g_{1j}(\tau,\varphi_{2})}\|^{2}_{H}\\\
&=2\|g_{0}(\tau,\tilde{\varphi}_{1})-g_{0}(\tau,\tilde{\varphi}_{2})\|^{2}_{H}+2\sum_{j=1}^{d}\|g_{1j}(\tau,\tilde{\varphi}_{1})-g_{1j}(\tau,\tilde{\varphi}_{2})\|^{2}_{H}\\\
&\leq\tilde{\beta}^{2}\|\tilde{\varphi}_{1}-\tilde{\varphi}_{2}\|^{2}_{H},\end{split}$

where $\tilde{\beta}^{2}=2(1+d)\beta^{2}$. Hence, we obtain

$\|\tilde{f}(\tilde{\varphi}_{1})-\tilde{f}(\tilde{\varphi}_{2})\|_{H}\leq\tilde{\beta}\|\tilde{\varphi}_{1}-\tilde{\varphi}_{2}\|_{H}.$
(30)

Suppose $\varphi_{1},\varphi_{2}\in\mathcal{H}$ are such that
$\varphi_{1}=F(\tilde{\varphi_{1}})$ and $\varphi_{2}=F(\tilde{\varphi_{2}}).$
Here, the map $F:\mathcal{H}\to\mathcal{H}$ is defined by
$\varphi=F(\tilde{\varphi}),$ where $\varphi$ is a solution to the Cauchy
problem

$\partial_{\tau}A^{-1}\varphi+\alpha(\cdot,\tau,\varphi)=f(\tau,\tilde{\varphi}),\qquad\varphi(0)=\varphi_{0}.$

Letting
$\varphi=\varphi_{1}-\varphi_{2}=F(\tilde{\varphi_{1}})-F(\tilde{\varphi_{2}})$,
we obtain

$\partial_{\tau}A^{-1}(\varphi_{1}-\varphi_{2})+\alpha(\cdot,\tau,\varphi_{1})-\alpha(\cdot,\tau,\varphi_{2})=f(\tilde{\varphi}_{1})-f(\tilde{\varphi}_{2}).$
(31)

Next, multiplying (31) by $\varphi_{1}-\varphi_{2}$ and taking the scalar
product in the space $H$, we obtain

$\displaystyle(\partial_{\tau}A^{-1}(\varphi_{1}-\varphi_{2}),\varphi_{1}-\varphi_{2})$
$\displaystyle+$
$\displaystyle(\alpha(\cdot,\tau,\varphi_{1})-\alpha(\cdot,\tau,\varphi_{2}),\varphi_{1}-\varphi_{2})$
(32) $\displaystyle=$
$\displaystyle(f(\tau,\tilde{\varphi}_{1})-f(\tau,\tilde{\varphi}_{2}),\varphi_{1}-\varphi_{2}).$

Using (30) and the fact that $A^{-1/2}$ is self-adjoint in $H$, then (32)
gives

$\begin{split}\frac{1}{2}\frac{d}{d\tau}&\|A^{-1/2}(\varphi_{1}-\varphi_{2})\|^{2}_{H}+\omega\|\varphi_{1}-\varphi_{2}\|^{2}_{H}\\\
&\leq\langle
f(\tau,\tilde{\varphi}_{1})-f(\tau,\tilde{\varphi}_{2}),\varphi_{1}-\varphi_{2}\rangle=\langle
A^{1/2}(f(\tau,\tilde{\varphi}_{1})-f(\tau,\tilde{\varphi}_{2})),A^{-1/2}(\varphi_{1}-\varphi_{2})\rangle\\\
&\leq\|A^{1/2}(f(\tau,\tilde{\varphi}_{1})-f(\tau,\tilde{\varphi}_{2}))\|_{H}\|\varphi_{1}-\varphi_{2}\|_{V^{\prime}}=\|\tilde{f}(\tilde{\varphi}_{1})-\tilde{f}(\tilde{\varphi}_{2})\|_{H}\|\varphi_{1}-\varphi_{2}\|_{V^{\prime}}\\\
&\leq\tilde{\beta}\|\tilde{\varphi_{1}}-\tilde{\varphi_{2}}\|_{H}\|\varphi_{1}-\varphi_{2}\|_{V^{\prime}}.\end{split}$

This implies

$\frac{1}{2}\frac{d}{d\tau}\|\varphi_{1}-\varphi_{2}\|^{2}_{V^{\prime}}+\omega\|\varphi_{1}-\varphi_{2}\|^{2}_{H}\leq\tilde{\beta}\|\tilde{\varphi_{1}}-\tilde{\varphi_{2}}\|_{H}\|\varphi_{1}-\varphi_{2}\|_{V^{\prime}}.$

Then, integrating on a small time interval $[0,T]$ from $0$ to $t$ and noting
that $\varphi_{1}(0)=\varphi_{2}(0)=\varphi_{0}$, we obtain

$\begin{split}\frac{1}{2}\|\varphi_{1}(\tau)-\varphi_{2}(\tau)\|^{2}_{V^{\prime}}&+\omega\int_{0}^{\tau}\|\varphi_{1}(s)-\varphi_{2}(s)\|^{2}_{H}ds\\\
&\leq\tilde{\beta}\int_{0}^{\tau}\|\tilde{\varphi_{1}}(s)-\tilde{\varphi_{2}}(s)\|_{H}\|\varphi_{1}(s)-\varphi_{2}(s)\|_{V^{\prime}}ds\\\
&\leq\tilde{\beta}\max\limits_{\tau\in[0,T]}~{}\|\varphi_{1}(\tau)-\varphi_{2}(\tau)\|_{V^{\prime}}\int_{0}^{T}\|\tilde{\varphi_{1}}(\tau)-\tilde{\varphi_{2}}(\tau)\|_{H}d\tau.\end{split}$

Taking the maximum over $\tau\in[0,T]$ and using the fact that for any
$a,b\in\mathbb{R},~{}ab\leq\frac{1}{2}a^{2}+\frac{1}{2}b^{2}$, we obtain

$\begin{split}\frac{1}{2}(\max\limits_{\tau\in[0,T]}~{}&\|\varphi_{1}(\tau)-\varphi_{2}(\tau)\|_{V^{\prime}})^{2}+\omega\int_{0}^{T}\|\varphi_{1}(\tau)-\varphi_{2}(\tau)\|^{2}_{H}d\tau\\\
&\leq\tilde{\beta}\max\limits_{\tau\in[0,T]}~{}\|\varphi_{1}(\tau)-\varphi_{2}(\tau)\|_{V^{\prime}}\int_{0}^{T}\|\tilde{\varphi_{1}}(\tau)-\tilde{\varphi_{2}}(\tau)\|_{H}d\tau\\\
&\leq\frac{1}{2}(\max\limits_{\tau\in[0,T]}~{}\|\varphi_{1}(\tau)-\varphi_{2}(\tau)\|_{V^{\prime}})^{2}+\frac{\tilde{\beta}^{2}}{2}(\int_{0}^{T}\|\tilde{\varphi_{1}}(\tau)-\tilde{\varphi_{2}}(\tau)\|_{H}d\tau)^{2}.\end{split}$

Using the Cauchy–Schwartz inequality, we obtain
$\omega\int_{0}^{T}\|\varphi_{1}(\tau)-\varphi_{2}(\tau)\|^{2}_{H}d\tau\leq\frac{\tilde{\beta}^{2}}{2}\int_{0}^{T}d\tau\int_{0}^{T}\|\tilde{\varphi_{1}}(\tau)-\tilde{\varphi_{2}}(\tau)\|^{2}_{H}d\tau=\frac{\tilde{\beta}^{2}T}{2}\int_{0}^{T}\|\tilde{\varphi_{1}}(\tau)-\tilde{\varphi_{2}}(\tau)\|^{2}_{H}d\tau.$
This implies that

$\|F(\tilde{\varphi_{1}})-F(\tilde{\varphi_{2}})\|^{2}_{\mathcal{H}}\leq\frac{\tilde{\beta}^{2}T}{2\omega}\|\tilde{\varphi_{1}}-\tilde{\varphi_{2}}\|^{2}_{\mathcal{H}}.$

Thus, for sufficiently small value of $T$ such that
$\frac{\tilde{\beta}^{2}T}{2\omega}<1$, the operator $F$ is a contraction on
the space $\mathcal{H}$. Therefore, by the Banach fixed point theorem, $F$ has
a unique fixed point in $\mathcal{H}$. It is worth noting that $\tilde{\beta}$
and $\omega$ are given such that they are independent of $T$. If $T>0$ is
arbitrary, then we can apply a simple continuation argument. In other words,
if the solution exists in $(0,T_{0})$ interval with
$\frac{\tilde{\beta}^{2}T_{0}}{2\omega}<1$, then starting from the initial
condition $\varphi_{0}=\varphi(T_{0}/2)$, we can continue the solution
$\varphi$ from the interval $(0,T_{0})$ over the interval
$(0,T_{0})\cup(T_{0}/2,T_{0}/2+T_{0})\equiv(0,3T_{0}/2)$. Continuing in this
manner, we obtain the existence and uniqueness of a solution
$\varphi\in{\mathcal{H}}$ defined on the time interval $(0,T)$.

Finally, the solution belongs to the space $\mathcal{V}$ because the right-
hand side, i.e., the function
$\hat{f}(\tau)=g_{0}(\tau,\varphi(\cdot,\tau))+\nabla\cdot\bm{g}_{1}(\tau,\varphi(\cdot,\tau))$
belongs to ${\mathcal{V}}^{\prime}$. Applying Theorem 3 we conclude
$\varphi\in{\mathcal{V}}$, as claimed. $\diamondsuit$

The following result demonstrates the absolute continuity and a-priori energy
estimate property of the solution. Based on the assumption of the previous
theorem, we have
$\alpha(\cdot,0),g_{0}(\cdot,0),g_{1j}(\cdot,0)\in\mathcal{H}$. Here, the
space $\mathcal{X}=L^{\infty}((0,T);V^{\prime})$ is endowed with the norm

$\|\varphi\|^{2}_{\mathcal{X}}=\sup_{\tau\in[0,T]}\|\varphi(\tau)\|^{2}_{V^{\prime}},\;\forall\varphi\in\mathcal{X}.$

Again, the following result and its are contained in our recent paper [66,
Theorem 3].

######  Theorem 5.

[66, Theorem 3] Suppose that the functions $\alpha,g_{0},g_{1j}$ satisfy the
assumptions of Theorem 4. Then, the unique solution $\varphi\in\mathcal{V}$ to
the Cauchy problem $(\ref{eq:hatf})$ is absolutely continuous, i.e.,
$\varphi\in C([0,T];H)$. Moreover, there exists a constant $\tilde{C}>0$, such
that the unique solution satisfies the following inequality:

$\|\varphi\|^{2}_{\mathcal{X}}+\|\varphi\|^{2}_{\mathcal{H}}\leq\tilde{C}\bigl{(}\|\varphi_{0}\|^{2}_{V^{\prime}}+\|\alpha(\cdot,0)\|^{2}_{\mathcal{H}}+\|g_{0}(\cdot,0)\|^{2}_{\mathcal{H}}+\sum_{j=1}^{d}\|g_{1j}(\cdot,0)\|^{2}_{\mathcal{H}}\bigr{)}.$
(33)

###### Proof.

Since $\hat{f}\in\mathcal{V}^{\prime}$, where
$\hat{f}=g_{0}+\nabla\cdot\bm{g}_{1}$ and
$\mathcal{A}(\tau,\varphi)\in\mathcal{V}^{\prime}$, then
$\partial_{\tau}\varphi\in\mathcal{V}^{\prime}$. Therefore, for each
$\varphi_{0}\in H$, we have $\varphi\in W$, where $W$ is the Banach space
$W=\\{\varphi,\varphi\in\mathcal{V},\partial_{\tau}\varphi\in\mathcal{V}^{\prime}\\}$.
According to [63, Proposition 1.2], we have $W\hookrightarrow C([0,T];H)$.
Hence, the unique solution $\varphi$ to the Cauchy problem (4) belongs to the
space $C([0,T];H)$, as claimed.

Next, we show that the unique solution satisfies a-priori energy estimate
(33). Let $\varphi$ be a unique solution to the Cauchy problem (27). Multiply
(29) by $\varphi$ and take the scalar product in $H$ to obtain

$(\partial_{\tau}A^{-1}\varphi,\varphi)_{H}+(\alpha(\cdot,\tau,\varphi),\varphi)_{H}=(A^{-1}g_{0}(\tau,\varphi)+A^{-1}\nabla\cdot\bm{g}_{1}(\tau,\varphi),\varphi).$
(34)

Using the Lipschitz continuity of $g_{0},\bm{g}_{1}$, and strong monotonicity
of $\alpha$, we obtain

$\displaystyle\frac{1}{2}\frac{d}{d\tau}\|\varphi\|^{2}_{V^{\prime}}+\omega\|\varphi\|^{2}_{H}$
$\displaystyle=$
$\displaystyle(\partial_{\tau}A^{-1}\varphi,\varphi)+\omega\|\varphi\|^{2}_{H}$
$\displaystyle\leq$
$\displaystyle(\partial_{\tau}A^{-1}\varphi,\varphi)+(\alpha(\cdot,\varphi)-\alpha(\cdot,0),\varphi)$
$\displaystyle=$
$\displaystyle(A^{-1}(g_{0}(\cdot,\varphi)+\nabla\cdot\bm{g}_{1}(\tau,\varphi))-\alpha(\cdot,0),\varphi)$
$\displaystyle=$
$\displaystyle(A^{-1}(g_{0}(\cdot,\varphi)-g_{0}(\cdot,0)+\nabla\cdot\bm{g}_{1}(\cdot,\varphi)-\nabla\cdot\bm{g}_{1}(\cdot,0)),\varphi)$
$\displaystyle+(A^{-1}(g_{0}(\cdot,0)+\nabla\cdot\bm{g}_{1}(\cdot,0)),\varphi)-(\alpha(\cdot,0),\varphi)$
$\displaystyle=$
$\displaystyle(A^{-1/2}(g_{0}(\cdot,\varphi)-g_{0}(\cdot,0)+\nabla\cdot\bm{g}_{1}(\cdot,\varphi)-\nabla\cdot\bm{g}_{1}(\cdot,0)),A^{-1/2}\varphi)$
$\displaystyle+(A^{-1/2}(g_{0}(\cdot,0)+\nabla\cdot\bm{g}_{1}(\cdot,0)),A^{-1/2}\varphi)-(\alpha(\cdot,0),\varphi)$
$\displaystyle\leq$
$\displaystyle\beta(1+d)\|\varphi\|_{H}\|\varphi\|_{V^{\prime}}+\|A^{-1/2}(g_{0}(\cdot,0)+\nabla\cdot\bm{g}_{1}(\cdot,0))\|_{H}\|\varphi\|_{V^{\prime}}$
$\displaystyle+\|\alpha(\cdot,0)\|_{H}\|\varphi\|_{H}$ $\displaystyle\leq$
$\displaystyle\frac{\omega}{4}\|\varphi\|^{2}_{H}+\frac{\beta^{2}(1+d)^{2}}{\omega}\|\varphi\|^{2}_{V^{\prime}}+\frac{1}{2}\|A^{-1/2}(g_{0}(\cdot,0)+\nabla\cdot\bm{g}_{1}(\cdot,0))\|^{2}_{H}$
$\displaystyle+\frac{1}{2}\|\varphi\|^{2}_{V^{\prime}}+\frac{1}{\omega}\|\alpha(\cdot,0)\|^{2}_{H}+\frac{\omega}{4}\|\varphi\|^{2}_{H}.$

Hence, there exist constants $C_{0},C_{1}>0$ such that

$\displaystyle\frac{d}{d\tau}\|\varphi\|^{2}_{V^{\prime}}+\omega\|\varphi\|^{2}_{H}$
$\displaystyle\leq$ $\displaystyle
C_{1}\|\varphi\|^{2}_{V^{\prime}}+C_{0}\bigl{(}\|g_{0}(\cdot,0)\|^{2}_{H}+\sum_{j=1}^{d}\|g_{1j}(\cdot,0)\|^{2}_{H}+\|\alpha(\cdot,0)\|^{2}_{H}\bigr{)}.$

Solving the differential inequality $y^{\prime}(\tau)\leq
C_{1}y(\tau)+r(\tau)$, where
$y(\tau)=\|\varphi(\cdot,\tau)\|^{2}_{V^{\prime}}$ and
$r(\tau)=C_{0}\bigl{(}\|g_{0}(\cdot,\tau,0)\|^{2}_{H}+\sum_{j=1}^{d}\|g_{1j}(\cdot,\tau,0)\|^{2}_{H}+\|\alpha(\cdot,\tau,0)\|^{2}_{H}\bigr{)}$,
we obtain

$y(\tau)\leq e^{C_{1}T}\bigl{(}y(0)+\int_{0}^{T}r(s)ds\bigr{)},$

and the proof of the Theorem follows. ∎

##  5\. Applications to Option Pricing

The classical linear Black–Scholes model and its multidimensional
generalizations have been widely used in the financial market analysis. It is
well-known that the price $V=V(t,S)$ of an option on an underlying asset price
$S$ at time $t\in[0,T]$ can be obtained as a solution to the linear
Black–Scholes parabolic equation of the form (1). Generally, th underlying
asset price is assumed to follow the geometric Brownian motion $dS/S=\mu
dt+\sigma dW$. Here, $\\{W_{t},t\geq 0\\}$ is the standard Wiener process. The
terminal condition $\Phi(S)$ represents the payoff diagram at maturity $t=T$,
$\Phi(S)=(S-K)^{+}$ (call option case) or $\Phi(S)=(K-S)^{+}$ (put option
case).

For the multidimensional case, where the option price
$V(t,S_{1},\cdots,S_{n})$ depends on the vector of $n$ underlying stochastic
assets $S=(S_{1},\cdots,S_{n})$ with the volatilities $\sigma_{i}$ and mutual
correlations $\varrho_{ij},i,j=1,\cdots,n$, the Black–Scholes pricing equation
can be expressed as follows:

$\frac{\partial V}{\partial
t}+\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}\rho_{ij}\sigma_{i}\sigma_{j}S_{i}S_{j}\frac{\partial^{2}V}{\partial
S_{i}\partial S_{j}}+r\sum_{i=1}^{n}S_{i}\frac{\partial V}{\partial
S_{i}}-rV=0,\quad V(T,S)=\Phi(S).$ (35)

Equations (1) and (35) can be transformed into equation (4.) defined on the
whole space $\mathbb{R}^{n}$ (c.f., Ševčovič, Stehlíková, Mikula [59, Chapter
4, Section 5]).

According to stock markets observations, the models (1) and (35) were derived
under some restrictive assumptions, e.g., completeness and frictionless of the
financial market, perfect replication of a portfolio and its liquidity, and
absence of transaction costs. However, these assumptions are often violated in
financial markets. In the past few decades, several attempts have been made to
investigate the effects of nontrivial transaction costs [41, 6, 42, 60]. For
instance, Schönbucher and Willmott [55], Frey and Patie [25], Frey and Stremme
[26] investigated the feedback and illiquid market effects due to large
traders choosing given stock-trading strategies. Jandačka and Ševčovič
recently investigated the effects of the risk arising from an unprotected
portfolio. Barles and Soner [9] analyzed the option pricing models based on
utility maximization. The common feature of these generalizations of the
linear Black–Scholes equation (1) is that the constant volatility $\sigma$ is
replaced by a nonlinear function depending on the second derivative
$\partial_{S}^{2}V$ of the option price $V$. Among these generalizations, Frey
and Stremme [32] derived a nonlinear Black–Scholes model by assuming that the
underlying asset dynamics takes into account the presence of feedback effects
due to the influence of a large trader choosing particular stock-trading
strategy (see also [55, 25, 27]).

Recently, Cruz and Ševčovič [20] generalized the Black–Scholes equation in two
important directions. First, they employed the ideas of Frey and Stremme [32]
to incorporate the effect of a large trader into the model. Second, they
relaxed the assumption on liquidity of market by assuming that the underlying
asset price follows a Lévy stochastic process with jumps to obtain the
following nonlinear PIDE:

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\frac{\partial V}{\partial
t}+\frac{1}{2}\frac{\sigma^{2}S^{2}}{\left(1-\varrho
S\partial_{S}\phi\right)^{2}}\frac{\partial^{2}V}{\partial
S^{2}}+rS\frac{\partial V}{\partial S}-rV$ (36) $\displaystyle+$
$\displaystyle\int_{\mathbb{R}}\left(V(t,S+H)-V(t,S)-H\frac{\partial
V}{\partial S}\right)\nu(\mathrm{d}z),$

where the shift function $H=H(\phi,S,z)$ depends on the large investor stock-
trading strategy function $\phi=\phi(t,S)$. Moreover, this shift function is a
solution to the following implicit algebraic equation:

$H=\rho S(\phi(t,S+H)-\phi(t,S))+S(e^{z}-1).$ (37)

The large trader strategy function $\phi$ may depend on the derivative
$\partial_{S}V$ of the option price $V$, e.g., $\phi(t,S)=\partial_{S}V(t,S)$.
However, in our application, we assume the trading strategy function
$\phi(t,S)$ is prescribed and globally Hölder continuous. Next, we present the
analysis of this equation depending the behavior of the parameter $\rho=0$.

If $\rho=0$, then $H=S(e^{z}-1)$. Thus, equation (36) can be reduced to a
linear PIDE of the form (4.) in the one-dimensional space ($n=1$). This is
obtained using the standard transformation $\tau=T-t,x=\ln(\frac{S}{K})$ and
setting $V(t,S)=e^{-r\tau}u(\tau,x)$.

However, if $\rho>0$, then (36) can be transformed into a nonlinear parabolic
PIDE. Indeed, suppose that the transformed large trader stock-trading strategy
$\psi(\tau,x)=\phi(t,S)$. Then, $V(t,S)$ solves equation (36) if and only if
the transformed function $u(\tau,x)$ is a solution to the following nonlinear
parabolic equation:

$\displaystyle\frac{\partial u}{\partial\tau}$ $\displaystyle=$
$\displaystyle\frac{\sigma^{2}}{2}\frac{1}{(1-\rho\partial_{x}\psi)^{2}}\frac{\partial^{2}u}{\partial^{2}x}+\left(r-\frac{\sigma^{2}}{2}\frac{1}{(1-\rho\partial_{x}\psi)^{2}}-\delta(\tau,x)\right)\frac{\partial
u}{\partial x}$ (38) $\displaystyle+$
$\displaystyle\int_{\mathbb{R}}\left(u(\tau,x+\xi)-u(\tau,x)-\xi\frac{\partial
u}{\partial x}(\tau,x)\right)\nu(\mathrm{d}z),\quad u(0,x)=\Phi(Ke^{x}),$

$\tau\in[0,T],x\in\mathbb{R}$. The shift function $\xi(\tau,x,z)$ is a
solution to the following algebraic equation:

$e^{\xi}=e^{z}+\rho(\psi(\tau,x+\xi)-\psi(\tau,x)),$ (39)

and
$\delta(\tau,x)=\int_{\mathbb{R}}(e^{\xi}-1-\xi)\nu(\mathrm{d}z)=\int_{\mathbb{R}}(e^{z}-1-\xi+\rho(\psi(\tau,x+\xi)-\psi(\tau,x)))\nu(\mathrm{d}z)$.

For small values of $0<\rho\ll 1$, we can construct the first order asymptotic
expansion $\xi(\tau,x,z)=\xi_{0}(\tau,x,z)+\rho\xi_{1}(\tau,x,z)$. For
$\rho=0$, we obtain $\xi_{0}(\tau,x,z)=z$. Hence,

$e^{z+\rho\xi_{1}}=e^{z}+\rho(\psi(\tau,x+z+\rho\xi_{1})-\psi(\tau,x)).$

Taking the first derivative of the above implicit equation with respect to
$\rho$ and evaluating it at the origin $\rho=0$, we obtain
$\xi_{1}=e^{-z}(\psi(\tau,x+z)-\psi(\tau,x))$, i.e.,

$\xi(\tau,x,z)=z+\rho e^{-z}(\psi(\tau,x+z)-\psi(\tau,x)).$ (40)

Consequently, we obtain the following lemma.

######  Lemma 6.

[58, Lemma 1] Assume that the stock-trading strategy $\phi=\phi(t,S)$ is a
globally $\omega$-Hölder continuous function, $0<\omega\leq 1$. Then, the
transformed function $\psi(\tau,x)=\phi(t,S)$ is $\omega$-Hölder continuous,
and the first order asymptotic expansion $\xi(\tau,x,z)$ of the nonlinear
algebraic equation (39) is $\omega$-Hölder continuous in all variables.
Furthermore, there exists a constant $C_{0}>0$ such that
$\sup_{\tau,x}|\xi(\tau,x,z)|\leq C_{0}|z|^{\omega}(1+e^{|z|})$ for any
$z\in\mathbb{R}$.

###  5.1. Linearization of PIDE

In what follows, we consider a simplified linear approximation of (36) by
setting $\rho=0$ in the diffusion function, but we keep the shift function $H$
depending on the parameter $\rho$. Then, the transformed Cauchy problem for
the solution $u$ with the first order approximation of the shift function
$\xi$ is given as follows:

$\displaystyle\frac{\partial u}{\partial\tau}$ $\displaystyle=$
$\displaystyle\frac{\sigma^{2}}{2}\frac{\partial^{2}u}{\partial^{2}x}+\left(r-\frac{\sigma^{2}}{2}+\delta(\tau,x)\right)\frac{\partial
u}{\partial x}$ (41)
$\displaystyle+\int_{\mathbb{R}}\left(u(\tau,x+\xi)-u(\tau,x)-\xi\frac{\partial
u}{\partial x}(\tau,x)\right)\nu(\mathrm{d}z),$

$\tau\in[0,T],x\in\mathbb{R}$, where
$\xi(\tau,x,z)=z+\rho(\psi(\tau,x+z))-\psi(\tau,x))$.

Note that the call/put option payoff functions
$\Phi(S)=\Phi(Ke^{x})=(S-K)^{+}=K(e^{x}-1)^{+}$ /
$\Phi(S)=\Phi(Ke^{x})=(K-S)^{+}=K(1-e^{x})^{+}$ do not belong to the Banach
space $X^{\gamma}$. According to [19], the procedure on how to overcome this
problem and formulate existence and uniqueness of a solution to the PIDE (41)
is based on shifting the solution $u$ by $u^{BS}$. Here,
$u^{BS}(\tau,x)=e^{r\tau}V^{BS}(T-\tau,Ke^{x})$ is an explicitly given
solution to the linear Black–Scholes equation without the PIDE part. In other
words, $u^{BS}$ solves the following linear parabolic equation:

$\frac{\partial
u^{BS}}{\partial\tau}-\frac{\sigma^{2}}{2}\frac{\partial^{2}u^{BS}}{\partial
x^{2}}-\left(r-\frac{\sigma^{2}}{2}\right)\frac{\partial u^{BS}}{\partial
x}=0,\quad u^{BS}(0,x)=\Phi(Ke^{x}),\ \tau\in(0,T),x\in\mathbb{R}.$ (42)

Recall that $u^{BS}(\tau,x)=Ke^{x+r\tau}N(d_{1})-KN(d_{2})$ (call option
case), where $d_{1,2}=(x+(r\pm\sigma^{2}/2)\tau)/(\sigma\sqrt{\tau})$ (c.f.,
[41, 59]). Here,
$N(d)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{d}e^{-\xi^{2}/2}\mathrm{d}\xi$ is
the cumulative density function of the normal distribution. The next result
and its proof are based on the recent paper by Ševčovič and Udeani [58]

######  Theorem 7.

[58, Theorem 2] Assume the transformed stock-trading strategy function
$\psi(\tau,x)$ is globally $\omega$-Hölder continuous in both variables.
Suppose that $\nu$ is a Lévy measure with the shape parameters
$\alpha<3,D\in\mathbb{R}$, where either $\mu>0$, or $\mu=0$ and $D>1$. Let
$X^{\gamma}$ be the space of Bessel potentials space
${\mathscr{L}}^{p}_{2\gamma}(\mathbb{R})$, where
$\frac{\alpha-1}{2\omega}<\gamma<\frac{p+1}{2p}$ and
$\frac{1}{2}\leq\gamma<1$. Let $T>0$. Then, the linear PIDE (41) has a unique
mild solution $u$ with the property that the difference $U=u-u^{BS}$ belongs
to the space $C([0,T],X^{\gamma})$.

###### Proof.

We first outline the idea of the proof. The initial condition
$u(0,\cdot)\not\in X^{\gamma}$ because of two reasons. It is not smooth for
$x=0$, and it grows exponentially for $x\to\infty$ (call option) or
$x\to-\infty$ (put option). The shift function $U=u-u^{BS}$ satisfies
$U(0,\cdot)\equiv 0$, and so the initial condition $U(0,\cdot)$ belongs to
$X^{\gamma}$. However, the shift function $u^{BS}$ enters the governing PIDE
as it includes the term $f(u^{BS}(\tau,\cdot))$ in the right-hand side. Since
$u^{BS}(0,x)$ is not sufficiently smooth for $x=0$, the shift term
$f(u^{BS}(\tau,\cdot))$ is singular for $\tau\to 0^{+}$. Following the ideas
of [19], for the shift term $f(u^{BS}(\tau,\cdot))$, we can provide Hölder
estimates, which are sufficient for proving the main result of this theorem
(c.f., [19, Lemma 4.1]). Furthermore, the exponential growth of the function
$u^{BS}$ will be overcome since $\tilde{f}(e^{x})=0$, where
$\tilde{f}(u)=f(u)-\delta(\tau,\cdot)\partial_{x}u$, i.e.,

$\tilde{f}(u)(x)=\int_{\mathbb{R}}\left(u(x+\xi)-u(x)-(e^{\xi}-1)\partial_{x}u(x)\,\right)\nu(\mathrm{d}z).$

Next, we present more details of the proof. The function $u^{BS}$ solves the
linear PDE (42). Thus, the difference $U=u-u^{BS}$ of the solution $u$ to (41)
and $u^{BS}$ satisfies the PIDE with the right-hand side:

$\displaystyle\frac{\partial U}{\partial\tau}$ $\displaystyle=$
$\displaystyle\frac{\sigma^{2}}{2}\frac{\partial^{2}U}{\partial
x^{2}}+\left(r-\frac{\sigma^{2}}{2}-\delta(\tau,x)\right)\frac{\partial
U}{\partial x}+f(U)+f(u^{BS})-\delta(\tau,x)\frac{\partial u^{BS}}{\partial
x}$ $\displaystyle=$
$\displaystyle\frac{\sigma^{2}}{2}\frac{\partial^{2}U}{\partial
x^{2}}+f(U)+g(\tau,x,\partial_{x}U)+h(\tau,\cdot),$

$U(0,x)=0,\ x\in\mathbb{R},\tau\in(0,T)$. Here
$g(\tau,x,\partial_{x}U)=(r-\sigma^{2}/2-\delta(\tau,x))\partial_{x}U$, and
$h(\tau,\cdot)=\tilde{f}(u^{BS}(\tau,\cdot))$. According to Proposition 2,
$f:X^{\gamma}\to X$ is a bounded linear mapping. Consequently, it is Lipschitz
continuous, provided that $1/2\leq\gamma<1$ and $\gamma>(\alpha-1)/(2\omega)$.
Clearly, $\tilde{f}(e^{x})=0$. Hence,

$\tilde{f}(u^{BS})=\tilde{f}(u^{BS}-Ke^{r\tau+x}),\quad\hbox{and}\ \
\partial_{\tau}\tilde{f}(u^{BS})=\tilde{f}(\partial_{\tau}(u^{BS}-Ke^{r\tau+x})).$

Now, it follows from [19, Lemma 4.1] that the following estimate holds true:

$\|h(\tau_{1},\cdot)-h(\tau_{2},\cdot)\|_{L^{p}}=\|\tilde{f}(u^{BS}(\tau_{1},\cdot))-\tilde{f}(u^{BS}(\tau_{2},\cdot))\|_{L^{p}}\leq
C_{0}|\tau_{1}-\tau_{2}|^{-\gamma+\frac{p+1}{2p}},$
$\|h(\tau,\cdot)\|_{L^{p}}=\|\tilde{f}(u^{BS})(\tau,\cdot))\|_{L^{p}}\leq
C_{0}|\tau^{-(2\gamma-1)\left(\frac{1}{2}-\frac{1}{2p}\right)},$

for any $0<\tau_{1},\tau_{2},\tau\leq T$. The function $h:[0,T]\to X\equiv
L^{p}(\mathbb{R})$ is $((p+1)/(2p)-\gamma)$-Hölder continuous because
$\gamma<\frac{p+1}{2p}$. Moreover,

$\int_{0}^{T}\|h(\tau,\cdot)\|_{L^{p}}d\tau=\int_{0}^{T}\|\tilde{f}(u^{BS}(\tau,\cdot))\|_{L^{p}}d\tau\leq
C_{0}\int_{0}^{T}\tau^{-(2\gamma-1)\left(\frac{1}{2}-\frac{1}{2p}\right)}d\tau<\infty,$

because $(2\gamma-1)\left(\frac{1}{2}-\frac{1}{2p}\right)<1$. Recall that the
crucial part of the proof of [19, Lemma 4.1] was based on the estimates:

$\|\tilde{f}(u^{BS}(\tau,\cdot))\|_{L^{p}}\leq
C_{0}\|v(\tau,\cdot)\|_{X^{\gamma-1/2}},\quad\text{and}\
\|\partial_{\tau}\tilde{f}(u^{BS}(\tau,\cdot))\|_{L^{p}}\leq
C_{0}\|\partial_{\tau}v(\tau,\cdot)\|_{X^{\gamma-1/2}},$

where
$v(\tau,x)=\partial_{x}\left(u^{BS}(\tau,x)-Ke^{r\tau+x}\right)=Ke^{r\tau+x}(N(d_{1}(\tau,x))-1)$.
This estimate is satisfied because of Proposition 2 under the assumptions made
on $\gamma$. The proof for the case of a put option is similar. The final
estimate on the Hölder continuity of the mapping $h$ follows from careful
estimates of the solution $u^{BS}$ derived in the proof of [19, Lemma 4.1].
Now, the proof follows from Theorem 2 and Proposition 3. ∎

##  6\. Feedback effects under jump-diffusion asset price dynamics

Suppose that a large trader uses a stock-holding strategy $\alpha_{t}$ and
$S_{t}$ is a cadlag process (right continuous with limits to the left). In
what follows, we will identify $S_{t}$ with $S_{t^{-}}$. We assume $S_{t}$ has
the following dynamics:

$\displaystyle\mathrm{d}S_{t}=\mu S_{t}\mathrm{d}t+\sigma
S_{t}\mathrm{d}W_{t}+\rho
S_{t}\mathrm{d}\alpha_{t}+\int_{\mathbb{R}}S_{t}(e^{x}-1)J_{X}(\mathrm{d}t,\mathrm{d}x),$
(43)

which can seen as a perturbation of the classical jump-diffusion model. For
instance, if a large trader does not trade, then $\alpha_{t}=0$ or the market
liquidity parameter $\rho$ is set to zero, then the stock price $S_{t}$
follows the classical jump-diffusion model.

We will assume the following structural hypothesis in this chapter:

######  Assumption 1.

[, Assumption 1] Assume the trading strategy $\alpha_{t}=\phi(t,S_{t})$ and
the parameter $\rho\geq 0$ satisfy $\rho L<1$, where
$L=\sup_{S>0}|S\frac{\partial\phi}{\partial S}|$.

Next, we show an explicit formula for the dynamics of $S_{t}$ satisfying (43)
under certain regularity assumptions made on the stock-holding function
$\phi(t,S)$.

######  Proposition 4.

Suppose that the stock-holding strategy $\alpha_{t}=\phi(t,S_{t})$ satisfies
Assumption 1, where $\phi\in C^{1,2}([0,T]\times\mathbb{R^{+}})$. If the
process $S_{t},t\geq 0,$ satisfies the implicit stochastic equation (43), then
the process $S_{t}$ is driven by the following SDE:

$\displaystyle\mathrm{d}S_{t}=b(t,S_{t})S_{t}\mathrm{d}t+v(t,S_{t})S_{t}\mathrm{d}W_{t}+\int_{\mathbb{R}}H(t,x,S_{t})J_{X}(\mathrm{d}t,\mathrm{d}x),$
(44)

where

$\displaystyle b(t,S)=\frac{1}{1-\rho S\frac{\partial\phi}{\partial
S}(t,S)}\left(\mu+\rho\left(\frac{\partial\phi}{\partial
t}+\frac{1}{2}v(t,S)^{2}S^{2}\frac{\partial^{2}\phi}{\partial
S^{2}}\right)\right),$ (45) $\displaystyle v(t,S)=\frac{\sigma}{1-\rho
S\frac{\partial\phi}{\partial S}(t,S)},$ (46) $\displaystyle
H(t,x,S)=S(e^{x}-1)+\rho S\left[\phi(t,S+H(t,x,S))-\phi(t,S)\right].$ (47)

###### Proof.

First, the SDE (44) can be expressed for $S_{t}$ as follows:

$\displaystyle\mathrm{d}S_{t}$ $\displaystyle=$
$\displaystyle\left(b(t,S_{t})S_{t}+\int_{|x|<1}H(t,x,S_{t})\nu(\mathrm{d}x)\right)\mathrm{d}t+v(t,S_{t})S_{t}\mathrm{d}W_{t}$
$\displaystyle+\int_{|x|\geq
1}H(t,x,S_{t})J_{X}(\mathrm{d}t,\mathrm{d}x)+\int_{|x|<1}H(t,x,S_{t})\tilde{J}_{X}(\mathrm{d}t,\mathrm{d}x).$

Since the function $\phi(t,S)$ is smooth, by applying Itô formula (1) to the
process $\phi(t,S_{t})$, we obtain

$\displaystyle\mathrm{d}\alpha_{t}$ $\displaystyle=$
$\displaystyle\left(\frac{\partial\phi}{\partial
t}+\frac{1}{2}v(t,S_{t})^{2}S_{t}^{2}\frac{\partial^{2}\phi}{\partial
S^{2}}\right)\mathrm{d}t+\frac{\partial\phi}{\partial S}\mathrm{d}S_{t}$
$\displaystyle+\int_{\mathbb{R}}\phi(t,S_{t}+H(t,x,S_{t}))-\phi(t,S_{t})-H(t,x,S_{t})\frac{\partial\phi}{\partial
S}(t,S_{t})J_{X}(\mathrm{d}t,\mathrm{d}x).$

Substituting the differential $\mathrm{d}\alpha_{t}$ into (43), we obtain

$\displaystyle\mathrm{d}S_{t}$ $\displaystyle=$ $\displaystyle\mu
S_{t}\mathrm{d}t+\sigma
S_{t}\mathrm{d}W_{t}+\int_{\mathbb{R}}S_{t}(e^{x}-1)J_{X}(\mathrm{d}t,\mathrm{d}x)+\rho
S_{t}\frac{\partial\phi}{\partial S}\mathrm{d}S_{t}$ $\displaystyle+\rho
S_{t}\left(\frac{\partial\phi}{\partial
t}+\frac{1}{2}v(t,S_{t})^{2}S_{t}^{2}\frac{\partial^{2}\phi}{\partial
S^{2}}\right)\mathrm{d}t$ $\displaystyle+\rho
S_{t}\int_{\mathbb{R}}\phi(t,S_{t}+H(t,x,S_{t}))-\phi(t,S_{t})-H(t,x,S_{t})\frac{\partial\phi}{\partial
S}(t,S_{t})J_{X}(\mathrm{d}t,\mathrm{d}x).$

Next, rearrange terms in (6.) to obtain

$\displaystyle(1-\rho S_{t}\frac{\partial\phi}{\partial
S}(t,S_{t}))\mathrm{d}S_{t}=(\mu S_{t}+\rho S_{t}(\frac{\partial\phi}{\partial
t}+\frac{1}{2}v(t,S_{t})^{2}S_{t}^{2}\frac{\partial^{2}\phi}{\partial
S^{2}}))\mathrm{d}t$ $\displaystyle+\sigma S_{t}\mathrm{d}W_{t}-\rho
S_{t}\int_{\mathbb{R}}H(t,x,S_{t})\frac{\partial\phi}{\partial
S}(t,S_{t})J_{X}(\mathrm{d}t,\mathrm{d}x)$ (50)
$\displaystyle+\int_{\mathbb{R}}S_{t}(e^{x}-1)+\rho
S_{t}\left(\phi(t,S_{t}+H(t,x,S_{t}))-\phi(t,S_{t})\right)J_{X}(\mathrm{d}t,\mathrm{d}x).$

By comparing terms in (44) and (50), we obtain the expressions (45), (46), and
the implicit equation for the function $H$:

$\displaystyle H(t,x,S)$ $\displaystyle=$ $\displaystyle\frac{1}{1-\rho
S\frac{\partial\phi}{\partial S}(t,S)}\left(S(e^{x}-1)+\rho
S\left(\phi(t,S+H(t,x,S))-\phi(t,S)\right)\right)$ (51)
$\displaystyle-\frac{1}{1-\rho S\frac{\partial\phi}{\partial S}(t,S)}\rho
S\frac{\partial\phi}{\partial S}(t,S)H(t,x,S).$

Simplifying this expression for $H$, we establish (47), as claimed. ∎

Since the function $H$ is given implicitly by equation (47), we can expand its
solution in terms of a small parameter $\rho$ as follows:

$H(t,x,S)=H^{0}(t,x,S)+\rho H^{1}(t,x,S)+O(\rho^{2})~{}as~{}\rho\to 0,$

to deduce the following proposition:

######  Proposition 5.

Assume $\rho$ is sufficiently small. Then, the first-order approximation of
the function $H(t,x,S)$ is given as follows:

$\displaystyle H(t,x,S)=S(e^{x}-1)+\rho
S\left(\phi(t,Se^{x})-\phi(t,S)\right)+O(\rho^{2})\quad\text{as}\ \rho\to 0.$
(52)

The next proposition and its proof is based on the recent paper by Cruz and
Ševčovič [20].

######  Proposition 6.

[20, Proposition 3.4] Assume that the asset price process $S_{t}=e^{X_{t}+rt}$
satisfies SDE (44), where the Lévy measure $\nu$ is such that $\int_{|x|\geq
1}e^{2x}\nu\left(\mathrm{d}x\right)<\infty$. Let the price of a derivative
security $V(t,S)$ be define by

$V(t,S)=\mathbb{E}\left[e^{-r(T-t)}\Phi(S_{T})|S_{t}=S\right]=e^{-r(T-t)}\mathbb{E}\left[\Phi(Se^{r(T-t)+X_{T-t}})\right].$
(53)

Suppose that the payoff function $\Phi$ is Lipschitz continuous and the
function $\phi$ has a bounded derivative. Then, $V(t,S)$ is a solution to the
PIDE of the form:

$\displaystyle 0=\frac{\partial V}{\partial
t}+\frac{1}{2}v(t,S)^{2}S^{2}\frac{\partial^{2}V}{\partial
S^{2}}+rS\frac{\partial V}{\partial S}-rV$
$\displaystyle\qquad+\int_{\mathbb{R}}V(t,S+H(t,x,S))-V(t,S)-H(t,x,S)\frac{\partial
V}{\partial S}(t,S)\nu(\mathrm{d}x),$ (54)

where $v(t,S)$ and $H(t,x,S)$ are given by $\eqref{coeffcondv}$ and
$\eqref{coeffcondone}$, respectively.

###### Proof.

Recall that the asset price dynamics of $S_{t}$ under the measure $\mathbb{Q}$
is given by

$\displaystyle\mathrm{d}S_{t}=rS_{t}\mathrm{d}t+v(t,S_{t})S_{t}\mathrm{d}W_{t}+\int_{\mathbb{R}}H(t,x,S_{t})\tilde{J}_{X}(\mathrm{d}t,\mathrm{d}x).$
(55)

Applying the Itô’s lemma to $V(t,S_{t})$, we obtain
$\mathrm{d}(V(t,S_{t})e^{-rt})=a(t)\mathrm{d}t+\mathrm{d}M_{t}$, where

$\displaystyle a(t)$ $\displaystyle=$ $\displaystyle\frac{\partial V}{\partial
t}+\frac{1}{2}v(t,S_{t})^{2}S_{t}^{2}\frac{\partial^{2}V}{\partial
S^{2}}+rS_{t}\frac{\partial V}{\partial S}-rV$
$\displaystyle+\int_{\mathbb{R}}V(t,S_{t}+H(t,x,S_{t}))-V(t,S_{t})-H(t,x,S_{t})\frac{\partial
V}{\partial S}(t,S_{t})\nu(\mathrm{d}x),$

and

$\mathrm{d}M_{t}=e^{-rt}S_{t}v(t,S_{t})\frac{\partial V}{\partial
S}\mathrm{d}W_{t}+e^{-rt}\int_{\mathbb{R}}V(t,S_{t}+H(t,x,S_{t}))-V(t,S_{t})\tilde{J}_{X}(\mathrm{d}t,\mathrm{d}x).$

Our aim is to show that $M_{t}$ is a martingale. Consequently, we have
$a\equiv 0$ a.s., and $V$ is a solution to (54) ([17, Proposition 8.9]). To
show that the term
$\int_{0}^{T}e^{-rt}\int_{\mathbb{R}}V(t,S_{t}+H(t,x,S_{t}))-V(t,S_{t})\tilde{J}_{S}(\mathrm{d}t,\mathrm{d}y)$
is a martingale, it is suffices to show that

$\displaystyle\mathbb{E}\left[\int_{0}^{T}e^{-2rt}\left(\int_{\mathbb{R}}V(t,S_{t}+H(t,x,S_{t}))-V(t,S_{t})\nu(\mathrm{d}x)\right)^{2}\mathrm{d}t\right]<\infty.$
(56)

Since $\sup_{0\leq t\leq T}\mathbb{E}\left[e^{X_{T-t}}\right]<\infty$, and the
payoff function $\Phi$ is Lipschitz continuous, $V(t,S)$ is Lipschitz
continuous with the Lipschitz constant $C>0$. Moreover, since the function
$\phi(t,S)$ has bounded derivatives, we have

$S\left|\phi(t,S+H(t,x,S))-\phi(t,S)\right|\leq
S\left|\frac{\partial\phi}{\partial S}\right||H(t,x,S)|\leq L|H(t,x,S)|.$

(see Assumption 1). Since $H(t,x,S)=S(e^{x}-1)+\rho
S(\phi(t,S+H(t,x,S))-\phi(t,S))$, we obtain $|H(t,x,S)|^{2}\leq
S^{2}(e^{x}-1)^{2}/(1-\rho L)^{2}$. Again, since $V$ is Lipschitz continuous
with the Lipschitz constant $C>0$, we have

$\displaystyle\mathbb{E}\left[\int_{0}^{T}e^{-2rt}\left(\int_{\mathbb{R}}V(t,S_{t}+H(t,x,S_{t}))-V(t,S_{t})\nu(\mathrm{d}x)\right)^{2}\mathrm{d}t\right]$
$\displaystyle\leq\frac{C^{2}}{(1-\rho
L)^{2}}\mathbb{E}\left[\int_{0}^{T}\int_{\mathbb{R}}e^{-2rt}|S_{t}|^{2}(e^{x}-1)^{2}\nu(\mathrm{d}x)\mathrm{d}t\right]<\infty,$

since $\sup_{t\in\left[0,T\right]}\mathbb{E}\left[S_{t}^{2}\right]<\infty$.
Here, $C_{0}=\int_{\mathbb{R}}(e^{x}-1)^{2}\nu(\mathrm{d}x)<\infty$ because of
the assumptions made on the measure $\nu$. Next, we show that
$\int_{0}^{T}e^{-rt}S_{t}v(t,S_{t})\frac{\partial V}{\partial
S}(t,S_{t})\mathrm{d}W_{t}$ is a martingale. Since
$S\frac{\partial\phi}{\partial S}(t,S)$ is bounded, we have

$\displaystyle 0<v(t,S)=\frac{\sigma}{1-\rho S\frac{\partial\phi}{\partial
S}(t,S)}\leq\frac{\sigma}{1-\rho L}\equiv C_{1}<\infty.$

Therefore, $\mathbb{E}[\int_{0}^{T}e^{-2rt}(\frac{\partial V}{\partial
S}(t,S_{t})v(t,S_{t})S_{t})^{2}\mathrm{d}t]\leq
C^{2}C_{1}^{2}\int_{0}^{T}e^{-2rt}\mathbb{E}[S^{2}_{t}]\mathrm{d}t<\infty$
because $S_{t}$ is a martingale. Thus, $M_{t}$ is also a martingale, and
$a\equiv 0$. Hence, $V$ is a solution to PIDE (54), as claimed. ∎

######  Remark 3.

If $\rho=0$, then $H(t,x,S)=S(e^{x}-1)$. Moreover, equation (54) becomes

$\frac{\partial V}{\partial
t}+\frac{\sigma^{2}}{2}S^{2}\frac{\partial^{2}V}{\partial
S^{2}}+rS\frac{\partial V}{\partial
S}-rV+\int_{\mathbb{R}}V(t,Se^{x})-V(t,S)-S(e^{x}-1)\frac{\partial V}{\partial
S}(t,S)\nu(\mathrm{d}x)=0,$ (57)

which is the well-known classical PIDE. If there are no jumps ($\nu=0$) and a
trader follows the delta hedging strategy, i.e.,
$\phi(t,S)=\partial_{S}V(t,S)$, then equation (54) reduces to the Frey–Stremme
option pricing model:

$\frac{\partial V}{\partial t}+\frac{1}{2}\frac{\sigma^{2}}{\left(1-\varrho
S\partial^{2}_{S}V\right)^{2}}S^{2}\frac{\partial^{2}V}{\partial
S^{2}}+rS\frac{\partial V}{\partial S}-rV=0$ (58)

(c.f. [27]). Finally, if $\rho=0$ and $\nu=0$, equation (54) reduces to the
classical linear Black–Scholes equation.

For simplicity, we assume the interest rate is zero, i.e., $r=0$. Then, the
function $V(t,S)$ is a solution to the following PIDE:

$\displaystyle\frac{\partial V}{\partial t}$ $\displaystyle+$
$\displaystyle\frac{1}{2}v(t,S)^{2}S^{2}\frac{\partial^{2}V}{\partial S^{2}}$
(59) $\displaystyle+$
$\displaystyle\int_{\mathbb{R}}V(t,S+H(t,x,S))-V(t,S)-H(t,x,S)\frac{\partial
V}{\partial S}(t,S)\nu(\mathrm{d}x)=0.$

Next, let the tracking error of a trading strategy $\alpha_{t}=\phi(t,S_{t})$
be
$e_{T}^{M}:=\Phi(S_{T})-V_{0}=V(T,S_{T})-V_{0}-\int_{0}^{T}\alpha_{t}\mathrm{d}S_{t}$.
By applying Itô’s formula to $V(t,S_{t})$ and using $\eqref{equationsimple}$,
we obtain

$\displaystyle
V(T,S_{T})-V_{0}=V(T,S_{T})-V(0,S_{0})=\int_{0}^{T}\mathrm{d}V(t,S_{t})$
$\displaystyle=\int_{0}^{T}\frac{\partial V}{\partial
S}\mathrm{d}S_{t}+\int_{0}^{T}\frac{\partial V}{\partial
t}+\frac{1}{2}v(t,S_{t})^{2}S_{t}^{2}\frac{\partial^{2}V}{\partial^{2}S}\mathrm{d}t$
$\displaystyle\qquad+\int_{0}^{T}\int_{\mathbb{R}}V(t,S_{t}+H(t,x,S_{t}))-V(t,S_{t})-H(t,x,S_{t})\frac{\partial
V}{\partial S}J_{X}(\mathrm{d}t,\mathrm{d}x)$
$\displaystyle=\int_{0}^{T}\frac{\partial V}{\partial
S}\mathrm{d}S_{t}-\int_{0}^{T}\int_{\mathbb{R}}V(t,S_{t}+H(t,x,S_{t}))-V(t,S_{t})-H(t,x,S_{t})\frac{\partial
V}{\partial S}\nu(\mathrm{d}x)\mathrm{d}t$
$\displaystyle\qquad+\int_{0}^{T}\int_{\mathbb{R}}V(t,S_{t}+H(t,x,S_{t}))-V(t,S_{t})-H(t,x,S_{t})\frac{\partial
V}{\partial S}J_{X}(\mathrm{d}t,\mathrm{d}x)$
$\displaystyle=\int_{0}^{T}\frac{\partial V}{\partial
S}\mathrm{d}S_{t}+\int_{0}^{T}\int_{\mathbb{R}}V(t,S_{t}+H(t,x,S_{t}))-V(t,S_{t})-H(t,x,S_{t})\frac{\partial
V}{\partial S}\tilde{J}_{X}(\mathrm{d}t,\mathrm{d}x).$

Using expression (55) for the dynamics of the asset price $S_{t}$ (with
$r=0$), the tracking error $e_{T}^{M}$ can be expressed as follows:

$\displaystyle e_{T}^{M}$ $\displaystyle=$ $\displaystyle
V(T,S_{T})-V_{0}-\int_{0}^{T}\alpha_{t}\mathrm{d}S_{t}=\int_{0}^{T}\left(\frac{\partial
V}{\partial S}(t,S_{t})-\alpha_{t}\right)\mathrm{d}S_{t}$
$\displaystyle+\int_{0}^{T}\int_{\mathbb{R}}V(t,S_{t}+H(t,x,S_{t}))-V(t,S_{t})-H(t,x,S_{t})\frac{\partial
V}{\partial S}\tilde{J}_{X}(\mathrm{d}t,\mathrm{d}x)$ $\displaystyle=$
$\displaystyle\int_{0}^{T}v(t,S_{t})S_{t}\left(\frac{\partial V}{\partial
S}-\alpha_{t}\right)\mathrm{d}W_{t}$
$\displaystyle+\int_{0}^{T}\int_{\mathbb{R}}V(t,S_{t}+H(t,x,S_{t}))-V(t,S_{t})-\alpha_{t}H(t,x,S_{t})\tilde{J}_{X}(\mathrm{d}t,\mathrm{d}x).$

######  Remark 4.

For the delta hedging strategy $\alpha_{t}=\phi(t,S_{t})=\frac{\partial
V}{\partial S}(t,S_{t})$, the tracking error function $e_{T}^{M}$ can be
expressed as follows:

$e_{T}^{M}=\int_{0}^{T}\int_{\mathbb{R}}V(t,S_{t}+H(t,x,S_{t}))-V(t,S_{t})-H(t,x,S_{t})\frac{\partial
V}{\partial S}(t,S_{t})\tilde{J}_{X}(\mathrm{d}t,\mathrm{d}x).$

It is obvious that the tracking error for the delta hedging strategy need not
be zero for $\nu\not\equiv 0$.

The next proposition presents a criterion that can be used to find the optimal
hedging strategy. The proposition and its proof is based on the recent paper
by Cruz and Ševčovič [20].

######  Proposition 7.

[20, Proposition 3.5] The trading strategy $\alpha_{t}=\phi(t,S_{t})$ of a
large trader minimizing the variance
$\mathbb{E}\left[(\epsilon_{T}^{M})^{2}\right]$ of the tracking error is given
by the implicit equation:

$\displaystyle\phi(t,S_{t})$ $\displaystyle=$
$\displaystyle\beta^{\rho}(t,S_{t})\bigl{[}v(t,S_{t})^{2}S_{t}^{2}\frac{\partial
V}{\partial S}(t,S_{t})$ (61)
$\displaystyle+\int_{\mathbb{R}}\left(V(t,S_{t}+H(t,x,S_{t}))-V(t,S_{t}))H(t,x,S_{t}\right)\nu(\mathrm{d}x)\bigr{]},$

where
$\beta^{\rho}(t,S_{t})=1/[v(t,S_{t})^{2}S_{t}^{2}+\int_{\mathbb{R}}H(t,x,S_{t})^{2}\nu(\mathrm{d}x)]$
and $H(t,x,S)=S(e^{x}-1)+\rho S[\phi(t,S+H(t,x,S))-\phi(t,S)]$.

###### Proof.

Using equation (6.) for the tracking error $\epsilon_{T}^{M}$ and Itô’s
isometry, we have

$\displaystyle\mathbb{E}\left[(\epsilon_{T}^{M})^{2}\right]$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[\int_{0}^{T}v(t,S_{t})^{2}S_{t}^{2}\left(\frac{\partial
V}{\partial S}(t,S_{t})-\alpha_{t}\right)^{2}\mathrm{d}t\right]$
$\displaystyle+\mathbb{E}\left[\int_{0}^{T}\int_{\mathbb{R}}\left(V(t,S_{t}+H(t,x,S_{t}))-V(t,S_{t})-\alpha_{t}H(t,x,S_{t})\right)^{2}\nu(\mathrm{d}x)\mathrm{d}t\right].$

The minimizer $\alpha_{t}$ of the above convex quadratic minimization problem
satisfies the first-order necessary conditions
$\mathrm{d}(\mathbb{E}\left[\epsilon_{T}^{2}\right],\alpha_{t})=0$, that is,

$\displaystyle 0$ $\displaystyle=$
$\displaystyle-2\mathbb{E}\left[\int_{0}^{T}\left(v(t,S_{t})^{2}S_{t}^{2}\left(\frac{\partial
V}{\partial S}(t,S_{t})-\alpha_{t}\right)\right.\right.$
$\displaystyle\left.\left.+\int_{\mathbb{R}}H(t,x,S_{t})\bigl{(}V(t,S_{t}+H(t,x,S_{t}))-V(t,S_{t})-\alpha_{t}H(t,x,S_{t})\bigr{)}\nu(\mathrm{d}x)\right)\omega_{t}\mathrm{d}t\right]$

for any variation $\omega_{t}$. Thus, the tracking error minimizing strategy
$\alpha_{t}$ is given by (61). ∎

######  Remark 5.

It is worth noting that the optimal trading strategy minimizing the variance
of the tracking error need not satisfy the structural Assumption 1. For
instance, if $\nu=0$, then the tracking error minimizer is the delta hedging
strategy $\phi=\partial_{S}V$. For a call or put option, its gamma, i.e.,
$\partial^{2}_{S}V(t,S)$, becomes infinite as $t\to T$ and $S=K$. However,
given a level $L>0$, we can minimize the tracking error
$\mathbb{E}\left[\epsilon_{T}^{2}\right]$ under the additional constraint
$\sup_{S>0}|S\frac{\partial\phi}{\partial S}(t,S)|\leq L$. In other words, we
can solve the following convex constrained nonlinear optimization problem

$\min_{\phi}\ \ \mathbb{E}\left[\epsilon_{T}^{2}\right]\ \ s.t.\ \
|S\partial_{S}\phi|\leq L$

instead of the unconstrained minimization problem proposed in Proposition 7.

######  Remark 6.

Note that if $\nu=0$ and $\rho\geq 0$, the trading strategy $\alpha_{t}$
reduces to the Black–Scholes delta hedging strategy, i.e.,
$\alpha_{t}=\frac{\partial V}{\partial S}(t,S_{t})$. Meanwhile, if
$\nu\not\equiv 0$ and $\rho=0$, then the optimal trading strategy becomes
$\alpha_{t}=\phi^{0}(t,S_{t})$, where

$\phi^{0}(t,S_{t})=\beta^{0}(t,S_{t})\left(\sigma^{2}S_{t}^{2}\frac{\partial
V}{\partial
S}(t,S_{t})+\int_{\mathbb{R}}S_{t}(e^{x}-1)\left(V(t,S_{t}e^{x})-V(t,S_{t})\right)\nu(\mathrm{d}x)\right),$

where
$\beta^{0}(t,S_{t})=1/[\sigma^{2}S_{t}^{2}+\int_{\mathbb{R}}S^{2}_{t}(e^{x}-1)^{2}\nu(\mathrm{d}x)]$.

We conclude this section with the following proposition providing the first-
order approximation of the tracking error minimizing trading strategy for
small values of the parameter $\rho\ll 1$. In what follows, we derive the
first-order approximation of $\phi^{\rho}(t,S_{t})$ as
$\phi^{\rho}(t,S_{t})=\phi^{0}(t,S_{t})+\rho\phi^{1}(t,S_{t})+O(\rho^{2})$ as
$\rho\to 0$. Clearly, the first-order Taylor expansion of the volatility
function $v(t,S)$ is given by

$v(t,S)^{2}=\frac{\sigma^{2}}{(1-\rho
S\partial_{S}\phi)^{2}}=\sigma^{2}+2\rho\sigma^{2}S\frac{\partial\phi^{0}}{\partial
S}(t,S)+O(\rho^{2}),\ \ \text{as}\ \rho\to 0.$

According to Proposition 5 (see (52)), we have $H(t,x,S)=H^{0}(t,x,S)+\rho
H^{1}(t,x,S)+O(\rho^{2})$, where

$H^{0}(t,x,S)=S(e^{x}-1),\qquad
H^{1}(t,x,S)=S[\phi^{0}(t,Se^{x})-\phi^{0}(t,S)].$ (62)

The function $\beta^{\rho}$ can be expanded as follows:
$\beta^{\rho}(t,S)=\beta^{0}(t,S)+\rho\beta^{(1)}(t,S)+O(\rho^{2})$,

$\displaystyle\beta^{0}(t,S)$ $\displaystyle=$ $\displaystyle
1/[\sigma^{2}S^{2}+S^{2}\int_{\mathbb{R}}(e^{x}-1)^{2}\nu(\mathrm{d}x)],\quad$
(63) $\displaystyle\beta^{(1)}(t,S)$ $\displaystyle=$
$\displaystyle-(\beta^{0}(t,S))^{2}\left[2\sigma^{2}S^{3}\frac{\partial\phi^{0}}{\partial
S}(t,S)+2S^{2}\int_{\mathbb{R}}(e^{x}-1)[\phi^{0}(t,Se^{x})-\phi^{0}(t,S)]\nu(\mathrm{d}x)\right].$

Using the first-order expansions of the functions $v^{2},\beta^{\rho}$, and
$H$, we obtain the following results.

######  Proposition 8.

[20, Proposition 3.6] For small values of the parameter $\rho\ll 1$, the
tracking error variance minimizing strategy $\alpha_{t}=\phi^{\rho}(t,S_{t})$
is given by

$\displaystyle\phi^{\rho}(t,S_{t})=\phi^{0}(t,S_{t})+\rho\phi^{(1)}(t,S_{t})+O(\rho^{2}),\
\ \text{as}\ \rho\to 0,$ (64)

where

$\displaystyle\phi^{(1)}(t,S)$ $\displaystyle=$
$\displaystyle\beta^{0}(t,S)\left[2\sigma^{2}S^{3}\frac{\partial V}{\partial
S}(t,S)\frac{\partial\phi^{0}}{\partial S}(t,S)\right.$
$\displaystyle+\left.\int_{\mathbb{R}}\left(V(t,Se^{x})-V(t,S)+\frac{\partial
V}{\partial
S}(t,Se^{x})H^{0}(t,x,S)\right)H^{1}(t,x,S)\nu(\mathrm{d}x)\right]$
$\displaystyle+\beta^{(1)}(t,S)\left[\sigma^{2}S^{2}\frac{\partial V}{\partial
S}(t,S)+\int_{\mathbb{R}}\left(V(t,Se^{x})-V(t,S)\right)H^{0}(t,x,S)\nu(\mathrm{d}x)\right],$

and the functions $H^{0},H^{1},\beta^{0}$, and $\beta^{(1)}$ are defined as in
(62) and (63).

###  6.1. Numerical simulation for the underlying PIDE

In this section, we illustrate the behavior of the solutions to the linear
PIDE with various Lévy measures. Specifically, we consider European put
options, i.e., $\Phi(S)=(K-S)^{+}$. The goal of the numerical simulation is to
compare the solution to the linear Black–Scholes equation with solutions to
the Merton and variance gamma PIDE models. The common model parameters were
chosen as follows: $\sigma=0.23,K=100,T=1$, and $r\in\\{0,0.1\\}$. For the
underlying Lévy process, we consider the variance gamma process with
parameters $\theta=-0.43,\kappa=0.27$ and the Merton processes with parameters
$\lambda=0.1$, $m=-0.2$, and $\delta=0.15$. First, we employ the finite
difference discretization scheme proposed and analyzed by Cruz and Ševčovič
[20] to calculate the numerical solution to the equation. Their scheme is
based on a uniform spatial finite difference discretization with a spatial
step $\Delta x=0.01$, and implicit time discretization with a step $\Delta
t=0.005$. Then, we set the total number of spatial discretization steps as
$N=400$ and the number of time discretization steps as $M=200$. We restricted
the spatial computational domain to $x\in[-L,L]$, where $L=4$. For more
details about the discretization scheme, see the recent paper by Cruz and
Ševčovič [20].

Figure 6.1. shows the comparison of European put option prices between PIDE
and linear Black–Scholes models. Figures 6.1. (a) and (b) show the plots of
put option prices $V(0,S)$ for $S\in[80,125]$ for the interest rates $r=0$ and
$r=0.1$, respectively. Table 6.1. summarizes the numerical values of option
prices for two different values of the interest rate $r=0.1$ and $r=0$. The
option price for the Merton and variance gamma models are higher than that of
the classical Black-Scholes model. This is based on the idea prices of put or
call options should be higher on underlying assets following stochastic
processes with jumps than those following a continuous geometric Brownian
motion.

a) b)

Figure 1.:  Comparison of European put option prices for the Black–Scholes
(BS) model and PIDE models under variance gamma (VG) and Merton (Mer) PIDE
processes. Source: own computations based on the paper [20] by J. Cruz and D.
Ševčovič.

Table 1.: European put option prices $V(0,S)$ for the Black-Scholes and PIDE
models under Variance Gamma and Merton processes for $r=0$ and $r=0.1$.
Source: J. Cruz and D. Ševčovič, [20, 18].

| BS | PIDE-VG | PIDE-Merton | Payoff
---|---|---|---|---
$S$ | $r=0$ | $r=0.1$ | $r=0$ | $r=0.1$ | $r=0$ | $r=0.1$ |
85.2144 | 15.2547 | 7.35166 | 19.2687 | 14.9855 | 17.1692 | 12.9056 | 14.7856
88.692 | 12.2484 | 5.24145 | 17.2948 | 13.3899 | 14.8335 | 10.9901 | 11.308
92.3116 | 9.42895 | 3.51944 | 15.428 | 11.8822 | 12.6423 | 9.21922 | 7.68837
96.0789 | 6.90902 | 2.21106 | 13.674 | 10.4691 | 10.6201 | 7.61307 | 3.92106
100. | 4.78444 | 1.29196 | 12.0372 | 9.15576 | 8.78655 | 6.18483 | 0.
104.081 | 3.1099 | 0.69843 | 10.52 | 7.94499 | 7.155 | 4.94044 | 0.
108.329 | 1.88555 | 0.34773 | 9.12343 | 6.83762 | 5.73137 | 3.87864 | 0.
112.75 | 1.0604 | 0.15881 | 7.84623 | 4.51403 | 5.83246 | 2.99166 | 0.

##  7. Hamilton-Jacobi-Bellman equation

In this section, we present the motivation for studying the fully nonlinear
parabolic equation (24), which can be viewed as a parabolic PIDE in some
sense. We also present the relationship between the nonlinear parabolic
equation and the transformed equation using the so-called Ricatti transform.

The motivation for studying the nonlinear parabolic equation of the form (24)
arises from dynamic stochastic programming for $d=1$. The fully nonlinear HJB
equation describing optimal portfolio selection strategy is represented by the
following nonlinear parabolic equation:

$\displaystyle\partial_{t}V+\max_{{\bm{\theta}}\in\triangle}\left(\mu(x,t,{\bm{\theta}})\,\partial_{x}V+\frac{1}{2}\sigma(x,t,{\bm{\theta}})^{2}\,\partial_{x}^{2}V\right)=0\,,$
(65) $\displaystyle V(x,T)=u(x),$ (66)

where $x\in\mathbb{R},t\in[0,T)$. A solution $V=V(x,t)$ to the parabolic
equation (65) is subject to the terminal condition $V(x,T)=u(x)$. According to
Kilianová and Ševčovič [35, 34, 36], such HJB equation of the form (65) arises
from the dynamic stochastic programming, where the goal of an investor is to
maximize the conditional expected value of the terminal utility of the
portfolio:

$\max_{{\bm{\theta}}|_{[0,T)}}\mathbb{E}\left[u(x_{T}^{\bm{\theta}})\,\big{|}\,x_{0}^{\bm{\theta}}=x_{0}\right],$
(67)

on a finite time horizon $[0,T]$. Here, $u:\mathbb{R}\to\mathbb{R}$ is an
increasing terminal utility function, and $x_{0}$ is a given initial state
condition of the process $\\{x_{t}^{\bm{\theta}}\\}$ at $t=0$. The underlying
stochastic process $\\{x_{t}^{{\bm{\theta}}}\\}$ with a drift
$\mu(x,t,{\bm{\theta}})$ and volatility $\sigma(x,t,{\bm{\theta}})$ is assumed
to satisfy the following Itô’s SDE:

$dx_{t}^{\bm{\theta}}=\mu(x_{t}^{\bm{\theta}},t,{\bm{\theta}}_{t})dt+\sigma(x_{t}^{\bm{\theta}},t,{\bm{\theta}}_{t})dW_{t}\,,$
(68)

where the control process $\\{{\bm{\theta}}_{t}\\}$ is adapted to the process
$\\{x_{t}\\}$, and $\\{W_{t}\\}$ is the standard one-dimensional Wiener
process. The control parameter ${\bm{\theta}}$ belongs to a given compact
subset $\triangle$ in $\mathbb{R}^{n}$. An example of such a subset is the
compact convex simplex
$\triangle\equiv\mathcal{S}^{n}=\\{{\bm{\theta}}\in\mathbb{R}^{n}\ |\
{\bm{\theta}}\geq\mathbf{0},\mathbf{1}^{T}{\bm{\theta}}=1\\}\subset\mathbb{R}^{n}$,
where $\mathbf{1}=(1,\cdots,1)^{T}\in\mathbb{R}^{n}$.

Consider the value function

$V(x,t):=\sup_{{\bm{\theta}}|_{[t,T)}}\mathbb{E}\left[u(x_{T}^{\bm{\theta}})|x_{t}^{\bm{\theta}}=x\right].$
(69)

Then, according to Bertsekas [11], the value function $V=V(x,t)$ solves the
fully nonlinear HJB parabolic equation (65) and $V(x,T):=u(x)$.

Several attempts have been made for solving the HJB equation (65). Macová and
Ševčovič [43] analyzed solutions to a fully nonlinear parabolic equation
modeling the problem of optimal portfolio construction. Consequently, they
formulated the problem of optimal stock to bond proportion in the management
of a pension fund portfolio could be formulated in terms of the solutions to
the HJB equation. Federicol et al. [23] investigated the utility maximization
problem for an investment-consumption portfolio when the current utility
depends on the wealth process, regularity of solutions to the HJB equation.
They defined a dual problem and treated it by means of dynamic programming to
show that the viscosity solutions of the associated HJB equation belong to a
class of smooth function. Recently, Ishimura and Ševčovič [30] constructed and
analyzed solutions to the class of HJB equation (65) with range bounds on the
optimal response variable. They constructed monotone traveling wave solutions
and identified parametric regions for which the traveling wave solutions have
positive or negative wave speed. Abe and Ishimura [1] employed the Riccati
transformation method for solving the full nonlinear HJB equations, which was
later studied and generalized by Kilianová and Ševčovič [35]. They
investigated solutions of a fully nonlinear HJB equation for a constrained
dynamic stochastic optimal allocation problem. However, no attempt has been
made in solving the fully nonlinear HJB parabolic equation arising from
portfolio optimization in high-dimensional space using the monotone operator
technique. The monotone operator method is essential because it does not only
give constructive proof for existence theorems, but it also leads to various
comparison results, which are effective tools for studying qualitative
properties of solutions. In this chapter, we consider the case when the
utility function $u$ is increasing, as a consequence,
$\partial_{x}V(x,\tau)>0$.

###  7.1. Static Markowitz model for portfolio optimization

This subsection presents the motivation for studying such nonlinear parabolic
equation (65). It describes the mathematical formulation of the classical
Markowitz model for portfolio optimization. In the static portfolio
optimization, this model aims to maximize the mean return of the set of
stochastic returns $X^{i},i=1,\ldots,n$, under the constraint that variance of
the portfolio is bounded by a given constant $\sigma_{0}^{2}/2$. Given a
vector $\bm{\theta}=(\theta^{1},\ldots,\theta^{n})^{T}$ of weights, we
construct a portfolio $X=\sum_{i=1}^{n}\theta^{i}X^{i}$. Let
$\bm{\mu}\in\mathbb{R}^{n},\mu^{i}=\mathbb{E}(X^{i})$ be the vector of mean
returns of stochastic asset returns and $\bm{\Sigma}$ be their covariance
matrix, $\bm{\Sigma}_{ij}=cov(X^{i}X^{j})$, then
$\mathbb{E}(X)=\bm{\mu}^{T}\bm{\theta}$, and the variance
$\mathbb{D}(X)=\bm{\theta}^{T}\bm{\Sigma}\bm{\theta}$. The Markowitz optimal
portfolio optimization problem can be formulated as the following convex
optimization problem:

$\displaystyle\max_{\bm{\theta}\in\mathbb{R}^{n}}\bm{\mu}^{T}\bm{\theta}\qquad\hbox{--
maximize the mean return},$ $\displaystyle s.t.\ \
\frac{1}{2}\bm{\theta}^{T}\bm{\Sigma}\bm{\theta}\leq\frac{1}{2}\sigma_{0}^{2}\qquad\hbox{--
the variance is prescribed},$ $\displaystyle\ \ \ \ \ \
\sum_{i=1}^{n}\theta^{i}=1\qquad\hbox{-- weights sum up to 100\%},$
$\displaystyle\ \ \ \ \ \ \bm{\theta}\geq 0\qquad\hbox{-- no short positions
allowed}.$

The corresponding Lagrange function for the minimization of
$-\bm{\mu}^{T}\bm{\theta}$ has the following form:

${\mathcal{L}}(\bm{\theta},\varphi,\lambda,\xi)=-\bm{\mu}^{T}\bm{\theta}+{\varphi\frac{1}{2}\bm{\theta}^{T}\bm{\Sigma}\bm{\theta}}+\lambda
1^{T}\bm{\theta}+\xi^{T}\bm{\theta},$ (70)

where $\varphi\in\mathbb{R},\lambda\in\mathbb{R},\xi\in\mathbb{R}^{n}$, and
$\xi\geq 0$ are Lagrange multipliers. It is worth noting that the same
Lagrange function corresponds to the minimization problem:

$\displaystyle\alpha(\varphi):=\min_{\bm{\theta}\in\triangle}-\bm{\mu}^{T}\bm{\theta}+{\varphi\frac{1}{2}\bm{\theta}^{T}\bm{\Sigma}\bm{\theta}}$
$\displaystyle\ \ \text{where}\ \
\triangle\equiv\\{\bm{\theta}\in\mathbb{R}^{n},\ \sum_{i=1}^{n}\theta^{i}=1,\
\ \bm{\theta}\geq 0\\},$

provided the Lagrange multiplier $\varphi>0$ is known and fixed. Figure 7.1.
shows the optimal asset selection for German DAX30 stock index for various
values of $\varphi>0$. The optimal value is denoted by $\alpha(\varphi)$. The
value of the Lagrange multiplier $\varphi$ can be viewed as a measure of
investor’s risk-aversion (see Fig. 7.1.). Therefore, the higher the value of
the risk aversion, the more portfolio is diversified among less risky assets
with smaller mean returns.

$\varphi\approx 1$ $\varphi\approx 4$ $\varphi\approx 6$ $\varphi\approx 8$

Figure 2.: Optimal asset selection for German DAX30 stock index for various
values of $\varphi>0$.

###  7.2. Riccati transformation of the HJB equation and application to
optimal portfolio selection problem

####  7.2.1. Riccati transformation

This section presents how the HJB equation (65) can be transformed into a
quasilinear PDE using the so-called Ricatti transformation techniques. Such
transformed parabolic equation correspond to the Cauchy problem (24), which is
obtained after some pertubation in the main operator.

The Riccati transformation $\varphi$ of the value function $V$ can be defined
based on the approached introduced by Abe and Ishimura [1], Ishimura and
Ševčovič [30], Ševčovič and Macová [43], and Kilianová and Ševčovič [35] as
follows:

$\varphi(x,\tau)=-\frac{\partial_{x}^{2}V(x,t)}{\partial_{x}V(x,t)},\quad\hbox{where}\
\ \tau=T-t.$ (71)

Suppose the value function $V(x,t)$ is increasing in the $x$-variable. In
other words, assume that the terminal utility function $u(x)$ is increasing.
Then, the HJB equation (65) can be rewritten as follows:

$\partial_{t}V-\alpha(\cdot,\varphi)\partial_{x}V=0,\qquad
V(\cdot,T)=u(\cdot),$ (72)

where $\alpha(x,\tau,\varphi)$ is the value function of the following
parametric optimization problem:

$\alpha(x,\tau,\varphi)=\min_{{\bm{\theta}}\in\triangle}\left(-\mu(x,t,{\bm{\theta}})+\frac{\varphi}{2}\sigma(x,t,{\bm{\theta}})^{2}\right),\quad\tau=T-t\,.$
(73)

######  Remark 7.

It is worth noting that the optimization problem (73) is related to the
classical Markowitz model on optimal portfolio selection problem formulated as
maximization of the mean return
$\mu({\bm{\theta}})\equiv{\bm{\mu}}^{T}{\bm{\theta}}$ under the volatility
constraint
$\frac{1}{2}\sigma({\bm{\theta}})^{2}\equiv\frac{1}{2}{\bm{\theta}}^{T}{\bm{\Sigma}}{\bm{\theta}}\leq\frac{1}{2}\sigma^{2}_{0}$,
i.e.,

$\displaystyle\max_{{\bm{\theta}}\in\triangle}{\bm{\mu}}^{T}{\bm{\theta}},\quad
s.t.\ \
\frac{1}{2}{\bm{\theta}}^{T}{\bm{\Sigma}}{\bm{\theta}}\leq\frac{1}{2}\sigma^{2}_{0},$

where the decision set is the simplex
$\triangle=\\{{\bm{\theta}}\in\mathbb{R}^{n}\ |\
{\bm{\theta}}\geq\mathbf{0},\mathbf{1}^{T}{\bm{\theta}}=1\\}$. The Lagrange
multiplier for the volatility constraint can be viewed as the parameter
$\varphi$ entering the parametric optimization problem (73).

Next, let $\partial_{x}\alpha$ be the total differential of the function
$\alpha(x,\tau,\varphi)$, where $\varphi=\varphi(x,\tau)$, i.e.,

$\partial_{x}\alpha(x,\tau,\varphi)=\alpha^{\prime}_{x}(x,\tau,\varphi)+\alpha^{\prime}_{\varphi}(x,\tau,\varphi)\,\partial_{x}\varphi.$

Here, $\alpha^{\prime}_{x}$ and $\alpha^{\prime}_{\varphi}$ are partial
derivatives of $\alpha$ with respect to $x$\- and $\varphi$\- variables,
respectively.

Kilianová and Ševčovič [34, Theorem 4.2] recently established the relationship
between the transformed function $\varphi$ and the value function $V$. The
reported that an increasing value function $V(x,t)$ in the $x$-variable is a
solution to the HJB equation (65) if and only if the transformed function
$\varphi(x,\tau)=-\partial_{x}^{2}V(x,t)/\partial_{x}V(x,t),\,\ t=T-\tau$, is
a solution to the Cauchy problem for the quasilinear parabolic PDE:

$\displaystyle\partial_{\tau}\varphi-\partial^{2}_{x}\alpha(\cdot,\varphi)=-\partial_{x}\left(\alpha(\cdot,\varphi)\varphi\right),$
(74)
$\displaystyle\varphi(x,0)=\varphi_{0}(x)\equiv-u^{\prime\prime}(x)/u^{\prime}(x),\quad(x,\tau)\in\mathbb{R}\times(0,T).$
(75)

Note that the Cauchy problem for the quasilinear parabolic PDE (74) is
equivalent to the nonlinear parabolic equation (24) in one-dimensional space.
This is obtainable after some shift/perturbation in the main operator of the
transformed equation (74). This nonlinear parabolic equation (24) presented in
an abstract setting can be viewed as nonlinear PIDE in some sense.

####  7.2.2. Properties of the value function as a diffusion function

In this subsection, we presents the qualitative properties of the value
function and sufficient conditions on the decision set $\triangle$ and
functions $\mu$ and $\sigma$ that guarantee higher smoothness of the value
function $\alpha$. Let $C^{k,1}(\mathcal{D})$ be the space consisting of all
$k$-differentiable functions defined on the domain
$\mathcal{D}\subset\mathbb{R}^{d+2}$, whose $k$-th derivative is globally