# Sandwiched Volterra Volatility model:
Markovian approximations and hedging

Giulia Di Nunno1,2
<EMAIL_ADDRESS>Anton Yurchenko-Tytarenko1
<EMAIL_ADDRESS>
(1Department of Mathematics, University of Oslo
2Department of Business and Management Science, NHH Norwegian School of
Economics, Bergen
)

###### Abstract

We consider stochastic volatility dynamics driven by a general Hölder
continuous Volterra-type noise and with unbounded drift. For such models, we
consider the explicit computation of quadratic hedging strategies. While the
theoretical solution is well-known in terms of the non-anticipating derivative
for all square integrable claims, the fact that these models are typically
non-Markovian provides a concrete difficulty in the direct computation of
conditional expectations at the core of the explicit hedging strategy. To
overcome this difficulty, we propose a Markovian approximation of the model
which stems from an adequate approximation of the kernel in the Volterra
noise. We study the approximation of the volatility, the prices as well as the
optimal mean-square hedge and provide the corresponding error estimates. We
complete the work with numerical simulations performed with different methods.

Keywords: stochastic volatility, sandwiched process, Hölder continuous noise,
hedging, Monte Carlo methods
MSC 2020: 91G30; 60H10; 60H35; 60G22
Acknowledgements. The present research is carried out within the frame and
support of the ToppForsk project nr. 274410 of the Research Council of Norway
with title STORM: Stochastics for Time-Space Risk Models.

##  1 Introduction

A significant number of modern financial market models is characterized by the
absence of the Markov property. Such a choice of modeling framework is
substantiated by a number of empirical studies indicating presence of long
memory in the market volatility. For instance, [15, 16] described the long-
range dependence and volatility clustering effects which led to the
development of models based on fractional Brownian motion with Hurst index
$H>1/2$ such as [12, 13, 14]. Other empirical studies (see e.g. [28]) conclude
that Hölder regularity of volatility is strictly less than $1/2$ and this
statement gave birth to another class of non-Markovian “rough volatility”
models such as rough Bergomi [8], SABR [25], Stein & Stein [29] or Heston [22]
models. A substantial advantage of such models in comparison to the Markovian
ones is the ability to grasp the implied volatility surface far more
accurately and reproduce e.g. the power law of at-the-money volatility skew
(see [4, 23, 24] or [40, Theorem 1]).

Another example of a non-Markovian market model – which the underlying model
in this paper – is the Sandwiched Volterra Volatility (SVV) model introduced
in [19]. Namely, we consider the financial market composed of the two primary
assets:

  * (i)

a risk-free bond (or bank account) with the price $e^{\int_{0}^{t}\nu(s)ds}$,
$t\in[0,T]$, where $\nu\in C([0,T])$ is deterministic;

  * (ii)

a risky asset $S=\\{S(t),~{}t\in[0,T]\\}$, the risk-free dynamics of which is
given by

$\displaystyle S(t)$
$\displaystyle=S(0)+\int_{0}^{t}\nu(s)S(s)ds+\int_{0}^{t}Y(s)S(s)\left(\rho
dB_{1}(s)+\sqrt{1-\rho^{2}}dB_{2}(s)\right),$ (1.1) $\displaystyle Y(t)$
$\displaystyle=Y(0)+\int_{0}^{t}b(s,Y(s))ds+Z(t),$ (1.2)

where $S(0)$ and $Y(0)$ are given constants,
$(B_{1},B_{2})=\\{(B_{1}(t),B_{2}(t)),~{}t\in[0,T]\\}$ is a standard
2-dimensional Brownian motion, $\rho\in(-1,1)$,
$Z(t)=\int_{0}^{t}\mathcal{K}(t,s)dB_{1}(s)$ is a Hölder continuous Gaussian
Volterra process. The discounted price of $S$ will be denoted by

$\displaystyle X(t):=e^{-\int_{0}^{t}\nu(s)ds}S(t),\quad t\in[0,T].$ (1.3)

The drift $b=b(t,y)$ in (1.2) is unbounded and has an explosive growth to
$+\infty$ whenever $y\to\varphi(t)+$ and explosive decay to $-\infty$ whenever
$y\to\psi(t)-$, where $0<\varphi<\psi$ are two deterministic Hölder continuous
functions. This structure ensures that the volatility process
$Y=\\{Y(t),~{}t\in[0,T]\\}$ is sandwiched between $\varphi$ and $\psi$, i.e.

$0<\varphi(t)<Y(t)<\psi(t),\quad t\in[0,T].$ (1.4)

The property (1.4) guarantees that the volatility $Y$ stays strictly positive
and that the price $S$ has moments of all orders. Note that the latter feature
turns out to be crucial from the financial perspective: as noted in [5,
Section 8], “several actively traded fixed-income derivatives require at least
$L^{2}$ solutions to avoid infinite model prices”. And still, it is often
absent even in classical stochastic volatility models such as the standard
Heston model [5, Proposition 3.1]. Another advantage of (1.1)–(1.3) is the
flexibility in the driving noise. In this work, $Z$ can be an arbitrary Hölder
continuous Gaussian Volterra process which allows to reproduce multiple
empirical phenomena: for example, taking

$\mathcal{K}(t,s):=\frac{1}{\Gamma\left(H+\frac{1}{2}\right)}(t-s)^{H-\frac{1}{2}}\mathbbm{1}_{0\leq
s\leq t\leq T}$

gives long memory if $H>\frac{1}{2}$ and roughness if $H<\frac{1}{2}$ covering
both modeling paradigms mentioned above. Furthermore, the setting of Gaussian
Volterra processes is general enough to combine them as advocated in e.g. [3,
Section 7.7] or [26].

While absence of Markovianity in stochastic volatility models is well
supported by empirical evidence, we have to highlight that it also brings
substantial difficulties from the point of view of stochastic methods,
especially in optimization problems. As an example, consider a mean-square
hedging problem of the form

$\inf_{u}J(u):=\inf_{u}\mathbb{E}\left[\left(F-\int_{0}^{T}u(s)dX(s)\right)^{2}\right],$
(1.5)

where the financial claim $F=F(X(T))\in L^{2}(\mathbb{P})$ and the infimum is
taken over all $\mathbb{F}$-adapted $X$-integrable strategies with
$\mathbb{F}=\\{\mathcal{F}_{t},~{}t\in[0,T]\\}$ being the filtration generated
by $(B_{1},B_{2})$. From the theoretical perspective, the problem (1.5) is
well understood: for martingale discounted prices $X$, the existence of the
solution to (1.5) is guaranteed by the celebrated Galtchouk-Kunita-Watanabe
decomposition theorem (see e.g. [36, 37]). Furthermore, if $X$ is additionally
square integrable, [17] gives the explicit representation of the optimal
hedging strategy: according to [17, Theorem 2.1], the solution to (1.5) can be
written as the non-anticipating derivative $\mathfrak{D}F$ defined as the
$L^{2}$-limit of simple processes

$\begin{gathered}\mathfrak{D}F:=L^{2}\text{-}\lim_{|\pi|\to 0}u_{\pi},\\\
u_{\pi}:=\sum_{k=0}^{n-1}\frac{\mathbb{E}\left[F\cdot(X(t_{k+1})-X(t_{k}))~{}|~{}\mathcal{F}_{t_{k}}\right]}{\mathbb{E}\left[(X(t_{k+1})-X(t_{k}))^{2}~{}|~{}\mathcal{F}_{t_{k}}\right]}\mathbbm{1}_{(t_{k},t_{k+1}]},\end{gathered}$
(1.6)

where $|\pi|:=\max_{k}(t_{k}-t_{k-1})$ denotes the mesh of the partition
$\pi=\\{0=t_{0}<t_{1}<...<t_{n}=T\\}$. Formula (1.6) is explicit in the sense
that the hedge is written only in terms of the discounted price model, the
information flow of reference, and the claim $F$. This formula is in the
spirit of the Clarck-Haussmann-Ocone (CHO) formula (see e.g. [35, 21]) but we
stress that (1.6) has several important advantages. Namely, the CHO formula
exploits the Malliavin derivative which is tailored for specific noises (e.g.
Brownian motion) and claims falling in the domain of the Malliavin operator.
In turn, formula (1.6) is available for all square integrable claims and all
square integrable martingales as discounted prices. Nevertheless, the
practical use of (1.6) is still limited in many long memory and rough
volatility models including the SVV model (1.2)–(1.3) due to the absence of
the Markov property. Indeed, analytical expressions for $\mathfrak{D}F$ are
usually impossible to obtain whereas dependence of $X$ on the past
significantly complicates numerical computation of the conditional
expectations in (1.6) by e.g. Monte Carlo methods.

In this paper, we tackle this problem by constructing a Markovian
approximation to the SVV model (1.1)–(1.3) and then exploit it for the
numerical computation of conditional expectations in (1.6). By doing this, we
produce an algorithm for the computation of quadratic hedging strategies. The
core of our approximation method lies in the following observation: since non-
Markovianity in (1.1)–(1.3) comes from integrating the kernel
$\mathcal{K}(t,s)$, we produce an approximation of $\mathcal{K}$ with a
sequence of degenerate kernels $\\{\mathcal{K}_{m},~{}m\geq 1\\}$

$\mathcal{K}_{m}(t,s)=\sum_{i=1}^{m}e_{m,i}(t)f_{m,i}(s),\quad 0\leq s\leq
t\leq T,$

where $e_{m,i}$ and $f_{m,i}$ are $L^{2}$ functions, so that the
$(m+2)$-dimensional process

$(X_{m},Y_{m},U_{m,1},...,U_{m,m})$

given by

$\displaystyle X_{m}(t)$
$\displaystyle=X(0)+\int_{0}^{t}Y_{m}(s)X_{m}(s)\left(\rho
dB_{1}(s)+\sqrt{1-\rho^{2}}dB_{2}(s)\right),$ (1.7) $\displaystyle Y_{m}(t)$
$\displaystyle=Y(0)+\int_{0}^{t}b(s,Y_{m}(s))ds+\sum_{i=1}^{m}e_{m,i}(t)U_{m,i}(t),$
$\displaystyle U_{m,1}(t)$ $\displaystyle=\int_{0}^{t}f_{m,1}(s)dB_{1}(s),$
$\displaystyle~{}~{}\vdots$ $\displaystyle U_{m,m}(t)$
$\displaystyle=\int_{0}^{t}f_{m,m}(s)dB_{1}(s),$

is Markovian w.r.t. the filtration
$\mathbb{F}=\\{\mathcal{F}_{t},~{}t\in[0,T]\\}$. Note that a similar idea was
already utilized in [10] for the Riemann–Liouville fractional Brownian motion,
in [1] for the rough Heston, and in [9] for the fractionally integrated
standard Heston models, as well as in [2] in the context of optimal control in
more general stochastic Volterra systems.

However, there are two main challenges related to application of such an
approach in the setting (1.1)–(1.3). First of all, the drift $b$ in (1.2) is
unbounded which creates substantial technical difficulties in estimating the
error between the original model and its approximation. Second, in order to
ensure existence and uniqueness of $Y_{m}$ from (1.7), it is necessary to
demand that the Hölder regularity of the process
$Z_{m}(t):=\sum_{i=1}^{m}e_{m,i}(t)U_{m,i}(t)$ is the same as the one of the
original noise $Z$ or higher.

In this paper, we deal with the first challenge by using the pathwise bounds
for sandwiched processes developed in [20] whereas the second issue is
addressed by imposing more restrictive requirements on the approximating
sequence $\\{\mathcal{K}_{m},~{}m\geq 1\\}$. As examples, we provide explicit
sequences for the rough fractional and the general Hölder continuous kernels.
Once obtained the approximation $X_{m}$ of the discounted price $X$, we plug
it into (1.6) and prove the $L^{1}$–convergence of

$u_{\pi}^{m}:=\sum_{k=0}^{n-1}\frac{\mathbb{E}\left[F_{m}(X_{m}(t_{k+1})-X_{m}(t_{k}))~{}|~{}\mathcal{F}_{t_{k}}\right]}{\mathbb{E}\left[(X_{m}(t_{k+1})-X_{m}(t_{k}))^{2}~{}|~{}\mathcal{F}_{t_{k}}\right]}\mathbbm{1}_{(t_{k},t_{k+1}]},\quad
F_{m}:=F(X_{m}(T))$ (1.8)

to the optimal hedge $\mathfrak{D}F$ when $\mathcal{K}(t,s)=\mathcal{K}(t-s)$.

Note that the conditional expectations in (1.8) simplify as

$\displaystyle\mathbb{E}$
$\displaystyle\left[F(X_{m}(T))(X_{m}(t_{k+1})-X_{m}(t_{k}))~{}|~{}\mathcal{F}_{t_{k}}\right]$
$\displaystyle=\mathbb{E}\left[F(X_{m}(T))(X_{m}(t_{k+1})-X_{m}(t_{k}))~{}|~{}X_{m}(t_{k}),~{}Y_{m}(t_{k}),~{}U_{m,1}(t_{k}),...,U_{m,m}(t_{k})\right]$
$\displaystyle=:\Phi_{1}(t_{k},X_{m}(t_{k}),Y_{m}(t_{k}),U_{m,1}(t_{k}),...,U_{m,m}(t_{k})),$
$\displaystyle\mathbb{E}$
$\displaystyle\left[(X_{m}(t_{k+1})-X_{m}(t_{k}))^{2}~{}|~{}\mathcal{F}_{t_{k}}\right]$
$\displaystyle=\mathbb{E}\left[(X_{m}(t_{k+1})-X_{m}(t_{k}))^{2}~{}|~{}X_{m}(t_{k}),~{}Y_{m}(t_{k}),~{}U_{m,1}(t_{k}),...,U_{m,m}(t_{k})\right]$
$\displaystyle=:\Phi_{2}(t_{k},X_{m}(t_{k}),Y_{m}(t_{k}),U_{m,1}(t_{k}),...,U_{m,m}(t_{k})),$

with $\Phi_{1}$ and $\Phi_{2}$ being two measurable functions. Therefore, the
problem of searching for the optimal hedge boils down to computation of
$\Phi_{1}$ and $\Phi_{2}$. We use numerical schemes for the SVV model from
[18, 19] and provide two Monte Carlo algorithms to estimate these conditional
expectations: the Nested Monte Carlo and the Least Squares Monte Carlo.

The paper is organized as follows. Section 2 contains the detailed descriprion
of the SVV model (1.1)–(1.3) and gathers some necessary known results. Section
3 provides estimates of an error between the original model $(X,Y)$ and
$(X_{m},Y_{m})$ obtained by replacing the kernel in the volatility with a
general approximant $\mathcal{K}_{m}$. In Section 4, we describe the actual
construction of the Markovian approximation and illustrate the approach with
two examples of rough fractional and general Hölder continuous kernels.
Section 5 concentrates on the mean-variance hedging problem (1.5), the
description of two Monte Carlo approaches for computation of the non-
anticipating derivative as well as simulations.

##  2 Market modeling with the SVV and quadratic hedging

Hereafter we specify the model (1.1)–(1.3) on the complete probability space
$(\Omega,\mathcal{F},\mathbb{P})$ in more detail and then collect some
fundamental results for the upcoming discussion and recall the concept of non-
anticipating derivative within quadratic hedging. Before proceeding to the
main content, let us introduce some notation used in the sequel.

######  Notation.

  1. 1.

We denote by $\mathbb{F}=\\{\mathcal{F}_{t},~{}t\in[0,T]\\}$, $T<\infty$, the
$\mathbb{P}$-augmented filtration generated by the 2-dimensional Brownian
motion $(B_{1},B_{2})=\\{(B_{1}(t),B_{2}(t)),~{}t\in[0,T]\\}$ from
(1.1)–(1.3).

  2. 2.

In what follows, by $C$ we will denote any positive deterministic constant the
exact value of which is not important in the context. Note that $C$ may change
from line to line (or even within one line).

  3. 3.

Throughout the paper, $L^{2}(\mathbb{P})$ will denote a standard space of
square-integrable random variables. For a given square-integrable martingale
$\eta=\\{\eta(t),~{}t\in[0,T]\\}$, the Hilbert space of stochastic processes
such that

$\lVert
u\rVert_{L^{2}(\mathbb{P}\times[\eta])}:=\left(\mathbb{E}\left[\int_{0}^{T}u^{2}(s)d[\eta](s)\right]\right)^{\frac{1}{2}}<\infty$

will be denoted by $L^{2}(\mathbb{P}\times[\eta])$.

###  2.1 Stochastic volatility process

First of all, we describe the driving noise $Z$ in (1.2). Consider a function
$\mathcal{K}$: $[0,T]^{2}\to\mathbb{R}$ that satisfies the following
assumption.

######  Assumption (K).

The function $\mathcal{K}$: $[0,T]^{2}\to\mathbb{R}$ is a Volterra kernel,
i.e. $\mathcal{K}(t,s)=0$ whenever $t<s$, and the following conditions hold:

  * (K1)

$\mathcal{K}$ is square integrable, i.e.

$\int_{0}^{T}\int_{0}^{T}\mathcal{K}^{2}(t,s)dsdt=\int_{0}^{T}\int_{0}^{t}\mathcal{K}^{2}(t,s)dsdt<\infty,$

and, moreover,

$\int_{0}^{T}\mathcal{K}^{2}(t,s)ds<\infty\quad\forall t\in[0,T];$

  * (K2)

there exists a constant $H\in(0,1)$ such that, for any $\lambda\in(0,H)$,

$\int_{0}^{t}(\mathcal{K}(t,u)-\mathcal{K}(s,u))^{2}du\leq\ell_{\lambda}^{2}|t-s|^{2\lambda},\quad
0\leq s\leq t\leq T,$

where $\ell_{\lambda}>0$ is a constant, possibly dependent on $\lambda$.

By means of $\mathcal{K}$, we now define a Gaussian Volterra process
$Z=\\{Z(t),~{}t\in[0,T]\\}$ as

$Z(t):=\int_{0}^{t}\mathcal{K}(t,s)dB_{1}(s),\quad t\in[0,T],$ (2.1)

and immediately remark that

  * 1)

(K1) ensures that the stochastic integral in (2.1) is well-defined for all
$t\in[0,T]$,

  * 2)

(K2) guarantees (see e.g. [7, Theorem 1 and Corollary 4]) that $Z$ has a
modification that is Hölder continuous of any order $\lambda\in(0,H)$.
Moreover, for any $\lambda\in(0,H)$, the positive random variable
$\Lambda=\Lambda_{\lambda}$ such that

$|Z(t)-Z(s)|\leq\Lambda|t-s|^{\lambda},\quad t,s\in[0,T],$ (2.2)

can be chosen to have moments of all orders. In what follows, such Hölder
continuous modification as well as $\Lambda$ with all the moments will be
used.

In this paper, we mostly deal with difference kernels in the Volterra noise
$Z$. Before proceeding further, let us give two relevant examples.

######  Example 2.1.

(Hölder continuous kernels) Let
$\mathcal{K}(t,s)=\mathcal{K}(t-s)\mathbbm{1}_{0\leq s\leq t\leq T}$ for some
function $\mathcal{K}\in C([0,T])$. Assume also that $\mathcal{K}$ is
$H$-Hölder continuous for some $H\in(0,1)$. Then

$\displaystyle\int_{0}^{t}$
$\displaystyle(\mathcal{K}(t,u)-\mathcal{K}(s,u))^{2}du$
$\displaystyle=\int_{0}^{s}(\mathcal{K}(t-u)-\mathcal{K}(s-u))^{2}du+\int_{s}^{t}(\mathcal{K}(t-u)-\mathcal{K}(0)+\mathcal{K}(0))^{2}du$
$\displaystyle\leq
C\int_{0}^{s}|t-s|^{2H}du+C\int_{0}^{t-s}u^{2H}du+C\int_{0}^{t-s}\mathcal{K}^{2}(0)du$
$\displaystyle\leq C|t-s|^{2\left(H\wedge\frac{1}{2}\right)},$

i.e. (K2) is satisfied and the process

$Z(t)=\int_{0}^{t}\mathcal{K}(t-s)dB_{1}(s),\quad t\in[0,T],$

has a modification that is Hölder continuous up to the order
$H\wedge\frac{1}{2}$. Furthermore, if additionally $\mathcal{K}(0)=0$, then it
is easy to see that

$\mathbb{E}[(Z(t)-Z(s))^{2}]=\int_{0}^{t}(\mathcal{K}(t,u)-\mathcal{K}(s,u))^{2}du\leq
C|t-s|^{2H},$

i.e. $Z$ has a modification that is Hölder continuous up to the order $H$.

######  Example 2.2.

(Fractional kernel) Let
$\mathcal{K}(t,s)=\frac{1}{\Gamma\left(H+\frac{1}{2}\right)}(t-s)^{H-\frac{1}{2}}\mathbbm{1}_{0\leq
s\leq t\leq T}$ with $H\in\left(0,1\right)$. The process

$Z(t)=\frac{1}{\Gamma\left(H+\frac{1}{2}\right)}\int_{0}^{t}(t-s)^{H-\frac{1}{2}}dB_{1}(s),\quad
t\in[0,T],$

is called the Riemann-Liouville fractional Brownian motion (RL-fBm) and it is
well-known (see e.g. [39, Lemma 3.1]) that, for any $s<t$,

$\displaystyle\mathbb{E}[(Z(t)-Z(s))^{2}]$
$\displaystyle=\frac{1}{\Gamma^{2}\left(H+\frac{1}{2}\right)}\int_{0}^{s}\left((t-u)^{H-\frac{1}{2}}-(s-u)^{H-\frac{1}{2}}\right)^{2}du$
$\displaystyle\quad+\frac{1}{\Gamma^{2}\left(H+\frac{1}{2}\right)}\int_{0}^{t-s}u^{2H-1}du$
$\displaystyle\leq C|t-s|^{2H}$

for some constant $C$. Thus (K2) holds and the RL-fBm has a modification that
is Hölder continuous up to the order $H$.

For a given pair $\varphi$, $\psi$: $[0,T]\to\mathbb{R}$ such that
$\varphi(t)<\psi(t)$, $t\in[0,T]$, and any $a_{1}$, $a_{2}\geq 0$, we
introduce the notation $\mathcal{D}_{a_{1},a_{2}}$ for the set

$\mathcal{D}_{a_{1},a_{2}}:=\\{(t,y)\in[0,T]\times\mathbb{R}_{+},~{}y\in(\varphi(t)+a_{1},\psi(t)-a_{2})\\}.$

Consider a stochastic process $Y=\\{Y(t),~{}t\in[0,T]\\}$ defined by (1.2),
where the initial value $Y(0)$ and the drift $b$ satisfy the following
assumption.

######  Assumption (Y).

There exist $H$-Hölder continuous functions $\varphi$, $\psi$:
$[0,T]\to\mathbb{R}$, $0<\varphi(t)<\psi(t)$, $t\in[0,T]$, with $H$ being the
same as in (K2) such that

  * (Y1)

$Y(0)$ is deterministic and $0<\varphi(0)<Y(0)<\psi(0)$,

  * (Y2)

$b$: $\mathcal{D}_{0,0}\to\mathbb{R}$ is continuous and, moreover, for any
$\varepsilon\in\left(0,\min\left\\{1,\frac{1}{2}\lVert\psi-\varphi\rVert_{\infty}\right\\}\right)$

$|b(t_{1},y_{1})-b(t_{2},y_{2})|\leq\frac{c_{1}}{\varepsilon^{p}}\left(|y_{1}-y_{2}|+|t_{1}-t_{2}|^{H}\right),\quad(t_{1},y_{1}),(t_{2},y_{2})\in\mathcal{D}_{\varepsilon,\varepsilon},$

where $c_{1}>0$ and $p>1$ are given constants,

  * (Y3)

$b(t,y)\geq\frac{c_{2}}{(y-\varphi(t))^{\gamma}},\quad(t,y)\in\mathcal{D}_{0,0}\setminus\mathcal{D}_{y_{*},0},$
$b(t,y)\leq-\frac{c_{2}}{(\psi(t)-y)^{\gamma}},\quad(t,y)\in\mathcal{D}_{0,0}\setminus\mathcal{D}_{0,y_{*}},$

where $y_{*}$, $c_{2}>0$ are given constants and $\gamma>\frac{1}{H}-1]$,

  * (Y4)

the partial derivative $\frac{\partial b}{\partial y}$ with respect to the
spacial variable exists, is continuous and

$-c_{3}\left(1+\frac{1}{(y-\varphi(t))^{q}}+\frac{1}{(\psi(t)-y)^{q}}\right)<\frac{\partial
b}{\partial y}(t,y)<c_{3},\quad(t,y)\in\mathcal{D}_{0,0},$

for some $c_{3}>0$ and $q>0$.

According to [20], Assumptions (K) and (Y) together ensure that the SDE (1.2)
has a unique solution that is sandwiched, i.e.

$\varphi(t)<Y(t)<\psi(t),\quad t\in[0,T].$ (2.3)

This and other relevant results from [20] regarding the process (1.2) are
summarized in the following statement.

######  Theorem 2.3.

Let Assumptions (K) and (Y) hold. Then

  * 1)

the SDE (1.2) has a unique strong solution $Y$ and (2.3) holds with
probability 1;

  * 2)

for any $\lambda\in\left(\frac{1}{1+\gamma},H\right)$, where $H$ is from (K2),
there exist deterministic constants $L_{1}$ and $L_{2}>0$ (depending only on
$Y(0)$, the shape of $b$ and $\lambda$) such that, with probability 1,

$\varphi(t)+\frac{L_{1}}{(L_{2}+\Lambda)^{\frac{1}{\gamma\lambda+\lambda-1}}}\leq
Y(t)\leq\psi(t)-\frac{L_{1}}{(L_{2}+\Lambda)^{\frac{1}{\gamma\lambda+\lambda-1}}},\quad\quad
t\in[0,T],$ (2.4)

where $\Lambda$ is from (2.2) and $\gamma$ is from (Y3);

  * 3)

for any $r\in\mathbb{R}$,

$\mathbb{E}\left[\sup_{t\in[0,T]}\frac{1}{(Y(t)-\varphi(t))^{r}}\right]<\infty,\quad\mathbb{E}\left[\sup_{t\in[0,T]}\frac{1}{(\psi(t)-Y(t))^{r}}\right]<\infty.$

Having in mind property (2.3), we will refer to the process
$Y=\\{Y(t),~{}t\in[0,T]\\}$ defined by (1.2) as a sandwiched process.

###  2.2 Price process

Let Assumptions (K) and (Y) hold, $Z$ be a Gaussian Volterra process (2.1),
and $Y=\\{Y(t),~{}t\in[0,T]\\}$ be the sandwiched process (1.2) such that
(2.3) holds for some functions $0<\varphi<\psi$. We now consider the model
introduced in Section 1 with a risk-free asset $e^{\int_{0}^{t}\nu(s)ds}$ and
a risky asset $S=\\{S(t),~{}t\in[0,T]\\}$ defined by (1.1), where $S(0)>0$ is
some deterministic constant and $\rho\in(-1,1)$. Clearly, the process $X$
defined by (1.3) satisfies the SDE

$X(t)=X(0)+\int_{0}^{t}Y(s)X(s)\left(\rho
dB_{1}(t)+\sqrt{1-\rho^{2}}dB_{2}(t)\right),\quad t\in[0,T],$ (2.5)

where, for notational simplicity, we denote $X(0):=S(0)$. Moreover, the
following Theorem holds.

######  Theorem 2.4.

([19, Theorem 2.6])

  * 1)

Equation (1.1) has a unique strong solution and

$S(t)=S(0)\exp\left\\{\int_{0}^{t}\left(\nu(s)-\frac{Y^{2}(s)}{2}\right)ds+\int_{0}^{t}Y(s)\left(\rho
dB_{1}(t)+\sqrt{1-\rho^{2}}dB_{2}(t)\right)\right\\}.$

In particular,

$X(t)=X(0)\exp\left\\{-\int_{0}^{t}\frac{Y^{2}(s)}{2}ds+\int_{0}^{t}Y(s)\left(\rho
dB_{1}(t)+\sqrt{1-\rho^{2}}dB_{2}(t)\right)\right\\}.$

  * 2)

For any $r\in\mathbb{R}$,

$\mathbb{E}\left[\sup_{t\in[0,T]}S^{r}(t)\right]<\infty,\quad\mathbb{E}\left[\sup_{t\in[0,T]}X^{r}(t)\right]<\infty.$
(2.6)

######  Remark 2.5.

Theorem 2.4 implies that $X=\\{X(t),~{}t\in[0,T]\\}$ is a square integrable
martingale w.r.t. the filtration
$\mathbb{F}=\\{\mathcal{F}_{t},~{}t\in[0,T]\\}$ generated jointly by $B_{1}$
and $B_{2}$.

######  Remark 2.6.

We stress that (2.6) does not hold in general even in classical stochastic
volatility models such as the Heston model (see e.g. [5]). This has
repercussions in the possibilities of numerical simulations as well as in
various applications such as pricing hedging or portfolio optimization. In our
case, the existence of moments of all orders is achieved thanks to the
boundedness of $Y$.

We conclude the Subsection with a result on regularity of the probability laws
of $S$ and $X$ presented in [19].

######  Theorem 2.7.

[19, Theorem 3.8] For any $t\in(0,T]$, the law of $S(t)$ (and consequently of
$X(t)=e^{-\int_{0}^{t}\nu(s)ds}S(t)$) has continuous and bounded density.

###  2.3 Quadratic hedging: some considerations

Let Assumptions (K) and (Y) hold, $S=\\{S(t),~{}t\in[0,T]\\}$ be defined by
(1.1) and $X(t):=e^{-\int_{0}^{t}\nu(s)ds}S(t)$. In what follows, we discuss
mean-square hedging problem and in this Subsection we provide some known
theoretical background for that. We are interested in European options with
rather complicated payoffs $F:=F(X(T))$ covering both classical contracts
(such as standard call options) and exotic ones with discontinuities in $F$
(digital, supershare, binary, truncated payoff options and so on). Namely, the
payoff function $F$: $\mathbb{R}_{+}\to\mathbb{R}$ satisfies the following
assumption.

######  Assumption (F).

The payoff function $F$: $\mathbb{R}_{+}\to\mathbb{R}$ can be represented as

$F=F_{1}+F_{2},$

where

  * (i)

$F_{1}$ is globally Lipschitz, i.e. there exists $C>0$ such that

$|F_{1}(t)-F_{1}(s)|\leq C|t-s|,\quad s,t\geq 0;$

  * (ii)

$F_{2}$: $\mathbb{R}_{+}\to\mathbb{R}$ is of bounded variation over
$\mathbb{R}_{+}$, i.e.

$V(F_{2}):=\lim_{x\to\infty}V_{[0,x]}(F_{2})<\infty,$

where

$V_{[0,x]}(F_{2}):=\sup\sum_{j=1}^{N}|F_{2}(x_{j})-F_{2}(x_{j-1})|$

and the supremum above is taken over all $N\geq 0$ and all partitions
$0=x_{0}<x_{1}<...<x_{N}=x$.

In what follows, we will need to approximate the discounted price $X(T)$ by a
random variable $X_{m}(T)$ with “better” properties and then compare
$F=F(X(T))$ with $F_{m}:=F(X_{m}(T))$. In order to do that, the following
Theorem will be used.

######  Theorem 2.8.

Let $F=F_{1}+F_{2}$ satisfy Assumption (F) and $\xi$, $\hat{\xi}\in
L^{r}(\mathbb{P})$, $1\leq r<\infty$. Then

  * 1)

there exists a constant $C>0$ depending only on $F_{1}$ and $r$ such that

$\mathbb{E}[|F_{1}(\xi)-F_{1}(\hat{\xi})|^{r}]\leq
C\mathbb{E}[|\xi-\hat{\xi}|^{r}];$

  * 2)

if $\xi$ has bounded density $f_{\xi}$, then

$\mathbb{E}[|F_{2}(\xi)-F_{2}(\hat{\xi})|^{r}]\leq 3^{r+1}V^{r}(F_{2})(\sup
f_{\xi})^{\frac{r}{r+1}}\left(\mathbb{E}\left[|\xi-\hat{\xi}|^{r}\right]\right)^{\frac{r}{r+1}};$
(2.7)

  * 3)

if $\xi$ has bounded density $f_{\xi}$, then there exists a constant $C>0$
depending only on $F_{1}$, $F_{2}$, $r$ and the density $f_{\xi}$ such that

$\mathbb{E}[|F(\xi)-F(\hat{\xi})|^{r}]\leq
C\left(\mathbb{E}[|\xi-\hat{\xi}|^{r}]+\left(\mathbb{E}\left[|\xi-\hat{\xi}|^{r}\right]\right)^{\frac{r}{r+1}}\right).$

###### Proof.

Item 1) is obvious and follows directly from the Lipschitz condition for
$F_{1}$. Item 2) is proved in [6, Theorem 2.4]. Item 3) is simply a
combination of 1) and 2). ∎

Next, consider a quadratic hedging problem of minimizing the functional $J$
defined by (1.5) over all $\mathbb{F}$-adapted stochastic processes $u\in
L^{2}(\mathbb{P}\times[X])$. By Theorem 2.4, the discounted price $X$ is a
square integrable martingale w.r.t. the filtration $\mathbb{F}$ and hence [17]
states that the unique minimizer of $J$ is equal to the non-anticipating
derivative $\mathfrak{D}F$ of $F$ w.r.t. $X$. We give the necessary details
regarding the non-anticipating derivatives below.

Let $\xi\in L^{2}(\mathbb{P})$ and $\eta=\\{\eta(t),~{}t\in[0,T]\\}$ be a
square-integrable martingale w.r.t. a filtration
$\\{\mathcal{G}_{t},~{}t\in[0,T]\\}$. For an arbitrary partition
$\pi=\\{0=t_{0}<t_{1}<...<t_{n}=T\\}$ of $[0,T]$ with the mesh
$|\pi|:=\max_{k}(t_{k}-t_{k-1})$, denote

$\displaystyle u_{\pi}$
$\displaystyle:=\sum_{k=0}^{n-1}u_{\pi,k}\mathbbm{1}_{(t_{k},t_{k+1}]},$ (2.8)
$\displaystyle u_{\pi,k}$
$\displaystyle:=\frac{\mathbb{E}\left[\xi(\eta(t_{k+1})-\eta(t_{k}))~{}|~{}\mathcal{G}_{t_{k}}\right]}{\mathbb{E}\left[(\eta(t_{k+1})-\eta(t_{k}))^{2}~{}|~{}\mathcal{G}_{t_{k}}\right]},\quad
k=0,1,...,n-1.$

######  Definition 2.9.

Consider a monotone sequence of partitions $\\{\pi_{M},~{}M\geq 1\\}$, such
that $|\pi_{M}|\to 0$ as $M\to\infty$. The
$L^{2}(\mathbb{P}\times[\eta])$-limit

$\mathfrak{D}\xi:=\lim_{|\pi_{M}|\to 0}u_{\pi_{M}}$ (2.9)

is called the non-anticipating derivative of $\xi$ w.r.t. $\eta$.

The next result shows that $\mathfrak{D}\xi$ is well-defined and
$\int_{0}^{T}\mathfrak{D}\xi(s)d\eta(s)$ gives the orthogonal
$L^{2}(\mathbb{P}\times[\eta])$-projection of $\xi$ on the subspace of
stochastic integrals w.r.t. $\eta$. By this, it is easy to see that the linear
operator $\mathfrak{D}$ is actually the dual of the Itô integral.

######  Theorem 2.10.

([17, Theorem 2.1]) The non-anticipating derivative $\mathfrak{D}\xi$ is well-
defined, i.e. the limit (2.9) exists and does not depend on the particular
choice of the partitions. Moreover, any $\xi\in L^{2}(\mathbb{P})$ admits a
unique representation of the form

$\xi=\xi_{0}+\int_{0}^{T}\mathfrak{D}\xi(s)d\eta(s),$

where $\xi_{0}\in L^{2}(\mathbb{P})$ is such that $\mathfrak{D}\xi_{0}=0$ and
$\mathbb{E}\left[\xi_{0}\int_{0}^{T}\mathfrak{D}\xi(s)d\eta(s)\right]=0$.

######  Notation.

Since the limit (2.9) does not depend on the particular choice of partition
$\\{\pi_{M},~{}M\geq 1\\}$, we will use the notation

$\mathfrak{D}\xi:=\lim_{|\pi|\to 0}u_{\pi},$

meaning that the limit in $L^{2}(\mathbb{P}\times[\eta])$ is taken along an
arbitrary sequence of monotone partitions with the mesh going to zero.

######  Corollary 2.11.

  * 1)

Eq. (2.6) and Assumption (F) imply that $F\in L^{2}(\mathbb{P})$. Therefore,
Theorem 2.10 holds and the non-anticipating derivative $u=\mathfrak{D}F$ of
$F$ w.r.t. $X$ is indeed the minimizer of $J$ from (1.5) and provides an
explicit representation of the optimal mean-square hedge.

  * 2)

Consider an arbitrary partition $\pi=\\{0=t_{0}<t_{1}<...<t_{n}=T\\}$. The
proof of [17, Theorem 2.1] implies that the pre-limit sum

$\displaystyle u_{\pi}$
$\displaystyle:=\sum_{k=0}^{n-1}u_{\pi,k}\mathbbm{1}_{(t_{k},t_{k+1}]},$
(2.10) $\displaystyle u_{\pi,k}$
$\displaystyle:=\frac{\mathbb{E}\left[F(X(t_{k+1})-X(t_{k}))~{}|~{}\mathcal{F}_{t_{k}}\right]}{\mathbb{E}\left[(X(t_{k+1})-X(t_{k}))^{2}~{}|~{}\mathcal{F}_{t_{k}}\right]},\quad
k=0,1,...,n-1,$ (2.11)

is the $L^{2}(\mathbb{P}\times[\eta])$-orthogonal projection of $\xi$ onto the
subspace generated by stochastic integrals of simple processes

$\sum_{k=0}^{n-1}\zeta_{k}\mathbbm{1}_{(t_{k},t_{k+1}]}$ (2.12)

w.r.t. $X$. Note that admissible portfolios in real markets are exactly of
this type since there is no technical possibility of real “continuous”
trading.

##  3 Effects of a change of Volterra kernels

The very definition of the non-anticipating derivative $u=\mathfrak{D}F$
describing the hedging portfolio provides a natural approximation of it.
Indeed, simple processes $u_{\pi}$ given by (2.10) converge in
$L^{2}(\mathbb{P}\times[X])$ to $\mathfrak{D}F$ and each $u_{\pi}$ is itself
the optimal hedge in the corresponding class of simple processes (2.12).
However, computation of the conditional expectations in (2.11) is a
challenging task that becomes even more complicated since the Volterra noise
$Z$ from (1.2) may have memory (and thus $X$ is in general not Markovian). The
numerical computation of conditional expectations in a non-Markovian context
is a renowned crucial problem in this context of hedging and represents a true
obstacle for effective implementation. In order to tackle this problem, one
may try to obtain a finite-dimensional Markovian approximation to the process
$X$ which would significantly simplify numerical algorithms. Our approach is
to replace the kernel $\mathcal{K}$ in (2.1) with an appropriate approximation
in $L^{2}$ that allows to obtain a Markovian structure. In this, we have taken
inspiration from [1], but in our case the drift $b$ in (1.2) is unbounded and
thus we need to obtain new methods for error estimation.

In this Section we study a general kernel approximation $\mathcal{K}_{m}$ to
$\mathcal{K}$ and provide all the necessary technical estimates for the
corresponding approximations: first of the stochastic volatility process and
prices (see Subsection 3.1) and then of the very hedging strategies (see
Subsection 3.2). We emphasize that the study of the estimates for hedging
strategies is rather involved and is specialized to the family of kernels of
difference type, i.e.

$\mathcal{K}(t,s)=\mathcal{K}(t-s)\mathbbm{1}_{s<t}.$

###  3.1 General approximation error estimates

As noted above, we want to replace the kernel $\mathcal{K}$ with its
appropriate $L^{2}$-approximations $\mathcal{K}_{m}$, $m\geq 1$, and then
consider stochastic processes $Z_{m}=\\{Z_{m}(t),~{}t\in[0,T]\\}$ and
$Y_{m}=\\{Y_{m}(t),~{}t\in[0,T]\\}$ defined by

$Z_{m}(t):=\int_{0}^{t}\mathcal{K}_{m}(t,s)dB_{1}(s),\quad t\in[0,T],$ (3.1)

and

$Y_{m}(t)=Y(0)+\int_{0}^{t}b(s,Y_{m}(s))ds+Z_{m}(t),\quad t\in[0,T].$ (3.2)

However, we need to make sure that the solution to (3.2) exists, i.e. Theorem
2.3 holds, for all $m\geq 1$. Moreover, we will need that bounds of the type
(2.4) hold for each $Y_{m}$ with constants $L_{1}$, $L_{2}$ that do not depend
on $m$. In order to ensure that, we introduce the following assumption.

######  Assumption (Km).

The sequence of kernels $\\{\mathcal{K}_{m},m\geq 1\\}$ is such that each
function $\mathcal{K}_{m}$: $[0,T]^{2}\to\mathbb{R}$ is a Volterra kernel,
i.e. $\mathcal{K}_{m}(t,s)=0$ whenever $t<s$, and

  * (Km1)

for each $m\geq 1$, $\mathcal{K}_{m}$ is square integrable, i.e.

$\int_{0}^{T}\int_{0}^{T}\mathcal{K}_{m}^{2}(t,s)dsdt=\int_{0}^{T}\int_{0}^{t}\mathcal{K}_{m}^{2}(t,s)dsdt<\infty,$

and, moreover,

$\int_{0}^{T}\mathcal{K}_{m}^{2}(t,s)ds<\infty\quad\forall t\in[0,T];$

  * (Km2)

for each $m\geq 1$ and $\lambda\in(0,H)$ with $H$ being from (K2),

$\int_{0}^{t}\left(\mathcal{K}_{m}(t,u)-\mathcal{K}_{m}(s,u)\right)^{2}du\leq\ell^{2}_{\lambda}(t-s)^{2\lambda},\quad
0\leq s\leq t\leq T,$

where $\ell_{\lambda}>0$ is a constant that possibly depends on $\lambda$ but
does not depend on $m$.

As in Section 2, we see here that Assumption (Km1) guarantees that each
$Z_{m}$ has a continuous modification that satisfies the $\lambda$-Hölder
property for all $\lambda\in(0,H)$ while (Km2) guarantees that moments of
Hölder seminorms of $Z$ and $Z_{m}$ are in fact uniformly bounded in
$L^{r}(\mathbb{P})$. The corresponding result is presented in the next lemma.

######  Lemma 3.1.

Let $\mathcal{K}$ satisfy Assumption (K) and the sequence
$\\{\mathcal{K}_{m},m\geq 1\\}$ satisfy Assumption (Km). Then for any constant
$\lambda\in\left(0,H\right)$ there exist positive random variables $\Lambda$,
$\Lambda_{m}$, $m\geq 1$, such that for each $m\geq 1$, $r>0$ and
$s,t\in[0,T]$

$|Z(t)-Z(s)|\leq\Lambda|t-s|^{\lambda},$
$|Z_{m}(t)-Z_{m}(s)|\leq\Lambda_{m}|t-s|^{\lambda},$

and

$\mathbb{E}\Lambda^{r}+\sup_{m\geq 1}\mathbb{E}\Lambda_{m}^{r}<\infty.$ (3.3)

###### Proof.

Fix an arbitrary $\lambda\in\left(0,H\right)$, choose $p>1$ such that
$\lambda+\frac{2}{p}<H$ and denote

$\displaystyle\Lambda$
$\displaystyle:=A_{\lambda,p}\left(\int_{0}^{T}\int_{0}^{T}\frac{|Z(t)-Z(s)|^{p}}{|t-s|^{\lambda
p+2}}dsdt\right)^{\frac{1}{p}},$ $\displaystyle\Lambda_{m}$
$\displaystyle:=A_{\lambda,p}\left(\int_{0}^{T}\int_{0}^{T}\frac{|Z_{m}(t)-Z_{m}(s)|^{p}}{|t-s|^{\lambda
p+2}}dsdt\right)^{\frac{1}{p}},\quad m\geq 1,$

with $A_{\lambda,p}:=2^{3+\frac{2}{p}}\left(\frac{\lambda p+2}{\lambda
p}\right)$. By the Garsia-Rodemich-Rumsey inequality (see [27] and [20, Lemma
1.1]), for any $m\geq 1$ and $s,t\in[0,T]$,

$|Z(t)-Z(s)|\leq\Lambda|t-s|^{\lambda},\quad|Z_{m}(t)-Z_{m}(s)|\leq\Lambda_{m}|t-s|^{\lambda},$

and, moreover, the proof of [7, Theorem 1] implies that
$\mathbb{E}\Lambda^{r}<\infty$, $\mathbb{E}\Lambda^{r}_{m}<\infty$ for each
$r>0$.

It remains to prove that $\sup_{m\geq 1}\mathbb{E}\Lambda^{r}_{m}<\infty$.
Take any $r>p$, and observe that Minkowski integral inequality, Gaussian
distribution of $Z_{m}(t)-Z_{m}(s)$ and (Km2) yield

$\displaystyle(\mathbb{E}[\Lambda_{m}^{r}])^{\frac{p}{r}}$ $\displaystyle\leq
A^{p}_{\lambda,p}\int_{0}^{T}\int_{0}^{T}\frac{\mathbb{(}E[|Z_{m}(t)-Z_{m}(s)|^{r}])^{\frac{p}{r}}}{|t-s|^{\lambda
p+2}}dsdt$
$\displaystyle=2^{\frac{p}{2}}\left(\frac{\Gamma\left(\frac{r+1}{2}\right)}{\sqrt{\pi}}\right)^{\frac{p}{r}}A^{p}_{\lambda,p}\int_{0}^{T}\int_{0}^{T}\frac{\left(\int_{0}^{t\vee
s}\left(\mathcal{K}_{m}(t,u)-\mathcal{K}_{m}(s,u)\right)^{2}du\right)^{\frac{p}{2}}}{|t-s|^{\lambda
p+2}}dsdt$ $\displaystyle\leq
2^{\frac{p}{2}}\left(\frac{\Gamma\left(\frac{r+1}{2}\right)}{\sqrt{\pi}}\right)^{\frac{p}{r}}A^{p}_{\lambda,p}\ell^{p}_{\lambda+\frac{2}{p}}\int_{0}^{T}\int_{0}^{T}\frac{|t-s|^{\lambda
p+2}}{|t-s|^{\lambda p+2}}dsdt$ $\displaystyle\leq
2^{\frac{p}{2}}T^{2}\left(\frac{\Gamma\left(\frac{r+1}{2}\right)}{\sqrt{\pi}}\right)^{\frac{p}{r}}A^{p}_{\lambda,p}\ell^{p}_{\lambda+\frac{2}{p}}$

and thus

$\sup_{m\geq 1}E[\Lambda_{m}^{r}]\leq
2^{\frac{r}{2}}T^{\frac{2r}{p}}\frac{\Gamma\left(\frac{r+1}{2}\right)}{\sqrt{\pi}}A^{r}_{\lambda,p}\ell^{r}_{\lambda+\frac{2}{p}}<\infty,$

which ends the proof. ∎

Next, consider a family of sandwiched process
$Y_{m}=\\{Y_{m}(t),~{}t\in[0,T]\\}$, $m\geq 1$, defined by equations of the
form 3.2. Note that the conditions of Theorem 2.3 are met for all $m\geq 1$,
i.e. each $Y_{m}$ is well-defined. Since constants $L_{1}$, $L_{2}$ in (2.4)
depend only on $Y(0)$, the shape of $b$ and
$\lambda\in\left(\frac{1}{1+\gamma},H\right)$ that can be chosen jointly for
all $m\geq 1$, we have that

$\varphi(t)+\frac{L_{1}}{(L_{2}+\Lambda_{m})^{\frac{1}{\gamma\lambda+\lambda-1}}}<Y_{m}(t)<\psi(t)-\frac{L_{1}}{(L_{2}+\Lambda_{m})^{\frac{1}{\gamma\lambda+\lambda-1}}},\quad
t\in[0,T],\quad m\geq 1.$ (3.4)

Moreover, the following result is true.

######  Lemma 3.2.

Let Assumptions (K), (Y) and (Km) hold. Then, for all
$\lambda\in\left(\frac{1}{1+\gamma},H\right)$ and $m\geq 1$,

$|Y(t)-Y(s)|\leq\Upsilon|t-s|^{\lambda},\quad|Y_{m}(t)-Y_{m}(s)|\leq\Upsilon_{m}|t-s|^{\lambda},\quad
t,s\in[0,T],$

where $\Upsilon$, $\Upsilon_{m}$, $m\geq 1$, are positive random variables
such that for any $r>0$

$\mathbb{E}[\Upsilon^{r}]+\sup_{m\geq 1}\mathbb{E}[\Upsilon^{r}_{m}]<\infty.$

###### Proof.

Fix $\lambda\in\left(\frac{1}{1+\gamma},H\right)$ and take the corresponding
$\Lambda_{m}$ from Lemma 3.1. By (Y2) and (3.4), for any $u\in[0,T]$

$\displaystyle|b(u,Y_{m}(u))|$
$\displaystyle\leq|b(u,Y_{m}(u))-b(0,Y(0))|+|b(0,Y(0))|$
$\displaystyle\leq\frac{c_{1}(L_{2}+\Lambda_{m})^{\frac{p}{\gamma\lambda+\lambda-1}}}{L_{1}^{p}}\left(|Y_{m}(u)-Y(0)|+u^{H}\right)+|b(0,Y(0)|$
$\displaystyle\leq\frac{c_{1}(L_{2}+\Lambda_{m})^{\frac{p}{\gamma\lambda+\lambda-1}}}{L_{1}^{p}}\left(\max_{t\in[0,T]}\psi(t)+Y(0)+T^{H}\right)+|b(0,Y(0)|$

and thus, for any $s,t\in[0,T]$, $s\leq t$,

$\displaystyle|Y_{m}(t)-Y_{m}(s)|$
$\displaystyle\leq\int_{s}^{t}|b(u,Y_{m}(u))|du+(Z_{m}(t)-Z_{m}(s))$
$\displaystyle\leq\frac{c_{1}(L_{2}+\Lambda_{m})^{\frac{p}{\gamma\lambda+\lambda-1}}}{L_{1}^{p}}\left(\max_{t\in[0,T]}\psi(t)+Y(0)+T^{H}\right)(t-s)$
$\displaystyle\quad+|b(0,Y(0)|(t-s)+\Lambda_{m}(t-s)^{\lambda}$
$\displaystyle\leq\Upsilon_{m}|t-s|^{\lambda},$

with
$\Upsilon_{m}:=T^{1-\lambda}\frac{c_{1}(L_{2}+\Lambda_{m})^{\frac{p}{\gamma\lambda+\lambda-1}}}{L_{1}^{p}}\left(\max_{t\in[0,T]}\psi(t)+Y(0)+T^{H}\right)+T^{1-\lambda}|b(0,Y(0)|+\Lambda_{m}$.
By (3.3), it is easy to see that for any $r>0$

$\sup_{m\geq 1}\mathbb{E}[\Upsilon^{r}_{m}]<\infty$

as required. It remains to note that the random variable $\Upsilon$ for the
process $Y$ can be defined in the same manner as $\Upsilon_{m}$. ∎

The next result gives a pathwise estimate of distance between $Y$ and $Y_{m}$.

######  Theorem 3.3.

Let the processes $Y$ and $Y_{m}$, $m\geq 1$, satisfy Assumptions (K), (Y) and
(Km). Then, for each $m\geq 1$, there exist a deterministic constant $C>0$
that does not depend on $m$ or $t$ and a random variable $\xi_{m}$ that does
not depend on $t$ such that, for any $r>0$,

$\sup_{m\geq 1}\mathbb{E}[\xi_{m}^{r}]<\infty$ (3.5)

and, for any $t\in[0,T]$,

$|Y(t)-Y_{m}(t)|\leq
C\left(|Z(t)-Z_{m}(t)|+\xi_{m}\int_{0}^{t}|Z(u)-Z_{m}(u)|du\right).$

###### Proof.

Fix $t\in[0,T]$ and consider a uniform partition $0=t_{0}<t_{1}<...<t_{N}=t$,
$t_{n}=\frac{n}{N}t$, of the interval $[0,t]$ such that the mesh
$\Delta_{N}:=\frac{t}{N}$ satisfies the condition

$c_{3}\Delta_{N}<1,$ (3.6)

for $c_{3}$ from (Y4). Denote $e_{n,m}:=Y(t_{n})-Y_{m}(t_{n})$ and observe
that

$\displaystyle e_{N,m}$
$\displaystyle=Y(t_{N-1})+\int_{t_{N-1}}^{t_{N}}b(s,Y(s))ds+Z(t_{N})-Z(t_{N-1})$
$\displaystyle\quad-
Y_{m}(t_{N-1})-\int_{t_{N-1}}^{t_{N}}b(s,Y_{m}(s))ds+Z_{m}(t_{N})-Z_{m}(t_{N-1})$
$\displaystyle=e_{N-1,m}+\int_{t_{N-1}}^{t_{N}}\left(b(s,Y(s))-b(t_{N},Y(t_{N}))\right)ds$
$\displaystyle\quad-\int_{t_{N-1}}^{t_{N}}\left(b(s,Y_{m}(s))-b(t_{N},Y_{m}(t_{N}))\right)ds$
$\displaystyle\quad+\left(b(t_{N},Y(t_{N}))-b(t_{N},Y_{m}(t_{N}))\right)\Delta_{N}$
$\displaystyle\quad+\left(Z(t_{N})-Z(t_{N-1})\right)-\left(Z_{m}(t_{N})-Z_{m}(t_{N-1})\right).$

Note that, for each $n=1,...,N$, there exists $\Theta_{n,m}$ between
$Y(t_{n})$ and $Y_{m}(t_{n})$ such that

$b(t_{n},Y(t_{n}))-b(t_{n},Y_{m}(t_{n}))=\frac{\partial b}{\partial
y}(t_{n},\Theta_{n,m})(Y(t_{n})-Y_{m}(t_{n}))=\frac{\partial b}{\partial
y}(t_{n},\Theta_{n,m})e_{n,m},$

and thus

$\displaystyle\bigg{(}1-\frac{\partial b}{\partial y}(t_{N},\Theta_{N,m})$
$\displaystyle\Delta_{N}\bigg{)}e_{N,m}$ (3.7)
$\displaystyle=e_{N-1,m}+\int_{t_{N-1}}^{t_{N}}\left(b(s,Y(s))-b(t_{N},Y(t_{N}))\right)ds$
$\displaystyle\quad-\int_{t_{N-1}}^{t_{N}}\left(b(s,Y_{m}(s))-b(t_{N},Y_{m}(t_{N}))\right)ds$
$\displaystyle\quad+\left(Z(t_{N})-Z(t_{N-1})\right)-\left(Z_{m}(t_{N})-Z_{m}(t_{N-1})\right).$

Observe that each $1-\frac{\partial b}{\partial
y}(t_{N},\Theta_{N,m})\Delta_{N}>0$ by (3.6) and denote

$\zeta_{0,m}:=1,\quad\zeta_{n,m}:=\prod_{k=1}^{n}\left(1-\frac{\partial
b}{\partial y}(t_{k},\Theta_{k,m})\Delta_{N}\right),$

$\widetilde{e}_{n,m}:=\zeta_{n,m}e_{n,m}$. By multiplying the left- and right-
hand sides of (3.7) by $\zeta_{N-1,m}$, we obtain:

$\displaystyle\widetilde{e}_{N,m}$
$\displaystyle=\widetilde{e}_{N-1,m}+\zeta_{N-1,m}\int_{t_{N-1}}^{t_{N}}\left(b(s,Y(s))-b(t_{N},Y(t_{N}))\right)ds$
$\displaystyle\quad-\zeta_{N-1,m}\int_{t_{N-1}}^{t_{N}}\left(b(s,Y_{m}(s))-b(t_{N},Y_{m}(t_{N}))\right)ds$
$\displaystyle\quad+\zeta_{N-1,m}\Big{(}\left(Z(t_{N})-Z(t_{N-1})\right)-\left(Z_{m}(t_{N})-Z_{m}(t_{N-1})\right)\Big{)}$
$\displaystyle=\widetilde{e}_{N-2,m}+\sum_{n=N-1}^{N}\zeta_{n-1,m}\int_{t_{n-1}}^{t_{n}}\left(b(s,Y(s))-b(t_{n},Y(t_{n}))\right)ds$
$\displaystyle\quad-\sum_{n=N-1}^{N}\zeta_{n-1,m}\int_{t_{n-1}}^{t_{n}}\left(b(s,Y_{m}(s))-b(t_{n},Y_{m}(t_{n}))\right)ds$
$\displaystyle\quad+\sum_{n=N-1}^{N}\zeta_{n-1,m}\left(\left(Z(t_{n})-Z(t_{n-1})\right)-\left(Z_{m}(t_{n})-Z_{m}(t_{n-1})\right)\right)$
$\displaystyle=\sum_{n=1}^{N}\zeta_{n-1,m}\int_{t_{n-1}}^{t_{n}}\left(b(s,Y(s))-b(t_{n},Y(t_{n}))\right)ds$
$\displaystyle\quad-\sum_{n=1}^{N}\zeta_{n-1,m}\int_{t_{n-1}}^{t_{n}}\left(b(s,Y_{m}(s))-b(t_{n},Y_{m}(t_{n}))\right)ds$
$\displaystyle\quad+\sum_{n=1}^{N}\zeta_{n-1,m}\left(\left(Z(t_{n})-Z(t_{n-1})\right)-\left(Z_{m}(t_{n})-Z_{m}(t_{n-1})\right)\right),$

i.e.

$\displaystyle e_{N,m}$
$\displaystyle=\sum_{n=1}^{N}\frac{\zeta_{n-1,m}}{\zeta_{N,m}}\int_{t_{n-1}}^{t_{n}}\left(b(s,Y(s))-b(t_{n},Y(t_{n}))\right)ds$
$\displaystyle\quad-\sum_{n=1}^{N}\frac{\zeta_{n-1,m}}{\zeta_{N,m}}\int_{t_{n-1}}^{t_{n}}\left(b(s,Y_{m}(s))-b(t_{n},Y_{m}(t_{n}))\right)ds$
$\displaystyle\quad+\sum_{n=1}^{N}\frac{\zeta_{n-1,m}}{\zeta_{N,m}}\left(\left(Z(t_{n})-Z(t_{n-1})\right)-\left(Z_{m}(t_{n})-Z_{m}(t_{n-1})\right)\right).$

Next, observe that, by (Y4) and (3.6), for any $n=1,...,N$

$\frac{\zeta_{n-1,m}}{\zeta_{N,m}}=\prod_{k=n}^{N}\left(1-\frac{\partial
b}{\partial
y}(t_{k},\Theta_{k,m})\Delta_{N}\right)^{-1}\leq\prod_{k=n}^{N}\left(1-c_{3}\Delta_{N}\right)^{-1}\leq(1-c_{3}\Delta_{N})^{-N}.$

Note that $(1-c_{3}\Delta_{N})^{-N}$ converges as $N\to\infty$ and hence is
bounded w.r.t. $N$, therefore there exists a (non-random) constant $C>0$ that
does not depend on $n$, $N$ or $m$ such that

$\frac{\zeta_{n-1,m}}{\zeta_{N,m}}\leq C.$

Using this, one can easily deduce that

$\displaystyle|e_{N,m}|$
$\displaystyle\leq\left|\sum_{n=1}^{N}\frac{\zeta_{n-1,m}}{\zeta_{N,m}}\int_{t_{n-1}}^{t_{n}}\left(b(s,Y(s))-b(t_{n},Y(t_{n}))\right)ds\right|$
(3.8)
$\displaystyle\qquad+\left|\sum_{n=1}^{N}\frac{\zeta_{n-1,m}}{\zeta_{N,m}}\int_{t_{n-1}}^{t_{n}}\left(b(s,Y_{m}(s))-b(t_{n},Y_{m}(t_{n}))\right)ds\right|$
$\displaystyle\qquad+\left|\sum_{n=1}^{N}\frac{\zeta_{n-1,m}}{\zeta_{N,m}}\left(\left(Z(t_{n})-Z(t_{n-1})\right)-\left(Z_{m}(t_{n})-Z_{m}(t_{n-1})\right)\right)\right|$
$\displaystyle\leq
C\sum_{n=1}^{N}\int_{t_{n-1}}^{t_{n}}\left|b(s,Y(s))-b(t_{n},Y(t_{n}))\right|ds$
$\displaystyle\qquad+C\sum_{n=1}^{N}\int_{t_{n-1}}^{t_{n}}\left|b(s,Y_{m}(s))-b(t_{n},Y_{m}(t_{n}))\right|ds$
$\displaystyle\qquad+\left|\sum_{n=1}^{N}\frac{\zeta_{n-1,m}}{\zeta_{N,m}}\left(\left(Z(t_{n})-Z(t_{n-1})\right)-\left(Z_{m}(t_{n})-Z_{m}(t_{n-1})\right)\right)\right|$

Next, fix an arbitrary $\lambda\in\left(\frac{1}{1+\gamma},H\right)$ and
observe that (2.4) and (Y2) imply

$\displaystyle\left|b(s_{1},Y(s_{1}))-b(s_{2},Y(s_{2}))\right|$
$\displaystyle\leq
C(L_{2}+\Lambda)^{\frac{p}{\gamma\lambda+\lambda-1}}\left(|s_{1}-s_{2}|^{H}+|Y(s_{1})-Y(s_{2})|+\right)$
$\displaystyle\leq
C(L_{2}+\Lambda)^{\frac{p}{\gamma\lambda+\lambda-1}}\left(|s_{1}-s_{2}|^{\lambda}+|Y(s_{1})-Y(s_{2})|\right)$

for all $s_{1},s_{2}\in[0,T]$. Therefore, by Lemma 3.2,

$\displaystyle\sum_{n=1}^{N}$
$\displaystyle\int_{t_{n-1}}^{t_{n}}\left|b(s,Y(s))-b(t_{n},Y(t_{n}))\right|ds$
(3.9) $\displaystyle\leq
C(L_{2}+\Lambda)^{\frac{p}{\gamma\lambda+\lambda-1}}\sum_{n=1}^{N}\int_{t_{n-1}}^{t_{n}}|s-t_{n}|^{\lambda}ds$
$\displaystyle\quad+C(L_{2}+\Lambda)^{\frac{p}{\gamma\lambda+\lambda-1}}\sum_{n=1}^{N}\int_{t_{n-1}}^{t_{n}}|Y(s)-Y(t_{n})|ds$
$\displaystyle\leq
C(L_{2}+\Lambda)^{\frac{p}{\gamma\lambda+\lambda-1}}\sum_{n=1}^{N}\int_{t_{n-1}}^{t_{n}}|s-t_{n}|^{\lambda}ds$
$\displaystyle\quad+C(L_{2}+\Lambda)^{\frac{p}{\gamma\lambda+\lambda-1}}\Upsilon\sum_{n=1}^{N}\int_{t_{n-1}}^{t_{n}}|s-t_{n}|^{\lambda}ds$
$\displaystyle\leq
C(L_{2}+\Lambda)^{\frac{p}{\gamma\lambda+\lambda-1}}(1+\Upsilon)\Delta_{N}^{\lambda},$

where $\Upsilon$ is from Lemma 3.2 and has all the moments. Moreover, (3.4)
and the same argument as above yield that

$\displaystyle\sum_{n=1}^{N}\int_{t_{n-1}}^{t_{n}}\left|b(s,Y_{m}(s))-b(t_{n},Y_{m}(t_{n}))\right|ds\leq
C(L_{2}+\Lambda_{m})^{\frac{p}{\gamma\lambda+\lambda-1}}(1+\Upsilon_{m})\Delta_{N}^{\lambda}$
(3.10)

with $\Upsilon_{m}$ being from Lemma 3.2. Finally, using Abel summation-by-
parts formula, one can write:

$\displaystyle\bigg{|}\sum_{n=1}^{N}\frac{\zeta_{n-1,m}}{\zeta_{N,m}}\Big{(}\left(Z(t_{n})-Z(t_{n-1})\right)-\left(Z_{m}(t_{n})-Z_{m}(t_{n-1})\right)\Big{)}\bigg{|}$
(3.11)
$\displaystyle=\bigg{|}\frac{\zeta_{N-1,m}}{\zeta_{N,m}}(Z(t_{N})-Z_{m}(t_{N}))-\sum_{n=1}^{N-1}(Z(t_{n})-Z_{m}(t_{n}))\left(\frac{\zeta_{n,m}}{\zeta_{N,m}}-\frac{\zeta_{n-1,m}}{\zeta_{N,m}}\right)\bigg{|}$
$\displaystyle=\bigg{|}\frac{\zeta_{N-1,m}}{\zeta_{N,m}}(Z(t_{N})-Z_{m}(t_{N}))+\sum_{n=1}^{N-1}\frac{\zeta_{n-1,m}}{\zeta_{N,m}}\frac{\partial
b}{\partial y}(t_{n},\Theta_{n,m})(Z(t_{n})-Z_{m}(t_{n}))\Delta_{N}\bigg{|}$
$\displaystyle\leq
C|Z(t_{N})-Z_{m}(t_{N})|+C\sum_{n=1}^{N-1}\left|\frac{\partial b}{\partial
y}(t_{n},\Theta_{n,m})\right||Z(t_{n})-Z_{m}(t_{n})|\Delta_{N}.$

Random variables $\Theta_{n,m}$ lie between $Y(t_{n})$ and $Y_{m}(t_{n})$,
hence

$\displaystyle\Theta_{n,m}-\varphi(t)\geq\frac{L_{1}}{(L_{2}+\Lambda)^{\frac{1}{\gamma\lambda+\lambda-1}}}\wedge\frac{L_{1}}{(L_{2}+\Lambda_{m})^{\frac{1}{\gamma\lambda+\lambda-1}}},$
$\displaystyle\psi(t)-\Theta_{n,m}\geq\frac{L_{1}}{(L_{2}+\Lambda)^{\frac{1}{\gamma\lambda+\lambda-1}}}\wedge\frac{L_{1}}{(L_{2}+\Lambda_{m})^{\frac{1}{\gamma\lambda+\lambda-1}}},$

and thus it is easy to see by (Y4) that

$\left|\frac{\partial b}{\partial y}(t_{n},\Theta_{n,m})\right|\leq
C\left(1+(L_{2}+\Lambda)^{\frac{q}{\gamma\lambda+\lambda-1}}+(L_{2}+\Lambda_{m})^{\frac{q}{\gamma\lambda+\lambda-1}}\right),$

where $C$ is, as always, a deterministic constant that does not depend on $n$,
$N$ or $m$. Therefore,

$\displaystyle\bigg{|}\sum_{n=1}^{N}$
$\displaystyle\frac{\zeta_{n-1,m}}{\zeta_{N,m}}\left(\left(Z(t_{n})-Z(t_{n-1})\right)-\left(Z_{m}(t_{n})-Z_{m}(t_{n-1})\right)\right)\bigg{|}$
$\displaystyle\leq
C\left(|Z(t_{N})-Z_{m}(t_{N})|+\xi_{m}\sum_{n=1}^{N-1}|Z(t_{n})-Z_{m}(t_{n})|\Delta_{N}\right),$

where

$\xi_{m}:=1+(L_{2}+\Lambda)^{\frac{q}{\gamma\lambda+\lambda-1}}+(L_{2}+\Lambda_{m})^{\frac{q}{\gamma\lambda+\lambda-1}}.$

Summarizing (3.8) and (3.9)–(3.11), we have that

$\displaystyle|Y(t)-Y_{m}(t)|$ $\displaystyle\leq
C(L_{2}+\Lambda)^{\frac{p}{\gamma\lambda+\lambda-1}}(1+\Upsilon)\Delta_{N}^{\lambda}$
$\displaystyle\quad+C(L_{2}+\Lambda_{m})^{\frac{p}{\gamma\lambda+\lambda-1}}(1+\Upsilon_{m})\Delta_{N}^{\lambda}$
$\displaystyle\quad+C|Z(t)-Z_{m}(t)|$
$\displaystyle\quad+C\xi_{m}\sum_{n=1}^{N-1}|Z(t_{n})-Z_{m}(t_{n})|\Delta_{N}$

and, moving $\Delta_{N}\to 0$, we obtain

$\displaystyle|Y(t)-Y_{m}(t)|$ $\displaystyle\leq
C\left(|Z(t)-Z_{m}(t)|+\xi_{m}\int_{0}^{t}|Z(s)-Z_{m}(s)|ds\right).$

It remains to notice that

$\sup_{m\geq 1}\mathbb{E}[\xi_{m}^{r}]<\infty$

for any $r>0$ due to (3.3) from Lemma 3.1.

∎

We are now ready to study the error estimates for the approximations of the
prices. For this, we first present a result on the moments of the
approximating processes. Let us consider the price process $S$ given by (1.1)
and denote

$W(t):=\rho B_{1}(t)+\sqrt{1-\rho^{2}}B_{2}(t),\quad t\in[0,T].$

For all $m\geq 1$, we define the approximated price processes $S_{m}$ as

$\displaystyle S_{m}(t)$
$\displaystyle=S(0)+\int_{0}^{t}\nu(s)S_{m}(s)ds+\int_{0}^{t}Y_{m}(s)S_{m}(s)dW(s),$
(3.12) $\displaystyle Y_{m}(t)$
$\displaystyle=Y(0)+\int_{0}^{t}b(s,Y_{m}(s))ds+Z_{m}(t),$

where $Z_{m}$ is given in (3.1). We also define the approximation for the
discounted process $X$ in (1.3) as

$X_{m}(t):=e^{-\int_{0}^{t}\nu(s)ds}S_{m}(t),\quad t\in[0,T].$ (3.13)

Note that Theorem 2.4 immediately implies that for each $m\geq 1$ and all
$r\in\mathbb{R}$

$\mathbb{E}\left[\sup_{t\in[0,T]}S_{m}^{r}(t)\right]<\infty,\quad\mathbb{E}\left[\sup_{t\in[0,T]}X_{m}^{r}(t)\right]<\infty.$

However, in the sequel we will require a slightly stronger result summarized
in the following Lemma.

######  Lemma 3.4.

Under Assumptions (K), (Y) and (Km), for any $r\in\mathbb{R}$

$\sup_{m\geq
1}\mathbb{E}\left[\sup_{t\in[0,T]}S^{r}_{m}(t)\right]<\infty,\quad\sup_{m\geq
1}\mathbb{E}\left[\sup_{t\in[0,T]}X^{r}_{m}(t)\right]<\infty.$

###### Proof.

It is sufficient to prove the result for $S_{m}$. By Theorem 2.4 and
continuity of $\nu$, we can see that

$\displaystyle S_{m}^{r}(t)$
$\displaystyle=\exp\left\\{r\int_{0}^{t}\nu(s)ds-\frac{r}{2}\int_{0}^{t}Y_{m}^{2}(s)ds+r\int_{0}^{t}Y_{m}(s)dW(s)\right\\}$
(3.14)
$\displaystyle\leq\exp\left\\{rT\max_{s\in[0,T]}\nu(s)+\frac{r}{2}(r-1)\int_{0}^{t}Y_{m}^{2}(s)ds\right\\}\times$
$\displaystyle\qquad\times\exp\left\\{\int_{0}^{t}rY_{m}(s)dW(s)-\frac{1}{2}\int_{0}^{t}(rY_{m}(s))^{2}ds\right\\}.$

Since $\max_{t\in[0,T]}Y_{m}(t)\leq\max_{s\in[0,T]}\psi(s)$ thanks to Theorem
2.3,

$\displaystyle\exp$
$\displaystyle\left\\{rT\max_{s\in[0,T]}\nu(s)+\frac{r}{2}(r-1)\int_{0}^{t}Y_{m}^{2}(s)ds\right\\}$
(3.15)
$\displaystyle\leq\exp\left\\{rT\max_{s\in[0,T]}\nu(s)+\frac{rT}{2}(r-1)\max_{s\in[0,T]}\psi^{2}(s)\mathbbm{1}_{\\{r>1\\}}\right\\}.$

Next, consider the process

$\mathcal{E}_{t}(rY_{m}\cdot
W):=\exp\left\\{\int_{0}^{t}rY_{m}(s)dW(s)-\frac{1}{2}\int_{0}^{t}(rY_{m}(s))^{2}ds\right\\},\quad
t\in[0,T].$

Note that

$\mathcal{E}_{t}(rY_{m}\cdot W)=1+r\int_{0}^{t}\mathcal{E}_{s}(rY_{m}\cdot
W)Y_{m}(s)dW(s),\quad t\in[0,T],$

and, since $Y_{m}$ is bounded, $\mathcal{E}(rY_{m}\cdot W)$ is a uniformly
integrable martingale due to Novikov’s condition. The Burkholder–Davis–Gundy
inequality yields

$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\mathcal{E}_{t}(rY_{m}\cdot
W)\right]$
$\displaystyle=1+r\mathbb{E}\left[\sup_{t\in[0,T]}\int_{0}^{t}\mathcal{E}_{s}(rY_{m}\cdot
W)Y_{m}(s)dW(s)\right]$ (3.16) $\displaystyle\leq
1+Cr\mathbb{E}\left[\left(\int_{0}^{T}\mathcal{E}^{2}_{s}(rY_{m}\cdot
W)Y_{m}^{2}(s)ds\right)^{\frac{1}{2}}\right]$ $\displaystyle\leq
1+Cr\sup_{s\in[0,T]}\psi(s)\left(\int_{0}^{T}\mathbb{E}\left[\mathcal{E}^{2}_{s}(rY_{m}\cdot
W)\right]ds\right)^{\frac{1}{2}}.$

Since, by the Novikov criterion, $\mathbb{E}\left[\mathcal{E}_{s}(2rY_{m}\cdot
W)\right]=1$, we can write

$\displaystyle\mathbb{E}\left[\mathcal{E}^{2}_{s}(rY_{m}\cdot W)\right]$
$\displaystyle=\mathbb{E}\left[\mathcal{E}_{s}(2rY_{m}\cdot
W)\exp\left\\{\int_{0}^{s}(rY_{m}(u))^{2}du\right\\}\right]$
$\displaystyle\leq\exp\left\\{r^{2}T\max_{u\in[0,T]}\psi^{2}(u)\right\\}\mathbb{E}\left[\mathcal{E}_{s}(2rY_{m}\cdot
W)\right]$
$\displaystyle=\exp\left\\{r^{2}T\max_{u\in[0,T]}\psi^{2}(u)\right\\}$

hence

$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\mathcal{E}_{t}(rY_{m}\cdot
W)\right]$ $\displaystyle\leq
1+C\sqrt{T}r\sup_{s\in[0,T]}\psi(s)\exp\left\\{\frac{r^{2}T}{2}\max_{u\in[0,T]}\psi^{2}(u)\right\\}.$
(3.17)

Therefore, using (3.14), (3.15) and (3.17), we immediately obtain that

$\mathbb{E}\left[\sup_{t\in[0,T]}S_{m}^{r}(t)\right]\leq C$

with $C>0$ being a constant that does not depend on $m$ and this finally
proves the required result. ∎

We are now finally ready to proceed to the main result of the Subsection.

######  Theorem 3.5.

Let Assumptions (K), (Y), (Km) hold and $r\geq 2$ be fixed.

  * 1)

If $\mathcal{K},\mathcal{K}_{m}\in L^{r}([0,T]^{2})$, $m\geq 1$, then there
exists a constant $C>0$ that does not depend on $m$ such that

$\mathbb{E}\left[\sup_{t\in[0,T]}|S(t)-S_{m}(t)|^{r}\right]\leq
C\lVert\mathcal{K}-\mathcal{K}_{m}\rVert^{r}_{L^{r}([0,T]^{2})}.$

  * 2)

Furthermore, if $\mathcal{K}$ and $\mathcal{K}_{m}$, $m\geq 1$, _are of a
difference type_ , i.e.

$\mathcal{K}(t,s)=\mathcal{K}(t-s)\mathbbm{1}_{s<t},\quad\mathcal{K}_{m}(t,s)=\mathcal{K}_{m}(t-s)\mathbbm{1}_{s<t},\quad
t,s\in[0,T],$

then there exists constant $C>0$ that does not depend on $m$ such that

$\mathbb{E}\left[\sup_{t\in[0,T]}|S(t)-S_{m}(t)|^{r}\right]\leq
C\lVert\mathcal{K}-\mathcal{K}_{m}\rVert^{r}_{L^{2}([0,T])}.$

###### Proof.

Item 1. For any $t\in[0,T]$, (1.1), (3.12) and the continuity of $\nu$ imply

$\displaystyle|S(t)-S_{m}(t)|^{r}$ $\displaystyle\leq
C\left(\int_{0}^{t}|S(s)-S_{m}(s)|^{r}ds+\left|\int_{0}^{t}(Y(s)S(s)-Y_{m}(s)S_{m}(s))dW(s)\right|^{r}\right)$
$\displaystyle\leq C\int_{0}^{t}\sup_{u\in[0,s]}|S(u)-S_{m}(u)|^{r}ds$
$\displaystyle\quad+C\left(\sup_{u\in[0,t]}\left|\int_{0}^{u}(Y(s)S(s)-Y_{m}(s)S_{m}(s))dW(s)\right|\right)^{r},$

whence

$\displaystyle\sup_{u\in[0,t]}|S(u)-S_{m}(u)|^{r}$ $\displaystyle\leq
C\int_{0}^{t}\sup_{u\in[0,s]}|S(u)-S_{m}(u)|^{r}ds$
$\displaystyle\quad+C\left(\sup_{u\in[0,t]}\left|\int_{0}^{u}(Y(s)S(s)-Y_{m}(s)S_{m}(s))dW(s)\right|\right)^{r}.$

By the Burkholder-Davis-Gundy inequality,

$\displaystyle\mathbb{E}\left[\sup_{u\in[0,t]}|S(u)-S_{m}(u)|^{r}\right]$
$\displaystyle\leq
C\int_{0}^{t}\mathbb{E}\left[\sup_{u\in[0,s]}|S(u)-S_{m}(u)|^{r}\right]ds$
(3.18)
$\displaystyle\quad+C\mathbb{E}\left[\left(\int_{0}^{t}\big{(}Y(s)S(s)-Y_{m}(s)S_{m}(s)\big{)}^{2}ds\right)^{\frac{r}{2}}\right]$
$\displaystyle\leq
C\int_{0}^{t}\mathbb{E}\left[\sup_{u\in[0,s]}|S(u)-S_{m}(u)|^{r}\right]ds$
$\displaystyle\quad+C\int_{0}^{t}\mathbb{E}\left[\big{|}Y(s)S(s)-Y_{m}(s)S_{m}(s)\big{|}^{r}\right]ds.$

Let us focus on the last summand of (3.18). Using the fact that

$\max_{m\geq 1}\max_{t\in[0,T]}|Y_{m}(t)|\leq\max_{t\in[0,T]}|\psi(t)|,$

we have

$\displaystyle\int_{0}^{t}$
$\displaystyle\mathbb{E}\left[|Y(s)S(s)-Y_{m}(s)S_{m}(s)|^{r}\right]ds$ (3.19)
$\displaystyle\leq
C\int_{0}^{t}\mathbb{E}\left[Y^{r}_{m}(s)|S(s)-S_{m}(s)|^{r}\right]ds$
$\displaystyle\quad+C\int_{0}^{t}\mathbb{E}\left[S^{r}(s)|Y(s)-Y_{m}(s)|^{r}\right]ds$
$\displaystyle\leq
C\int_{0}^{t}\mathbb{E}\left[\sup_{u\in[0,s]}|S(u)-S_{m}(u)|^{r}\right]ds$
$\displaystyle\quad+C\int_{0}^{t}\mathbb{E}\left[S^{r}(s)|Y(s)-Y_{m}(s)|^{r}\right]ds.$

Studying again the last summand of (3.19), we obtain by Theorem 3.3 that there
exist random variables $\xi_{m}$, $m\geq 1$, such that (3.5) holds and

$\displaystyle\int_{0}^{t}\mathbb{E}\left[S^{r}(s)|Y(s)-Y_{m}(s)|^{r}\right]ds$
$\displaystyle\leq
C\int_{0}^{t}\mathbb{E}\left[S^{r}(s)|Z(s)-Z_{m}(s)|^{r}\right]ds$ (3.20)
$\displaystyle\quad+C\int_{0}^{t}\mathbb{E}\left[S^{r}(s)\xi^{r}_{m}\int_{0}^{s}|Z(u)-Z_{m}(u)|^{r}du\right]ds.$

Also, by [19, Theorem 2.6],

$\mathbb{E}\left[\sup_{t\in[0,T]}S^{2r}(t)\right]<\infty,$

so the Hölder inequality as well as the Gaussianity of the random variable
$Z(u)-Z_{m}(u)$ yield

$\displaystyle\int_{0}^{t}$
$\displaystyle\mathbb{E}\left[S^{r}(s)|Z(s)-Z_{m}(s)|^{r}\right]ds$ (3.21)
$\displaystyle\leq\int_{0}^{t}\left(\mathbb{E}\left[S^{2r}(s)\right]\right)^{\frac{1}{2}}\left(\mathbb{E}\left[|Z(s)-Z_{m}(s)|^{2r}\right]\right)^{\frac{1}{2}}ds$
$\displaystyle\leq
C\int_{0}^{t}\left(\int_{0}^{s}(\mathcal{K}(s,u)-\mathcal{K}_{m}(s,u))^{2}du\right)^{\frac{r}{2}}ds$
$\displaystyle\leq
C\lVert\mathcal{K}-\mathcal{K}_{m}\rVert^{r}_{L^{r}([0,T]^{2})}.$

Additionally, (3.5) together with [19, Theorem 2.6] imply that

$\sup_{m\geq 1}\mathbb{E}[\xi^{2r}_{m}\sup_{t\in[0,T]}S^{2r}(t)]<\infty$

hence

$\displaystyle\int_{0}^{t}$
$\displaystyle\mathbb{E}\left[S^{r}(s)\xi^{r}_{m}\int_{0}^{s}|Z(u)-Z_{m}(u)|^{r}du\right]ds$
(3.22)
$\displaystyle\leq\int_{0}^{t}\int_{0}^{s}\left(\mathbb{E}\left[S^{2r}(s)\xi^{2r}_{m}\right]\right)^{\frac{1}{2}}\left(\mathbb{E}\left[|Z(u)-Z_{m}(u)|^{2r}\right]\right)^{\frac{1}{2}}duds$
$\displaystyle\leq
C\int_{0}^{t}\int_{0}^{s}\left(\mathbb{E}\left[|Z(u)-Z_{m}(u)|^{2r}\right]\right)^{\frac{1}{2}}duds$
$\displaystyle\leq
C\int_{0}^{t}\int_{0}^{s}\left(\int_{0}^{u}\left(\mathcal{K}(u,v)-\mathcal{K}_{m}(u,v)\right)^{2}dv\right)^{\frac{r}{2}}duds$
$\displaystyle\leq
C\lVert\mathcal{K}-\mathcal{K}_{m}\rVert^{r}_{L^{r}([0,T]^{2})}.$

Therefore, taking into account (3.18)–(3.22), we see that there exists a
constant $C>0$ that does not depend on $m$ or the particular choice of
$t\in[0,T]$ such that

$\displaystyle\mathbb{E}$
$\displaystyle\left[\sup_{u\in[0,t]}|S(u)-S_{m}(u)|^{r}\right]$ (3.23)
$\displaystyle\leq
C\int_{0}^{t}\mathbb{E}\left[\sup_{u\in[0,s]}|S(u)-S_{m}(u)|^{r}\right]ds+C\lVert\mathcal{K}-\mathcal{K}_{m}\rVert^{r}_{L^{r}([0,T]^{2})},$

and item 1) now follows from the Gronwall’s inequality.

Item 2. If $\mathcal{K}$ and $\mathcal{K}_{m}$ both have the form

$\mathcal{K}(t,s)=\mathcal{K}(t-s)\mathbbm{1}_{s<t},\quad\mathcal{K}_{m}(t,s)=\mathcal{K}_{m}(t-s)\mathbbm{1}_{s<t},\quad
t,s\in[0,T],$

then (3.21)–(3.22) can be rewritten as

$\displaystyle\int_{0}^{t}\mathbb{E}\left[S^{r}(s)|Z(s)-Z_{m}(s)|^{r}\right]ds$
$\displaystyle\leq
C\int_{0}^{t}\left(\int_{0}^{s}(\mathcal{K}(s-u)-\mathcal{K}_{m}(s-u))^{2}du\right)^{\frac{r}{2}}ds$
$\displaystyle\leq
C\int_{0}^{t}\lVert\mathcal{K}-\mathcal{K}_{m}\rVert^{r}_{L^{2}([0,T])}ds$
$\displaystyle\leq
C\lVert\mathcal{K}-\mathcal{K}_{m}\rVert^{r}_{L^{2}([0,T])}$

and

$\displaystyle\int_{0}^{t}$
$\displaystyle\mathbb{E}\left[S^{r}(s)\xi^{r}_{m}\int_{0}^{s}|Z(u)-Z_{m}(u)|^{r}du\right]ds$
$\displaystyle\leq
C\int_{0}^{t}\int_{0}^{s}\left(\int_{0}^{u}\left(\mathcal{K}(u-v)-\mathcal{K}_{m}(u-v)\right)^{2}dv\right)^{\frac{r}{2}}duds$
$\displaystyle\leq
C\int_{0}^{t}\int_{0}^{s}\lVert\mathcal{K}-\mathcal{K}_{m}\rVert^{r}_{L^{2}([0,T])}duds$
$\displaystyle\leq
C\lVert\mathcal{K}-\mathcal{K}_{m}\rVert^{r}_{L^{2}([0,T])}.$

Whence (3.23) becomes

$\displaystyle\mathbb{E}$
$\displaystyle\left[\sup_{u\in[0,t]}|S(u)-S_{m}(u)|^{r}\right]$
$\displaystyle\leq
C\int_{0}^{t}\mathbb{E}\left[\sup_{u\in[0,s]}|S(u)-S_{m}(u)|^{r}\right]ds+C\lVert\mathcal{K}-\mathcal{K}_{m}\rVert^{r}_{L^{2}([0,T])}.$

The required result then again follows from Gronwall’s inequality. ∎

Theorem 3.5 can be easily reformulated in terms of the discounted prices.
Namely, the following Corollary holds.

######  Corollary 3.6.

Let Assumptions (K), (Y) and (Km) hold. Then, for any $r\geq 2$,

  * 1)

there exists a constant $C>0$ that does not depend on $m$ such that

$\mathbb{E}\left[\sup_{t\in[0,T]}|X(t)-X_{m}(t)|^{r}\right]\leq
C\lVert\mathcal{K}-\mathcal{K}_{m}\rVert^{r}_{L^{r}([0,T]^{2})};$

  * 2)

if additionally $\mathcal{K}$ and $\mathcal{K}_{m}$, $m\geq 1$, have the form

$\mathcal{K}(t,s)=\mathcal{K}(t-s)\mathbbm{1}_{s<t},\quad\mathcal{K}_{m}(t,s)=\mathcal{K}_{m}(t-s)\mathbbm{1}_{s<t},\quad
t,s\in[0,T],$

then there exists constant $C>0$ that does not depend on $m$ such that

$\mathbb{E}\left[\sup_{t\in[0,T]}|X(t)-X_{m}(t)|^{r}\right]\leq
C\lVert\mathcal{K}-\mathcal{K}_{m}\rVert^{r}_{L^{2}([0,T])}.$

###  3.2 Error estimates for the non-anticipating derivative

Earlier on, we investigated the error between the original discounted price
$X$ and the price $X_{m}$ obtained by replacing the kernel in the volatility.
Hereafter, we give the error estimates between the non-anticipating derivative
$\mathfrak{D}F$ of $F=F(X(T))$ w.r.t. $X$ and the non-anticipating derivative
$\mathfrak{D}F_{m}$ of $F_{m}=F(X_{m}(T))$ w.r.t. $X_{m}$.

Throughout this Subsection, we always assume that Assumptions (K) and (Y)
hold, the sequence of kernels $\\{\mathcal{K}_{m},~{}m\geq 1\\}$ satisfies
Assumption (Km) and $\mathcal{K}$, $\mathcal{K}_{m}$, $m\geq 1$, are all
difference kernels, i.e.

$\mathcal{K}(t,s)=\mathcal{K}(t-s)\mathbbm{1}_{s<t},\quad\mathcal{K}_{m}(t,s)=\mathcal{K}_{m}(t-s)\mathbbm{1}_{s<t},\quad
t,s\in[0,T].$

We also recall that the payoff function $F=F_{1}+F_{2}$ that we consider
satisfies Assumption (F). To allow for compact writing, we denote

$W(t):=\rho B_{1}(t)+\sqrt{1-\rho^{2}}B_{2}(t),\quad t\in[0,T],$

where $\rho\in(-1,1)$ is from (1.1).

The main results rely also on some technical lemmas which we present
separately.

######  Lemma 3.7.

Let Assumptions (K), (Y) and (Km) hold. For any pair $0\leq t_{1}\leq
t_{2}\leq T$, there exist positive constants $C_{1}<C_{2}$ that do not depend
on $m$ or the particular choice of $t_{1},t_{2}$ such that

$C_{1}(t_{2}-t_{1})X^{2}(t_{1})\leq\mathbb{E}\left[(\Delta
X)^{2}~{}|~{}\mathcal{F}_{t_{1}}\right]\leq C_{2}(t_{2}-t_{1})X^{2}(t_{1})$

and

$C_{1}(t_{2}-t_{1})X_{m}^{2}(t_{1})\leq\mathbb{E}\left[(\Delta
X_{m})^{2}~{}|~{}\mathcal{F}_{t_{1}}\right]\leq
C_{2}(t_{2}-t_{1})X_{m}^{2}(t_{1}).$

Here above $\Delta X:=X(t_{2})-X(t_{1})$, $\Delta
X_{m}:=X_{m}(t_{2})-X_{m}(t_{1})$.

###### Proof.

We only give the proof for the process $X$; the one for $X_{m}$ is identical.
By the very definition of $X$ and Itô lemma, we have

$\displaystyle\mathbb{E}\left[(\Delta X)^{2}~{}|~{}\mathcal{F}_{t_{1}}\right]$
$\displaystyle=\mathbb{E}\left[2\int_{t_{1}}^{t_{2}}(X(s)-X(t_{1}))Y(s)X(s)dW(s)~{}\bigg{|}~{}\mathcal{F}_{t_{1}}\right]$
(3.24)
$\displaystyle\quad+\mathbb{E}\left[\int_{t_{1}}^{t_{2}}Y^{2}(s)X^{2}(s)ds~{}\bigg{|}~{}\mathcal{F}_{t_{1}}\right]$
$\displaystyle=\int_{t_{1}}^{t_{2}}\mathbb{E}\left[Y^{2}(s)X^{2}(s)~{}\big{|}~{}\mathcal{F}_{t_{1}}\right]ds.$

We remark that the above relies on the boundedness of $Y$ and the moments of
$X$.

Let now $s>t_{1}$ be fixed. Note that, with probability 1,

$\displaystyle\mathbb{E}\left[X^{2}(s)~{}\big{|}~{}\mathcal{F}_{t_{1}}\right]\min_{t\in[0,T]}\varphi(t)\leq\mathbb{E}\left[Y^{2}(s)X^{2}(s)~{}\big{|}~{}\mathcal{F}_{t_{1}}\right]\leq\mathbb{E}\left[X^{2}(s)~{}\big{|}~{}\mathcal{F}_{t_{1}}\right]\max_{t\in[0,T]}\psi(t)$
(3.25)

and

$\displaystyle\mathbb{E}$
$\displaystyle\left[X^{2}(s)~{}\big{|}~{}\mathcal{F}_{t_{1}}\right]$
$\displaystyle=\mathbb{E}\left[\exp\left\\{2\int_{0}^{s}Y(u)dW(u)-\int_{0}^{s}Y^{2}(u)du\right\\}~{}\big{|}~{}\mathcal{F}_{t_{1}}\right]$
$\displaystyle=\mathbb{E}\left[\exp\left\\{\int_{0}^{s}2Y(u)dW(u)-\frac{1}{2}\int_{0}^{s}(2Y(u))^{2}du+\int_{0}^{s}Y^{2}(u)du\right\\}~{}\big{|}~{}\mathcal{F}_{t_{1}}\right],$

whence, by the boundedness of $Y$,

$\displaystyle\mathbb{E}\left[X^{2}(s)~{}\big{|}~{}\mathcal{F}_{t_{1}}\right]$
$\displaystyle\leq
e^{T\max_{t\in[0,T]}\psi(t)}\mathbb{E}\left[\exp\left\\{\int_{0}^{s}2Y(u)dW(u)-\frac{1}{2}\int_{0}^{s}(2Y(u))^{2}du\right\\}~{}\big{|}~{}\mathcal{F}_{t_{1}}\right]$
$\displaystyle\mathbb{E}\left[X^{2}(s)~{}\big{|}~{}\mathcal{F}_{t_{1}}\right]$
$\displaystyle\geq\mathbb{E}\left[\exp\left\\{\int_{0}^{s}2Y(u)dW(u)-\frac{1}{2}\int_{0}^{s}(2Y(u))^{2}du\right\\}~{}\big{|}~{}\mathcal{F}_{t_{1}}\right].$

Next, by the Novikov criterion, the process

$\exp\left\\{\int_{0}^{s}2Y(u)dW(u)-\frac{1}{2}\int_{0}^{s}(2Y(u))^{2}du\right\\},\quad
s\in[0,T],$

is a martingale, and hence, for $s>t_{1}$,

$\displaystyle\mathbb{E}$
$\displaystyle\left[\exp\left\\{\int_{0}^{s}2Y(u)dW(u)-\frac{1}{2}\int_{0}^{s}(2Y(u))^{2}du\right\\}~{}\big{|}~{}\mathcal{F}_{t_{1}}\right]$
$\displaystyle=\exp\left\\{\int_{0}^{t_{1}}2Y(u)dW(u)-\frac{1}{2}\int_{0}^{t_{1}}(2Y(u))^{2}du\right\\}$
$\displaystyle=X^{2}(t_{1})\exp\left\\{-\int_{0}^{t_{1}}Y^{2}(s)ds\right\\}.$

Therefore,

$\displaystyle\mathbb{E}\left[\exp\left\\{\int_{0}^{s}2Y(u)dW(u)-\frac{1}{2}\int_{0}^{s}(2Y(u))^{2}du\right\\}~{}\big{|}~{}\mathcal{F}_{t_{1}}\right]$
$\displaystyle\leq X^{2}(t_{1}),$
$\displaystyle\mathbb{E}\left[\exp\left\\{\int_{0}^{s}2Y(u)dW(u)-\frac{1}{2}\int_{0}^{s}(2Y(u))^{2}du\right\\}~{}\big{|}~{}\mathcal{F}_{t_{1}}\right]$
$\displaystyle\geq X^{2}(t_{1})e^{-T\max_{t\in[0,T]}\psi(t)},$

and we can now write

$e^{-T\max_{t\in[0,T]}\psi(t)}X^{2}(t_{1})\leq\mathbb{E}\left[X^{2}(s)~{}\big{|}~{}\mathcal{F}_{t_{1}}\right]\leq
e^{T\max_{t\in[0,T]}\psi(t)}X^{2}(t_{1}).$ (3.26)

Finally, by (3.24)–(3.26),

$C_{1}(t_{2}-t_{1})X^{2}(t_{1})\leq\mathbb{E}\left[(\Delta
X)^{2}~{}|~{}\mathcal{F}_{t_{1}}\right]\leq C_{2}(t_{2}-t_{1})X^{2}(t_{1})$

where

$\displaystyle C_{1}$
$\displaystyle:=\min_{t\in[0,T]}\varphi(t)e^{-T\max_{t\in[0,T]}\psi(t)},$
$\displaystyle C_{2}$
$\displaystyle:=\max_{t\in[0,T]}\psi(t)e^{T\max_{t\in[0,T]}\psi(t)}.$

∎

######  Lemma 3.8.

Let Assumptions (K), (Y) and (Km) hold and both $\mathcal{K}$ and
$\mathcal{K}_{m}$, $m\geq 1$, have the form

$\mathcal{K}(t,s)=\mathcal{K}(t-s)\mathbbm{1}_{s<t},\quad\mathcal{K}_{m}(t,s)=\mathcal{K}_{m}(t-s)\mathbbm{1}_{s<t},\quad
t,s\in[0,T].$

For any $0\leq t_{1}\leq t_{2}\leq T$ and $r\geq 2$, there exists a constant
$C>0$ that does not depend on $m$ or the particular choice of
$t_{1},t_{2}\in[0,T]$ such that

$\mathbb{E}[|\Delta X-\Delta X_{m}|^{r}]\leq
C(t_{2}-t_{1})^{\frac{r}{2}}\lVert\mathcal{K}-\mathcal{K}_{m}\rVert^{r}_{L^{2}([0,T])}$

with $\Delta X:=X(t_{2})-X(t_{1})$, $\Delta X_{m}:=X_{m}(t_{2})-X_{m}(t_{1})$.

###### Proof.

Using uniform boundedness of $Y$ and $Y_{m}$, the Burkholder-Davis-Gundy and
Jensen inequalities as well as Theorem 3.3, one can write:

$\displaystyle\mathbb{E}\left[|\Delta X-\Delta X_{m}|^{r}\right]$
$\displaystyle=\mathbb{E}\left[\left|\int_{t_{1}}^{t_{2}}(Y(s)X(s)-Y_{m}(s)X_{m}(s))dW(s)\right|^{r}\right]$
$\displaystyle\leq
C\mathbb{E}\left[\left|\int_{t_{1}}^{t_{2}}(Y(s)X(s)-Y_{m}(s)X_{m}(s))^{2}ds\right|^{\frac{r}{2}}\right]$
$\displaystyle\leq
C(t_{2}-t_{1})^{\frac{r}{2}-1}\int_{t_{1}}^{t_{2}}\mathbb{E}\left[|Y(s)X(s)-Y_{m}(s)X_{m}(s)|^{r}\right]ds$
$\displaystyle\leq
C(t_{2}-t_{1})^{\frac{r}{2}-1}\int_{t_{1}}^{t_{2}}\mathbb{E}\left[|X(s)-X_{m}(s)|^{r}\right]ds$
$\displaystyle\quad+C(t_{2}-t_{1})^{\frac{r}{2}-1}\int_{t_{1}}^{t_{2}}\mathbb{E}\left[X^{r}(s)|Y(s)-Y_{m}(s)|^{r}\right]ds$
$\displaystyle\leq
C(t_{2}-t_{1})^{\frac{r}{2}-1}\int_{t_{1}}^{t_{2}}\mathbb{E}\left[|X(s)-X_{m}(s)|^{r}\right]ds$
$\displaystyle\quad+C(t_{2}-t_{1})^{\frac{r}{2}-1}\int_{t_{1}}^{t_{2}}\mathbb{E}\left[X^{r}(s)|Z(s)-Z_{m}(s)|^{r}\right]ds$
$\displaystyle\quad+C(t_{2}-t_{1})^{\frac{r}{2}-1}\int_{t_{1}}^{t_{2}}\mathbb{E}\left[X^{r}(s)\xi^{r}_{m}\int_{0}^{s}|Z(u)-Z_{m}(u)|^{r}du\right]ds,$

where the random variable $\xi_{m}$ does not depend on $t_{1}$ or $t_{2}$.
Now, since

$\mathbb{E}\left[\sup_{s\in[0,T]}|X(s)-X_{m}(s)|^{r}\right]\leq
C\lVert\mathcal{K}-\mathcal{K}_{m}\rVert^{r}_{L^{2}([0,T])}$

by Corollary 3.6, it is clear that

$\int_{t_{1}}^{t_{2}}\mathbb{E}\left[|X(s)-X_{m}(s)|^{r}\right]ds\leq
C(t_{2}-t_{1})\lVert\mathcal{K}-\mathcal{K}_{m}\rVert^{r}_{L^{2}([0,T])}.$

Next, (2.6) in Theorem 2.4 implies that there exists a constant $C$ that does
not depend on $t_{1}$ or $t_{2}$ such that

$\int_{t_{1}}^{t_{2}}\mathbb{E}\left[X^{2r}(s)\right]ds<C,$

which yields

$\displaystyle\int_{t_{1}}^{t_{2}}$
$\displaystyle\mathbb{E}\left[X^{r}(s)|Z(s)-Z_{m}(s)|^{r}\right]ds$
$\displaystyle\leq\left(\int_{t_{1}}^{t_{2}}\mathbb{E}\left[X^{2r}(s)\right]ds\right)^{\frac{1}{2}}\left(\int_{t_{1}}^{t_{2}}\mathbb{E}\left[|Z(s)-Z_{m}(s)|^{2r}\right]ds\right)^{\frac{1}{2}}$
$\displaystyle\leq
C\left(\int_{t_{1}}^{t_{2}}\mathbb{E}\left[|Z(s)-Z_{m}(s)|^{2r}\right]ds\right)^{\frac{1}{2}}$
$\displaystyle\leq
C(t_{2}-t_{1})\lVert\mathcal{K}-\mathcal{K}_{m}\rVert^{r}_{L^{2}([0,T])}.$

Similarly, by (2.6) and (3.5) we can write

$\displaystyle\int_{t_{1}}^{t_{2}}$
$\displaystyle\mathbb{E}\left[X^{r}(s)\xi^{r}_{m}\int_{0}^{s}|Z(u)-Z_{m}(u)|^{r}du\right]ds$
$\displaystyle\leq
C\int_{t_{1}}^{t_{2}}\int_{0}^{s}\left(\mathbb{E}\left[|Z(u)-Z_{m}(u)|^{2r}\right]\right)^{\frac{1}{2}}duds$
$\displaystyle\leq
C(t_{2}-t_{1})\lVert\mathcal{K}-\mathcal{K}_{m}\rVert^{r}_{L^{2}([0,T])},$

which finalizes the proof. ∎

We are now ready to prove the main results of this Subsection. Recall that the
payoff function $F=F_{1}+F_{2}$ satisfies Assumption (F), $F:=F(X(T))$.
Consider $F_{m}:=F(X_{m}(T))$ and observe that for any $r\in\mathbb{R}$

$\mathbb{E}\left[F^{r}\right]+\sup_{m\geq
0}\mathbb{E}\left[F_{m}^{r}\right]<\infty$ (3.27)

by Lemma 3.4. Denote also $\Delta X(t_{k}):=X(t_{k+1})-X(t_{k})$ and $\Delta
X_{m}(t_{k}):=X_{m}(t_{k+1})-X_{m}(t_{k})$ and consider the non-anticipating
derivatives given by Definition 2.9:

$\displaystyle\mathfrak{D}F$ $\displaystyle:=\lim_{|\pi|\to 0}u_{\pi},$
$\displaystyle\mathfrak{D}F_{m}$ $\displaystyle:=\lim_{|\pi|\to
0}u^{m}_{\pi},$

where the limit is taken in $L^{2}(\mathbb{P}\times[X])$ and, for a given
partition $\pi=\\{0=t_{0}<t_{1}<...<t_{n}=T\\}$,

$\displaystyle
u_{\pi}:=\sum_{k=0}^{n-1}u_{\pi,k}\mathbbm{1}_{(t_{k},t_{k+1}]},\quad
u^{m}_{\pi}:=\sum_{k=0}^{n-1}u^{m}_{\pi,k}\mathbbm{1}_{(t_{k},t_{k+1}]},$
(3.28)

with

$\displaystyle u_{\pi,k}:=\frac{\mathbb{E}\left[F\Delta
X(t_{k})~{}|~{}\mathcal{F}_{t_{k}}\right]}{\mathbb{E}\left[(\Delta
X(t_{k}))^{2}~{}|~{}\mathcal{F}_{t_{k}}\right]},\quad
u^{m}_{\pi,k}:=\frac{\mathbb{E}\left[F_{m}\Delta
X_{m}(t_{k})~{}|~{}\mathcal{F}_{t_{k}}\right]}{\mathbb{E}\left[(\Delta
X_{m}(t_{k}))^{2}~{}|~{}\mathcal{F}_{t_{k}}\right]}.$ (3.29)

######  Lemma 3.9.

Let Assumptions (K), (Y) and (Km) hold, the payoff function $F$ satisfy
Assumption (F) and both $\mathcal{K}$ and $\mathcal{K}_{m}$, $m\geq 1$, have
the form

$\mathcal{K}(t,s)=\mathcal{K}(t-s)\mathbbm{1}_{s<t},\quad\mathcal{K}_{m}(t,s)=\mathcal{K}_{m}(t-s)\mathbbm{1}_{s<t},\quad
t,s\in[0,T].$

Then there exists a constant $C>0$ that does not depend on $m$ such that for
any partition $\pi=\\{0=t_{0}<t_{1}<...<t_{n}=T\\}$

$\mathbb{E}[|u_{\pi,k}-u^{m}_{\pi,k}|]\leq\frac{C}{\sqrt{t_{k+1}-t_{k}}}\left(\left(\mathbb{E}\left[(F-F_{m})^{4}\right]\right)^{\frac{1}{4}}+\lVert\mathcal{K}-\mathcal{K}_{m}\rVert_{L^{2}([0,T])}\right),\quad
k=0,...,n-1,$

where $u_{\pi,k}$ and $u^{m}_{\pi,k}$ are defined by (3.29).

###### Proof.

Fix $k=0,...,n-1$ and denote $\Delta:=t_{k+1}-t_{k}$. It is easy to see that

$\displaystyle\mathbb{E}$ $\displaystyle[|u_{\pi,k}-u^{m}_{\pi,k}|]$ (3.30)
$\displaystyle\leq\mathbb{E}\left[\frac{|F-F_{m}||\Delta
X(t_{k})|}{\mathbb{E}\left[(\Delta
X(t_{k}))^{2}~{}|~{}\mathcal{F}_{t_{k}}\right]}\right]$
$\displaystyle\quad+\mathbb{E}\left[|F_{m}|\frac{\left|\Delta X(t_{k})-\Delta
X_{m}(t_{k})\right|}{\mathbb{E}\left[(\Delta
X(t_{k}))^{2}~{}|~{}\mathcal{F}_{t_{k}}\right]}\right]$
$\displaystyle\quad+\mathbb{E}\left[|F_{m}|\left|\Delta
X_{m}(t_{k})\right|\frac{\mathbb{E}\left[|(\Delta X_{m}(t_{k}))^{2}-(\Delta
X(t_{k}))^{2}|~{}|~{}\mathcal{F}_{t_{k}}\right]}{\mathbb{E}\left[(\Delta
X(t_{k}))^{2}~{}|~{}\mathcal{F}_{t_{k}}\right]\mathbb{E}\left[(\Delta
X_{m}(t_{k}))^{2}~{}|~{}\mathcal{F}_{t_{k}}\right]}\right]$
$\displaystyle=:I_{1}+I_{2}+I_{3}.$

Now we will deal with each of the above summands separately.

Step 1: estimation of $\boldsymbol{I_{1}}.$ By Lemma 3.7, there exist
constants $C_{1}$, $C_{2}$ (not depending on the partition or $m$) such that

$\mathbb{E}\left[(\Delta
X(t_{k}))^{2}~{}|~{}\mathcal{F}_{t_{k}}\right]\geq\Delta C_{1}X^{2}(t_{k})$

and

$\mathbb{E}\left[(\Delta X(t_{k}))^{2}\right]\leq\Delta
C_{2}\mathbb{E}\left[X^{2}(t_{k})\right].$

Hence, using Hölder inequality and (2.6), one can write

$\displaystyle I_{1}$ $\displaystyle:=\mathbb{E}\left[\frac{|F-F_{m}||\Delta
X(t_{k})|}{\mathbb{E}\left[(\Delta
X(t_{k}))^{2}~{}|~{}\mathcal{F}_{t_{k}}\right]}\right]$ (3.31)
$\displaystyle\leq\frac{C}{\Delta}\mathbb{E}\left[|F-F_{m}||\Delta
X(t_{k})|X^{-2}(t_{k})\right]$
$\displaystyle\leq\frac{C}{\Delta}\left(\mathbb{E}\left[(\Delta
X(t_{k}))^{2}\right]\right)^{\frac{1}{2}}\left(\mathbb{E}[X^{-8}(t_{k})]\right)^{\frac{1}{4}}\left(\mathbb{E}\left[|F-F_{m}|^{4}\right]\right)^{\frac{1}{4}}$
$\displaystyle\leq\frac{C}{\sqrt{\Delta}}\left(\mathbb{E}\left[X^{2}(t_{k})\right]\right)^{\frac{1}{2}}\left(\mathbb{E}\left[|F-F_{m}|^{4}\right]\right)^{\frac{1}{4}}$
$\displaystyle\leq\frac{C}{\sqrt{\Delta}}\left(\mathbb{E}\left[|F-F_{m}|^{4}\right]\right)^{\frac{1}{4}},$

where $C>0$ can be chosen to not depend on $m$ or the partition.

Step 2: estimation of $\boldsymbol{I_{2}}.$ By Hölder inequality, Lemma 3.7,
Lemma 3.8 and (3.27),

$\displaystyle I_{2}$
$\displaystyle:=\mathbb{E}\left[|F_{m}|\frac{\left|\Delta X(t_{k})-\Delta
X_{m}(t_{k})\right|}{\mathbb{E}\left[(\Delta
X(t_{k}))^{2}~{}|~{}\mathcal{F}_{t_{k}}\right]}\right]$ (3.32)
$\displaystyle\leq\frac{C}{\Delta}\left(\mathbb{E}[\left|\Delta
X(t_{k})-\Delta
X_{m}(t_{k})\right|^{2}]\right)^{\frac{1}{2}}\left(\mathbb{E}\left[F^{2}_{m}X^{-4}(t_{k})\right]\right)^{\frac{1}{2}}$
$\displaystyle\leq\frac{C}{\sqrt{\Delta}}\lVert\mathcal{K}-\mathcal{K}_{m}\rVert^{r}_{L^{2}([0,T])},$

where $C$ does not depend on $m$ or the partition.

Step 3: estimation of $\boldsymbol{I_{3}}.$ Using Hölder’s inequality and
(3.27), one can verify that

$\displaystyle I_{3}$ $\displaystyle:=\mathbb{E}\left[|F_{m}|\left|\Delta
X_{m}(t_{k})\right|\frac{\mathbb{E}\left[|(\Delta X_{m}(t_{k}))^{2}-(\Delta
X(t_{k}))^{2}|~{}|~{}\mathcal{F}_{t_{k}}\right]}{\mathbb{E}\left[(\Delta
X(t_{k}))^{2}~{}|~{}\mathcal{F}_{t_{k}}\right]\mathbb{E}\left[(\Delta
X_{m}(t_{k}))^{2}~{}|~{}\mathcal{F}_{t_{k}}\right]}\right]$ (3.33)
$\displaystyle\leq\frac{C}{\Delta^{2}}\left(\mathbb{E}\left[F_{m}^{4}\right]\right)^{\frac{1}{4}}\left(\mathbb{E}\left[|\Delta
X_{m}(t_{k})|^{4}\right]\right)^{\frac{1}{4}}\left(\mathbb{E}\left[|(\Delta
X_{m}(t_{k}))^{2}-(\Delta X(t_{k}))^{2}|^{2}\right]\right)^{\frac{1}{2}}$
$\displaystyle\leq\frac{C}{\Delta^{2}}\left(\mathbb{E}\left[|\Delta
X_{m}(t_{k})|^{4}\right]\right)^{\frac{1}{4}}\left(\mathbb{E}\left[|(\Delta
X_{m}(t_{k}))^{2}-(\Delta X(t_{k}))^{2}|^{2}\right]\right)^{\frac{1}{2}}.$

By the Burkholder–Davis–Gundy and Hölder inequalities, uniform boundedness of
$Y_{m}$ and (2.6),

$\displaystyle\left(\mathbb{E}\left[|\Delta
X_{m}(t_{k})|^{4}\right]\right)^{\frac{1}{4}}$
$\displaystyle=\left(\mathbb{E}\left[\left|\int_{t_{k}}^{t_{k+1}}Y_{m}(s)X_{m}(s)dW(s)\right|^{4}\right]\right)^{\frac{1}{4}}$
(3.34)
$\displaystyle\leq\left(\mathbb{E}\left[\left(\int_{t_{k}}^{t_{k+1}}Y^{2}_{m}(s)X^{2}_{m}(s)ds\right)^{2}\right]\right)^{\frac{1}{4}}$
$\displaystyle\leq\left(\Delta\int_{t_{k}}^{t_{k+1}}\mathbb{E}\left[Y^{4}_{m}(s)X^{4}_{m}(s)\right]ds\right)^{\frac{1}{4}}$
$\displaystyle\leq
C\sqrt{\Delta}\left(\mathbb{E}\left[\sup_{u\in[0,T]}X^{4}_{m}(u)\right]\right)^{\frac{1}{4}}$
$\displaystyle\leq C\sqrt{\Delta}.$

Further, one can verify using Itô formula that

$\displaystyle\mathbb{E}$ $\displaystyle\left[|(\Delta
X_{m}(t_{k}))^{2}-(\Delta X(t_{k}))^{2}|^{2}\right]$ (3.35) $\displaystyle\leq
C\mathbb{E}\left[\left(\int_{t_{k}}^{t_{k+1}}\Big{(}Y^{2}_{m}(s)X^{2}_{m}(s)-Y^{2}(s)X^{2}(s)\Big{)}ds\right)^{2}\right]$
$\displaystyle\quad+C\mathbb{E}\Bigg{[}\bigg{(}2\int_{t_{k}}^{t_{k+1}}\Big{(}(X_{m}(s)-X_{m}(t_{k}))Y_{m}(s)X_{m}(s)$
$\displaystyle\qquad\qquad\qquad\qquad-(X(s)-X(t_{k}))Y(s)X(s)\Big{)}dW(s)\bigg{)}^{2}\Bigg{]}.$

We study the last two summands separately. It is clear that

$\displaystyle\mathbb{E}$
$\displaystyle\left[\left(\int_{t_{k}}^{t_{k+1}}\Big{(}Y^{2}_{m}(s)X^{2}_{m}(s)-Y^{2}(s)X^{2}(s)\Big{)}ds\right)^{2}\right]$
(3.36)
$\displaystyle\leq\Delta\int_{t_{k}}^{t_{k+1}}\mathbb{E}\left[\Big{(}Y^{2}_{m}(s)X^{2}_{m}(s)-Y^{2}(s)X^{2}(s)\Big{)}^{2}\right]ds$
$\displaystyle\leq
C\Delta\int_{t_{k}}^{t_{k+1}}\mathbb{E}\left[\Big{(}X^{2}_{m}(s)-X^{2}(s)\Big{)}^{2}\right]ds$
$\displaystyle\quad+C\Delta\int_{t_{k}}^{t_{k+1}}\mathbb{E}\left[X^{2}(s)\Big{(}Y^{2}_{m}(s)-Y^{2}(s)\Big{)}^{2}\right]ds.$

Taking into account that processes $\\{X^{2}(t),~{}t\in[0,T]\\}$ and
$\\{X_{m}^{2}(t),~{}t\in[0,T]\\}$ satisfy the stochastic differential
equations

$\begin{gathered}dX^{2}(t)=Y^{2}(t)X^{2}(t)dt+2Y(t)X^{2}(t)dW(t),\\\
dX_{m}^{2}(t)=Y_{m}^{2}(t)X_{m}^{2}(t)dt+2Y_{m}(t)X_{m}^{2}(t)dW(t),\end{gathered}$

boundedness of $Y$, uniform boundedness of $\\{Y_{m},~{}m\geq 1\\}$ as well as
the same argument as in Theorem 3.5 imply that there exists a constant $C>0$
such that

$\mathbb{E}\left[\sup_{s\in[0,T]}\Big{(}X^{2}_{m}(s)-X^{2}(s)\Big{)}^{2}\right]ds\leq
C\lVert\mathcal{K}-\mathcal{K}_{m}\rVert^{2}_{L^{2}([0,T])},$

whereas, again like in the proof of Theorem 3.5, we have

$\displaystyle\mathbb{E}\left[X^{2}(s)\Big{(}Y^{2}_{m}(s)-Y^{2}(s)\Big{)}^{2}\right]$
$\displaystyle=\mathbb{E}\left[X^{2}(s)(Y_{m}(s)+Y(s))^{2}(Y_{m}(s)-Y(s))^{2}\right]$
$\displaystyle\leq C\mathbb{E}\left[X^{2}(s)(Y_{m}(s)-Y(s))^{2}\right]$
$\displaystyle\leq
C\lVert\mathcal{K}-\mathcal{K}_{m}\rVert^{2}_{L^{2}([0,T])}.$

Plugging these estimates in (3.36), we obtain that

$\displaystyle\mathbb{E}\left[\left(\int_{t_{k}}^{t_{k+1}}\Big{(}Y^{2}_{m}(s)X^{2}_{m}(s)-Y^{2}(s)X^{2}(s)\Big{)}ds\right)^{2}\right]$
$\displaystyle\leq
C\Delta^{2}\lVert\mathcal{K}-\mathcal{K}_{m}\rVert^{2}_{L^{2}([0,T])}.$

Next, for the second summand of (3.35) we have

$\displaystyle\mathbb{E}$
$\displaystyle\Bigg{[}\bigg{(}\int_{t_{k}}^{t_{k+1}}\Big{(}(X_{m}(s)-X_{m}(t_{k}))Y_{m}(s)X_{m}(s)$
(3.37)
$\displaystyle\qquad\qquad\qquad\qquad-(X(s)-X(t_{k}))Y(s)X(s)\Big{)}dW(s)\bigg{)}^{2}\Bigg{]}$
$\displaystyle=\mathbb{E}\Bigg{[}\int_{t_{k}}^{t_{k+1}}\Big{(}(X_{m}(s)-X_{m}(t_{k}))Y_{m}(s)X_{m}(s)$
$\displaystyle\qquad\qquad\qquad\qquad-(X(s)-X(t_{k}))Y(s)X(s)\Big{)}^{2}ds\Bigg{]}$
$\displaystyle\leq
C\int_{t_{k}}^{t_{k+1}}\mathbb{E}\left[(X(s)-X(t_{k}))^{2}(Y_{m}(s)X_{m}(s)-Y(s)X(s))^{2}\right]ds$
$\displaystyle\quad+C\int_{t_{k}}^{t_{k+1}}\mathbb{E}\left[\Big{(}(X_{m}(s)-X_{m}(t_{k}))-(X(s)-X(t_{k}))\Big{)}^{2}Y^{2}_{m}(s)X^{2}_{m}(s)\right]ds$
$\displaystyle\leq
C\int_{t_{k}}^{t_{k+1}}\left(\mathbb{E}\left[(X(s)-X(t_{k}))^{4}\right]\right)^{\frac{1}{2}}\left(\mathbb{E}\left[(Y_{m}(s)X_{m}(s)-Y(s)X(s))^{4}\right]\right)^{\frac{1}{2}}ds$
$\displaystyle\quad+C\int_{t_{k}}^{t_{k+1}}\left(\mathbb{E}\left[\Big{(}(X_{m}(s)-X_{m}(t_{k}))-(X(s)-X(t_{k}))\Big{)}^{4}\right]\right)^{\frac{1}{2}}\times$
$\displaystyle\qquad\qquad\qquad\qquad\times\left(\mathbb{E}\left[X^{4}_{m}(s)\right]\right)^{\frac{1}{2}}ds.$

Using the same argument as in (3.34), one can verify that there exists a
constant $C>0$ (not depending on $m$ or the partition) such that, for all
$s\in[t_{k},t_{k+1}]$,

$\displaystyle\mathbb{E}\left[(X(s)-X(t_{k}))^{4}\right]\leq C(s-t_{k})^{2}.$
(3.38)

Moreover, using (2.6) and Theorem 3.3, it is easy to show that for any
$s\in[0,T]$

$\displaystyle\mathbb{E}\left[X^{4}(s)(Y_{m}(s)-Y(s))^{4}\right]$
$\displaystyle\leq\left(\mathbb{E}\left[X^{8}(s)\right]\right)^{\frac{1}{2}}\left(\mathbb{E}\left[(Y_{m}(s)-Y(s))^{8}\right]\right)^{\frac{1}{2}}$
$\displaystyle\leq
C\lVert\mathcal{K}-\mathcal{K}_{m}\rVert_{L^{2}([0,T])}^{4},$

whence

$\displaystyle\mathbb{E}\left[(Y_{m}(s)X_{m}(s)-Y(s)X(s))^{4}\right]$
$\displaystyle\leq C\mathbb{E}\left[Y^{4}_{m}(s)(X_{m}(s)-X(s))^{4}\right]$
(3.39)
$\displaystyle\quad+C\mathbb{E}\left[X^{4}(s)(Y_{m}(s)-Y(s))^{4}\right]$
$\displaystyle\leq C\mathbb{E}\left[(X_{m}(s)-X(s))^{4}\right]$
$\displaystyle\quad+C\mathbb{E}\left[X^{4}(s)(Y_{m}(s)-Y(s))^{4}\right]$
$\displaystyle\leq
C\lVert\mathcal{K}-\mathcal{K}_{m}\rVert_{L^{2}([0,T])}^{4}.$

Next, by Lemma 3.8, for all $s\in[t_{k},t_{k+1}]$,

$\displaystyle\mathbb{E}\left[\Big{(}(X_{m}(s)-X_{m}(t_{k}))-(X(s)-X(t_{k}))\Big{)}^{4}\right]$
$\displaystyle\leq
C(s-t_{k})^{2}\lVert\mathcal{K}-\mathcal{K}_{m}\rVert_{L^{2}([0,T])}^{4}.$
(3.40)

Plugging the estimates (3.38)–(3.40) into (3.37), we finally obtain that there
exists a constant $C>0$ (not depending on $m$ or the partition) such that

$\displaystyle\mathbb{E}$
$\displaystyle\Bigg{[}\bigg{(}\int_{t_{k}}^{t_{k+1}}\Big{(}(X_{m}(s)-X_{m}(t_{k}))Y_{m}(s)X_{m}(s)$
$\displaystyle\qquad\qquad\qquad\qquad-(X(s)-X(t_{k}))Y(s)X(s)\Big{)}dW(s)\bigg{)}^{2}\Bigg{]}$
$\displaystyle\leq
C\int_{t_{k}}^{t_{k+1}}(s-t_{k})\lVert\mathcal{K}-\mathcal{K}_{m}\rVert_{L^{2}([0,T])}^{2}ds$
$\displaystyle\quad+C\int_{t_{k}}^{t_{k+1}}(s-t_{k})\lVert\mathcal{K}-\mathcal{K}_{m}\rVert_{L^{2}([0,T])}^{2}\left(\sup_{m\geq
1}\mathbb{E}\left[\sup_{t\in[0,T]}X^{4}_{m}(s)\right]\right)^{\frac{1}{2}}ds$
$\displaystyle\leq
C\Delta^{2}\lVert\mathcal{K}-\mathcal{K}_{m}\rVert_{L^{2}([0,T])}^{2}.$

Taking into account all of the above, we can finally write

$\displaystyle\mathbb{E}\left[|(\Delta X_{m}(t_{k}))^{2}-(\Delta
X(t_{k}))^{2}|^{2}\right]$ $\displaystyle\leq
C\Delta^{2}\lVert\mathcal{K}-\mathcal{K}_{m}\rVert_{L^{2}([0,T])}^{2}$

and whence we have the estimate

$\displaystyle I_{3}$ $\displaystyle:=\mathbb{E}\left[|F_{m}|\left|\Delta
X_{m}(t_{k})\right|\frac{\mathbb{E}\left[|(\Delta X_{m}(t_{k}))^{2}-(\Delta
X(t_{k}))^{2}|~{}|~{}\mathcal{F}_{t_{k}}\right]}{\mathbb{E}\left[(\Delta
X(t_{k}))^{2}~{}|~{}\mathcal{F}_{t_{k}}\right]\mathbb{E}\left[(\Delta
X_{m}(t_{k}))^{2}~{}|~{}\mathcal{F}_{t_{k}}\right]}\right]$
$\displaystyle\leq\frac{C}{\Delta^{2}}\left(\mathbb{E}\left[|\Delta
X_{m}(t_{k})|^{4}\right]\right)^{\frac{1}{4}}\left(\mathbb{E}\left[|(\Delta
X_{m}(t_{k}))^{2}-(\Delta X(t_{k}))^{2}|^{2}\right]\right)^{\frac{1}{2}}$
$\displaystyle\leq\frac{C}{\Delta^{2}}\sqrt{\Delta}\Delta\lVert\mathcal{K}-\mathcal{K}_{m}\rVert_{L^{2}([0,T])}$
$\displaystyle\leq\frac{C}{\sqrt{\Delta}}\lVert\mathcal{K}-\mathcal{K}_{m}\rVert_{L^{2}([0,T])},$

which finalizes the proof. ∎

######  Theorem 3.10.

Let Assumptions (K), (Y) and (Km) hold, the payoff function $F=F_{1}+F_{2}$
satisfies Assumption (F), the kernels $\mathcal{K}$ and $\mathcal{K}_{m}$,
$m\geq 1$, have the form

$\mathcal{K}(t,s)=\mathcal{K}(t-s)\mathbbm{1}_{s<t},\quad\mathcal{K}_{m}(t,s)=\mathcal{K}_{m}(t-s)\mathbbm{1}_{s<t},\quad
t,s\in[0,T],$

and $\lVert\mathcal{K}-\mathcal{K}_{m}\rVert_{L^{2}([0,T])}\to 0$,
$m\to\infty$. Then the following statements hold.

  * 1)

There exists a constant $C>0$ that does not depend on $m$ such that, for any
partition $\pi=\\{0=t_{0}<t_{1}<...<t_{n}=T\\}$ with the mesh
$|\pi|:=\max_{k}|t_{k+1}-t_{k}|$,

$\mathbb{E}\left[\int_{0}^{T}|u_{\pi}(s)-u^{m}_{\pi}(s)|ds\right]\leq
Cn\sqrt{|\pi|}\lVert\mathcal{K}-\mathcal{K}_{m}\rVert_{L^{2}([0,T])}^{\frac{1}{5}},$
(3.41)

where $u_{\pi}$ and $u^{m}_{\pi}$ are defined by (3.28). In particular, if the
partition $\pi$ is uniform, i.e. $t_{k}=\frac{kT}{n}$ and $|\pi|=\frac{T}{n}$,

$\mathbb{E}\left[\int_{0}^{T}|u_{\pi}(s)-u^{m}_{\pi}(s)|ds\right]\leq
C\sqrt{n}\lVert\mathcal{K}-\mathcal{K}_{m}\rVert_{L^{2}([0,T])}^{\frac{1}{5}}.$
(3.42)

  * 2)

If $\\{m_{n},~{}n\geq 1\\}$ is such that

$n\sqrt{|\pi|}\lVert\mathcal{K}-\mathcal{K}_{m_{n}}\rVert_{L^{2}([0,T])}^{\frac{1}{5}}\to
0,\quad n\to\infty,$

then

$\mathbb{E}\left[\int_{0}^{T}|\mathfrak{D}F(s)-u^{m_{n}}_{\pi}(s)|ds\right]\to
0,\quad n\to\infty.$

###### Proof.

By Theorem 2.7, the random variable $X(T)$ has continuous and bounded density,
whence conditions of Theorem 2.8 are fulfilled and

$\displaystyle\mathbb{E}[|F-F_{m}|^{4}]$
$\displaystyle=\mathbb{E}[|F(X(T))-F(X_{m}(T))|^{4}]$ $\displaystyle\leq
C\left(\mathbb{E}[|X(T)-X_{m}(T)|^{4}]+\left(\mathbb{E}\left[|X(T)-X_{m}(T)|^{4}\right]\right)^{\frac{4}{5}}\right)$
$\displaystyle\leq
C\left(\lVert\mathcal{K}-\mathcal{K}_{m}\rVert_{L^{2}([0,T])}^{4}+\lVert\mathcal{K}-\mathcal{K}_{m}\rVert_{L^{2}([0,T])}^{\frac{4}{5}}\right)$
$\displaystyle\leq
C\lVert\mathcal{K}-\mathcal{K}_{m}\rVert_{L^{2}([0,T])}^{\frac{4}{5}},$

where the constant $C$ does not depend on $m$ or the partition. Furthermore,
by Lemma 3.9,

$\displaystyle\mathbb{E}$
$\displaystyle\left[\int_{0}^{T}|u_{\pi}(s)-u^{m}_{\pi}(s)|ds\right]$
$\displaystyle=\sum_{k=0}^{n-1}\mathbb{E}\left[|u_{\pi,k}-u^{m}_{\pi,k}|\right](t_{k+1}-t_{k})$
$\displaystyle\leq
C\left(\sum_{k=0}^{n-1}\sqrt{t_{k+1}-t_{k}}\right)\left(\left(\mathbb{E}[|F-F_{m}|^{4}]\right)^{\frac{1}{4}}+\lVert\mathcal{K}-\mathcal{K}_{m}\rVert_{L^{2}([0,T])}\right)$
$\displaystyle\leq
Cn\sqrt{|\pi|}\left(\lVert\mathcal{K}-\mathcal{K}_{m}\rVert_{L^{2}([0,T])}^{\frac{1}{5}}+\lVert\mathcal{K}-\mathcal{K}_{m}\rVert_{L^{2}([0,T])}\right)$
$\displaystyle\leq
Cn\sqrt{|\pi|}\lVert\mathcal{K}-\mathcal{K}_{m}\rVert_{L^{2}([0,T])}^{\frac{1}{5}},$

which gives item 1).

To prove item 2), first note that

$\displaystyle\mathbb{E}\left[\int_{0}^{T}\left|\mathfrak{D}F(s)-u_{\pi}(s)\right|ds\right]$
$\displaystyle=\mathbb{E}\left[\int_{0}^{T}\left|\mathfrak{D}F(s)-u_{\pi}(s)\right|Y(s)X(s)\frac{1}{Y(s)X(s)}ds\right]$
$\displaystyle\leq\left(\mathbb{E}\left[\int_{0}^{T}\left|\mathfrak{D}F(s)-u_{\pi}(s)\right|^{2}Y^{2}(s)X^{2}(s)ds\right]\right)^{\frac{1}{2}}\times$
$\displaystyle\qquad\times\left(\mathbb{E}\left[\int_{0}^{T}\frac{1}{Y^{2}(s)X^{2}(s)}ds\right]\right)^{\frac{1}{2}}$
$\displaystyle\leq
C\lVert\mathfrak{D}F-u_{\pi}\rVert_{L^{2}(\mathbb{P}\times[X])}\to 0$

as $|\pi|\to 0$. Whence

$\displaystyle\mathbb{E}$
$\displaystyle\left[\int_{0}^{T}|\mathfrak{D}F(s)-u^{m_{n}}_{\pi}(s)|ds\right]$
$\displaystyle\leq\mathbb{E}\left[\int_{0}^{T}\left|\mathfrak{D}F(s)-u_{\pi}(s)\right|ds\right]$
$\displaystyle\qquad+\mathbb{E}\left[\int_{0}^{T}\left|u_{\pi}(s)-u^{m_{n}}_{\pi}(s)\right|ds\right]$
$\displaystyle\leq\mathbb{E}\left[\int_{0}^{T}\left|\mathfrak{D}F(s)-u_{\pi}(s)\right|ds\right]$
$\displaystyle\qquad+Cn\sqrt{|\pi|}\lVert\mathcal{K}-\mathcal{K}_{m_{n}}\rVert_{L^{2}([0,T])}^{\frac{1}{5}}$
$\displaystyle\to 0,\quad n\to\infty.$

By this, the proof is complete. ∎

We remark that that the exponent ${\frac{1}{5}}$ appears in (3.41) and (3.42)
exclusively due the estimate (2.7) of Theorem 2.8 that corresponds to the
(possibly discontinuous) component $F_{2}$ of $F$. If $F_{2}\equiv 0$, i.e.
when

$\displaystyle\mathbb{E}[|F-F_{m}|^{4}]$
$\displaystyle=\mathbb{E}[|F(X(T))-F(X_{m}(T))|^{4}]$ $\displaystyle\leq
C\mathbb{E}[|X(T)-X_{m}(T)|^{4}]$ $\displaystyle\leq
C\lVert\mathcal{K}-\mathcal{K}_{m}\rVert_{L^{2}([0,T])}^{4},$

Theorem 3.10 can be reformulated as follows.

######  Theorem 3.11.

Let Assumptions (K), (Y) and (Km) hold, the payoff function $F=F_{1}$ be
globally Lipschitz, $\mathcal{K}$ and $\mathcal{K}_{m}$, $m\geq 1$, have the
form

$\mathcal{K}(t,s)=\mathcal{K}(t-s)\mathbbm{1}_{s<t},\quad\mathcal{K}_{m}(t,s)=\mathcal{K}_{m}(t-s)\mathbbm{1}_{s<t},\quad
t,s\in[0,T],$

and $\lVert\mathcal{K}-\mathcal{K}_{m}\rVert_{L^{2}([0,T])}\to 0$,
$m\to\infty$. Then the following statements are true.

  * 1)

There exists a constant $C>0$ that does not depend on $m$ such that for any
partition $\pi=\\{0=t_{0}<t_{1}<...<t_{n}=T\\}$ with the mesh
$|\pi|:=\max_{k}|t_{k+1}-t_{k}|$

$\mathbb{E}\left[\int_{0}^{T}|u_{\pi}(s)-u^{m}_{\pi}(s)|ds\right]\leq
Cn\sqrt{|\pi|}\lVert\mathcal{K}-\mathcal{K}_{m}\rVert_{L^{2}([0,T])},$

where $u_{\pi}$ and $u^{m}_{\pi}$ are defined by (3.28). In particular, if
$\pi$ is uniform, i.e. $t_{k}=\frac{kT}{n}$ and $|\pi|=\frac{T}{n}$,

$\mathbb{E}\left[\int_{0}^{T}|u_{\pi}(s)-u^{m}_{\pi}(s)|ds\right]\leq
C\sqrt{n}\lVert\mathcal{K}-\mathcal{K}_{m}\rVert_{L^{2}([0,T])}.$ (3.43)

  * 2)

If $\\{m_{n},~{}n\geq 1\\}$ is such that

$n\sqrt{|\pi|}\lVert\mathcal{K}-\mathcal{K}_{m_{n}}\rVert_{L^{2}([0,T])}\to
0,\quad n\to\infty,$

then

$\mathbb{E}\left[\int_{0}^{T}|\mathfrak{D}F(s)-u^{m_{n}}_{\pi}(s)|ds\right]\to
0,\quad n\to\infty.$

##  4 Markovian approximation of the SVV model

So far, we have obtained estimates of errors in the approximation of the
volatility process, the prices and the optimal hedge generated by changing the
kernel, without any requirements of Markovianity. We now move to a more
specific problem of constructing kernels $\mathcal{K}_{m}$, $m\geq 1$,
providing a Markovian structure.

###  4.1 General construction

Let Assumptions (K) and (Y) hold. Assume that there exists a sequence of
kernels $\\{\mathcal{K}_{m},~{}m\geq 1\\}\subset L^{2}([0,T]^{2})$ such that

  * (i)

$\\{\mathcal{K}_{m},~{}m\geq 1\\}$ satisfies Assumption (Km);

  * (ii)

$\lVert\mathcal{K}-\mathcal{K}_{m}\rVert_{L^{2}([0,T]^{2})}\to 0$,
$m\to\infty$;

  * (iii)

each $\mathcal{K}_{m}$ has the form

$\mathcal{K}_{m}(t,s)=\sum_{i=1}^{m}e_{m,i}(t)f_{m,i}(s),\quad 0\leq s\leq
t\leq T.$

Consider now the sequence of processes $\\{X_{m},~{}m\geq 1\\}$ given by
(3.13). Corollary 3.6 implies that

$\mathbb{E}\left[\sup_{t\in[0,T]}|X(t)-X_{m}(t)|^{2}\right]\to 0,\quad
m\to\infty.$

We can also see that the $(m+2)$-dimensional process
$(X_{m},Y_{m},U_{m,1},...,U_{m,m})$ given by

$\displaystyle X_{m}(t)$
$\displaystyle=X(0)+\int_{0}^{t}Y_{m}(s)X_{m}(s)\left(\rho
dB_{1}(s)+\sqrt{1-\rho^{2}}dB_{2}(s)\right),$ (4.1) $\displaystyle Y_{m}(t)$
$\displaystyle=Y(0)+\int_{0}^{t}b(s,Y_{m}(s))ds+\sum_{i=1}^{m}e_{m,i}(t)U_{m,i}(t),$
$\displaystyle U_{m,1}(t)$ $\displaystyle=\int_{0}^{t}f_{m,1}(s)dB_{1}(s),$
$\displaystyle~{}~{}\vdots$ $\displaystyle U_{m,m}(t)$
$\displaystyle=\int_{0}^{t}f_{m,m}(s)dB_{1}(s),$

is Markovian w.r.t. the filtration
$\mathbb{F}=\\{\mathcal{F}_{t},~{}t\in[0,T]\\}$ generated by $(B_{1},B_{2})$.
In particular, conditional the expectations of the type (2.11) can be
rewritten for the process $X_{m}$ as

$\displaystyle\mathbb{E}$
$\displaystyle\left[F(X_{m}(T))(X_{m}(t_{k+1})-X_{m}(t_{k}))~{}|~{}\mathcal{F}_{t_{k}}\right]$
$\displaystyle=\mathbb{E}\left[F(X_{m}(T))(X_{m}(t_{k+1})-X_{m}(t_{k}))~{}|~{}X_{m}(t_{k}),~{}Y_{m}(t_{k}),~{}U_{m,1}(t_{k}),...,U_{m,m}(t_{k})\right],$
$\displaystyle\mathbb{E}$
$\displaystyle\left[(X_{m}(t_{k+1})-X_{m}(t_{k}))^{2}~{}|~{}\mathcal{F}_{t_{k}}\right]$
$\displaystyle=\mathbb{E}\left[(X_{m}(t_{k+1})-X_{m}(t_{k}))^{2}~{}|~{}X_{m}(t_{k}),~{}Y_{m}(t_{k}),~{}U_{m,1}(t_{k}),...,U_{m,m}(t_{k})\right],$

which significantly simplifies their actual computation. The difficulty here
lies in finding an explicit sequence of kernels $\mathcal{K}_{m}$, $m\geq 1$,
satisfying conditions (i)–(iii) above. In the remaining part of the Section,
we provide such approximating sequences for some widely used kernel types,
namely rough fractional and Hölder continuous kernels.

###  4.2 Rough fractional kernels

Let $H\in\left(0,\frac{1}{2}\right)$ and

$\mathcal{K}(t,s)=\mathcal{K}(t-s)\mathbbm{1}_{s<t}=\frac{(t-s)^{H-\frac{1}{2}}}{\Gamma\left(H+\frac{1}{2}\right)}\mathbbm{1}_{s<t},$
(4.2)

i.e. $Z(t)=\int_{0}^{t}\mathcal{K}(t,s)dB_{1}(s)$ is a RL-fBm as discussed in
Example 2.2. For such a kernel, an appropriate sequence
$\\{\mathcal{K}_{m},~{}m\geq 1\\}$ of $L^{2}$-approximations is presented in
[1, Section 3]. Namely, let $\mu$ be a measure on $\mathbb{R}_{+}$ defined as

$\mu(d\alpha):=\frac{\alpha^{-H-\frac{1}{2}}}{\Gamma\left(H+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}-H\right)}d\alpha,$
$\sigma_{m,i}:=\int_{\tau_{m,i-1}}^{\tau_{m,i}}\mu(d\alpha),\quad\alpha_{m,i}:=\frac{1}{\sigma_{m,i}}\int_{\tau_{m,i-1}}^{\tau_{m,i}}\alpha\mu(d\alpha),\quad
i=1,...,m,$

where $0=\tau_{m,0}<\tau_{m,1}<...<\tau_{m,m}$ is such that

$\tau_{m,m}\to\infty,\quad\sum_{i=1}^{m}\int_{\tau_{m,i-1}}^{\tau_{m,i}}(\alpha_{m,i}-\alpha)^{2}\mu(d\alpha)\to
0,\quad m\to\infty.$ (4.3)

Denote

$\mathcal{K}_{m}(t-s)=\sum_{i=1}^{m}\sigma_{m,i}e^{-\alpha_{m,i}(t-s)},\quad
0\leq s<t\leq T,\quad m\geq 1.$ (4.4)

By [1, Lemma 5.2], the sequence $\\{\mathcal{K}_{m},~{}m\geq 1\\}$ satisfies
Assumption (Km) and, moreover, by [1, Proposition 3.3],

$\lVert\mathcal{K}-\mathcal{K}_{m}\rVert_{L^{2}([0,T])}\to 0,\quad
m\to\infty.$

Then, by item 2) of Corollary 3.6, for any $r\geq 2$

$\mathbb{E}\left[\sup_{t\in[0,T]}|X(t)-X_{m}(t)|^{r}\right]\leq
C\lVert\mathcal{K}-\mathcal{K}_{m}\rVert_{L^{2}([0,T])}^{r}.$

The values $0=\tau_{m,0}<\tau_{m,1}<...<\tau_{m,m}$ satisfying (4.3) can be
chosen in several ways. In [1, Subsection 3.2], it is suggested to put

$\tau_{m,i}:=\frac{1}{T}\left(\frac{\sqrt{10}(1-2H)}{5-2H}\right)^{\frac{2}{5}}im^{-\frac{1}{5}},\quad
i=0,1,2,...,m.$ (4.5)

In this case,

$\lVert\mathcal{K}-\mathcal{K}_{m}\rVert_{L^{2}([0,T])}\leq
C_{H}m^{-\frac{4H}{5}},$

where $C_{H}$ is a positive constant that depends only on the Hurst parameter
$H\in\left(0,\frac{1}{2}\right)$. In this case, Theorem 3.10 and Theorem 3.11
read as follows.

######  Corollary 4.1.

Let $\mathcal{K}$ be defined by (4.2) for some
$H\in\left(0,\frac{1}{2}\right)$, Assumption (Y) hold, $\mathcal{K}_{m}$ be
defined by (4.4) with $\tau_{m,i}$, $i=0,...,m$, $m\geq 1$, given by (4.5).
Let $\pi=\\{0=t_{0}<t_{1}<...<t_{n}=T\\}$, $t_{k}=\frac{kT}{n}$, be a uniform
partition of $[0,T]$ and consider the corresponding $u_{\pi}$, $u^{m}_{\pi}$
from (3.28).

  * 1)

If $F=F_{1}+F_{2}$ satisfies Assumption (F) and $m_{n}:=n^{\alpha}$ for
$\alpha>\frac{25}{8H}$, then

$\mathbb{E}\left[\int_{0}^{T}|u_{\pi}(s)-u^{m_{n}}_{\pi}(s)|ds\right]\leq
Cn^{-\frac{4\alpha H}{25}+\frac{1}{2}}\to 0,\quad n\to\infty.$ (4.6)

  * 2)

If $F=F_{1}$ is globally Lipschitz and $m_{n}:=n^{\alpha}$ for
$\alpha>\frac{5}{8H}$, then

$\mathbb{E}\left[\int_{0}^{T}|u_{\pi}(s)-u^{m_{n}}_{\pi}(s)|ds\right]\leq
Cn^{-\frac{4\alpha H}{5}+\frac{1}{2}}\to 0,\quad n\to\infty.$ (4.7)

######  Remark 4.2.

Putting $e_{m,i}(t):=\sigma_{m,i}e^{-\alpha_{m,i}t}$,
$f_{m,i}(s):=e^{\alpha_{m,i}s}$, we obtain that each process of the form

$e_{m,i}(t)U_{m,i}(t)=\sigma_{m,i}\int_{0}^{t}e^{-\alpha_{m,i}(t-s)}dB_{1}(s)=:V_{m,i}(t),\quad
t\in[0,T],$

from (4.1) is an Ornstein-Uhlenbeck process satisfying the SDE

$V_{m,i}(t)=-\alpha_{m,i}\int_{0}^{t}V_{m,i}(s)ds+\sigma_{m,i}B_{1}(t),\quad
t\in[0,T].$

###  4.3 Hölder continuous kernels

Assume that $\mathcal{K}(t,s)=\mathcal{K}(t-s)\mathbbm{1}_{s<t}$ with
$\mathcal{K}(0)=0$ and $\mathcal{K}\in C^{H}([0,T])$ for some $H\in(0,1)$,
i.e.

$|\mathcal{K}(t)-\mathcal{K}(s)|\leq C_{H}|t-s|^{H},\quad s,t\in[0,T].$ (4.8)

######  Example 4.3.

The fractional kernel

$\mathcal{K}(t-s)=\frac{(t-s)^{H-\frac{1}{2}}}{\Gamma\left(H+\frac{1}{2}\right)},\quad
0\leq s\leq t\leq T,$

with $H\in\left(\frac{1}{2},1\right)$ is Hölder continuous of order
$H-\frac{1}{2}$ and thus fits in the framework of this Subsection. However,
note that the corresponding RL-fBM
$Z(t)=\frac{1}{\Gamma\left(H+\frac{1}{2}\right)}\int_{0}^{t}(t-s)^{H-\frac{1}{2}}dB_{1}(s)$
has higher regularity and is $\lambda$-Hölder continuous for all
$\lambda\in(0,H)$.

Let $\mathcal{K}_{m}$ be the corresponding Bernstein polynomial of order $m$
defined as

$\displaystyle\mathcal{K}_{m}(t)$
$\displaystyle=\frac{1}{T^{m}}\sum_{i=0}^{m}\mathcal{K}\left(\frac{Ti}{m}\right)\binom{m}{i}t^{i}(T-t)^{m-i}$
(4.9)
$\displaystyle=\sum_{i=0}^{m}\left(\frac{1}{T^{i}}\sum_{j=0}^{i}(-1)^{i-j}\mathcal{K}\left(\frac{Tj}{m}\right)\binom{m}{j}\binom{m-j}{i-j}\right)t^{i}$
$\displaystyle=\sum_{i=0}^{m}\kappa_{m,i}t^{i}.$

By [34, Proposition 2],

$|\mathcal{K}_{m}(t)-\mathcal{K}_{m}(s)|\leq C_{H}|t-s|^{H},\quad
s,t\in[0,T],$

where $C_{H}$ is the same as in (4.8) for all $m\geq 1$. Since
$\mathcal{K}(0)=0$, $\mathcal{K}_{m}(0)=\kappa_{m,0}=0$ and we can write for
any $0\leq s\leq t\leq T$

$\displaystyle\int_{0}^{t}(\mathcal{K}_{m}(t-u)$
$\displaystyle-\mathcal{K}_{m}(s-u))^{2}du+\int_{s}^{t}\mathcal{K}_{m}^{2}(t-u)du$
$\displaystyle\leq
C_{H}^{2}T|t-s|^{2H}+\int_{0}^{t-s}\mathcal{K}_{m}^{2}(v)dv$
$\displaystyle\leq C_{H}^{2}T|t-s|^{2H}+C_{H}^{2}\int_{0}^{t-s}v^{2H}dv$
$\displaystyle\leq C_{H}^{2}T|t-s|^{2H}+\frac{C_{H}^{2}}{2H+1}|t-s|^{2H+1}$
$\displaystyle\leq C|t-s|^{2H},$

where $C$ does not depend on $m$, i.e. Assumption (Km) holds. Moreover, [34,
Theorem 1] gives the estimate

$\sup_{t\in[0,T]}|\mathcal{K}(t)-\mathcal{K}_{m}(t)|\leq Cm^{-\frac{H}{2}},$

which implies that

$\lVert\mathcal{K}-\mathcal{K}_{m}\rVert_{L^{2}([0,T])}\leq
Cm^{-\frac{H}{2}}\to 0,\quad m\to\infty,$

and whence for any $r\geq 2$

$\mathbb{E}\left[\sup_{t\in[0,T]}|X(t)-X_{m}(t)|^{r}\right]\leq
Cm^{-\frac{rH}{2}}\to 0,\quad m\to\infty,$

by Corollary 3.6. In particular, Theorem 3.10 and Theorem 3.11 imply the
following result.

######  Corollary 4.4.

Let $\mathcal{K}$ be a Hölder continuous difference kernel of order
$H\in(0,1)$ such that $\mathcal{K}(0)=0$, Assumption (Y) hold and
$\mathcal{K}_{m}$ be defined by (4.9). Let
$\pi=\\{0=t_{0}<t_{1}<...<t_{n}=T\\}$ be a uniform partition of $[0,T]$,
$t_{k}=\frac{kT}{n}$, and consider the corresponding $u_{\pi}$, $u^{m}_{\pi}$
from (3.28).

  * 1)

If $F=F_{1}+F_{2}$ satisfies Assumption (F) and $m_{n}:=n^{\alpha}$ for
$\alpha>\frac{5}{H}$, then

$\mathbb{E}\left[\int_{0}^{T}|u_{\pi}(s)-u^{m_{n}}_{\pi}(s)|ds\right]\leq
Cn^{-\frac{\alpha H}{10}+\frac{1}{2}}\to 0,\quad n\to\infty.$ (4.10)

  * 2)

If $F=F_{1}$ is globally Lipschitz and $m_{n}:=n^{\alpha}$ for
$\alpha>\frac{1}{H}$, then

$\mathbb{E}\left[\int_{0}^{T}|u_{\pi}(s)-u^{m_{n}}_{\pi}(s)|ds\right]\leq
Cn^{-\frac{\alpha H}{2}+\frac{1}{2}}\to 0,\quad n\to\infty.$ (4.11)

######  Remark 4.5.

Since

$\displaystyle\mathcal{K}_{m}(t-s)$
$\displaystyle=\sum_{k=0}^{m}\kappa_{m,k}(t-s)^{k}=\sum_{k=0}^{m}\sum_{i=0}^{k}(-1)^{k-i}\binom{k}{i}\kappa_{m,k}t^{i}s^{k-i}$
$\displaystyle=\sum_{i=0}^{m}\left(\sum_{k=i}^{m}(-1)^{k-i}\binom{k}{i}\kappa_{m,k}s^{k-i}\right)t^{i}$

the processes $U_{m,i}$ from (4.1) will be of the form

$U_{m,i}(t)=t^{i}\int_{0}^{t}\left(\sum_{k=i}^{m}\binom{k}{i}\kappa_{m,k}s^{k-i}\right)dB_{1}(s),\quad
i=0,1,...,m.$

######  Remark 4.6.

Note that each process

$Z_{m}(t)=\int_{0}^{t}\mathcal{K}_{m}(t-s)dB_{1}(s)=\sum_{i=1}^{m}\kappa_{m,i}\int_{0}^{t}(t-s)^{i}dB_{1}(s),\quad
t\in[0,T],$

is Hölder continuous of any order $\lambda\in(0,1)$, i.e. the solution to the
SDE

$Y_{m}(t)=Y(0)+\int_{0}^{t}b(s,Y_{m}(s))ds+Z_{m}(t),\quad t\in[0,T],$

exists and is unique for any value $\gamma>0$ in (Y3).

##  5 Monte Carlo approximation of the optimal hedge and simulations

We complete our study with simulations. In order to simulate paths of
sandwiched processes, we use the drift-implicit Euler approximation scheme
from [18]. The original and the approximated discounted price processes $X$
and $X_{m}$ are simulated just like in [19]. We also present simulations of
the hedging strategies for a standard European call option. Note that
algorithms presented below also work for exotic contracts with discontinuous
payoffs, but slower convergence rates require longer computations. All
simulations were performed in R programming language on the system with Intel
Core i9-9900K CPU and 64 Gb RAM.

###  5.1 Nested Monte Carlo method

A straightforward way to compute the conditional expectations

$\displaystyle\mathbb{E}$
$\displaystyle\left[F(X_{m}(T))(X_{m}(t_{k+1})-X_{m}(t_{k}))~{}|~{}X_{m}(t_{k}),~{}Y_{m}(t_{k}),~{}U_{m,1}(t_{k}),...,U_{m,m}(t_{k})\right]$
$\displaystyle\quad=:\Phi_{1}\big{(}t_{k},X_{m}(t_{k}),Y_{m}(t_{k}),U_{m,1}(t_{k}),...,U_{m,m}(t_{k})\big{)},$
$\displaystyle\mathbb{E}$
$\displaystyle\left[(X_{m}(t_{k+1})-X_{m}(t_{k}))^{2}~{}|~{}X_{m}(t_{k}),~{}Y_{m}(t_{k}),~{}U_{m,1}(t_{k}),...,U_{m,m}(t_{k})\right]$
$\displaystyle\quad=:\Phi_{2}\big{(}t_{k},X_{m}(t_{k}),Y_{m}(t_{k}),U_{m,1}(t_{k}),...,U_{m,m}(t_{k})\big{)}$

is the Nested Monte Carlo (NMC) approach that can be summarized as follows
(see also Fig. 1):

Figure 1: Nested Monte Carlo approach, $t_{k}=0.8$, $T=1$. Given the values
$X_{m}(t_{k})$, $Y_{m}(t_{k})$, $U_{m,1}(t_{k})$, …, $U_{m,m}(t_{k})$, we
simulate a number of trajectories $X^{(i)}_{m}$ on $(t_{k},T]$ (blue), use
each of those to compute
$F(X^{(i)}_{m}(T))(X^{(i)}_{m}(t_{k+1})-X^{(i)}_{m}(t_{k}))$ and
$(X^{(i)}_{m}(t_{k+1})-X^{(i)}_{m}(t_{k}))^{2}$. The means of the latter
values are then used as approximations of the required conditional
expectations.

  * 1)

given $X_{m}(t_{k})=x$, $Y_{m}(t_{k})=y$, $U_{m,1}(t_{k})=u_{1}$, …,
$U_{m,m}(t_{k})=u_{m}$, simulate $N$ independent trajectories
$\\{X^{(i)}_{m}(t),~{}t\in(t_{k},T]\\}$, $i=1,...,N$;

  * 2)

for each trajectory, compute
$F(X^{(i)}_{m}(T))(X^{(i)}_{m}(t_{k+1})-X^{(i)}_{m}(t_{k}))$ and
$(X^{(i)}_{m}(t_{k+1})-X^{(i)}_{m}(t_{k}))^{2}$, $i=1,...,N$;

  * 3)

put

$\displaystyle\widehat{\Phi}_{1}(t_{k},x,y,u_{1},...,u_{m})$
$\displaystyle:=\frac{1}{N}\sum_{i=1}^{N}F(X^{(i)}_{m}(T))(X^{(i)}_{m}(t_{k+1})-X^{(i)}_{m}(t_{k})),$
$\displaystyle\widehat{\Phi}_{2}(t_{k},x,y,u_{1},...,u_{m})$
$\displaystyle:=\frac{1}{N}\sum_{i=1}^{N}(X^{(i)}_{m}(t_{k+1})-X^{(i)}_{m}(t_{k}))^{2}.$

######  Example 5.1.

(Hölder continuous kernel) Consider the SVV model

$\displaystyle X(t)$ $\displaystyle=X(0)+\int_{0}^{t}Y(s)X(s)\left(\rho
dB_{1}(s)+\sqrt{1-\rho^{2}}dB_{2}(s)\right),$ $\displaystyle Y(t)$
$\displaystyle=Y(0)+\int_{0}^{t}b(s,Y(s))ds+\int_{0}^{t}\mathcal{K}(t,s)dB_{1}(s),$

with $X(0)=5$, $Y(0)=1$, $\rho=0.5$,
$b(t,y)=\frac{1}{(y-0.01)^{4}}-\frac{1}{(5-y)^{4}}$ and
$\mathcal{K}(t,s)=\mathcal{K}(t-s)=(t-s)^{0.4}$. The approximation
$(X_{m},Y_{m},U_{m,0},U_{m,1},...,U_{m,m})$ is constructed using the Bernstein
polynomials as described in Subsection 4.3.

(a) Kernels $\mathcal{K}$ and $\mathcal{K}_{m}$

(b) Volterra noises $Z$ and $Z_{m}$

(c) Volatility processes $Y$ and $Y_{m}$

(d) Price processes $X$ and $X_{m}$

Figure 2: Approximation of the SVV model with Hölder continuous kernel

Figure 2 illustrates approximation of the SVV model; original kernel and
generated sample paths of the corresponding processes are depicted in black
whereas approximations using the Bernstein polynomials are depicted in red (of
order $m=10$) and blue (of order $m=30$). Note that the path of $X$ simulated
using the original noise (black) is not visible on Figure 2(d) due to the high
degree of overlapping with the approximations (red and blue trajectories).
Figure 3 contains a path of the process

$\displaystyle\widehat{u}_{\pi}(t)$
$\displaystyle:=\sum_{k=0}^{9}\frac{\widehat{\Phi}_{1}\left(\frac{k}{10},X_{m}\left(\frac{k}{10}\right),Y_{m}\left(\frac{k}{10}\right),U_{m,0}\left(\frac{k}{10}\right),...,U_{m,m}\left(\frac{k}{10}\right)\right)}{\widehat{\Phi}_{2}\left(\frac{k}{10},X_{m}\left(\frac{k}{10}\right),Y_{m}\left(\frac{k}{10}\right),U_{m,0}\left(\frac{k}{10}\right),...,U_{m,m}\left(\frac{k}{10}\right)\right)}\mathbbm{1}_{\left(\frac{k}{10},\frac{k+1}{10}\right]}(t),$
(5.1)

with the European call payoff function $F(x):=\max\\{x-4,0\\}$ and the values
$X_{m}\left(\frac{k}{10}\right)$, $Y_{m}\left(\frac{k}{10}\right)$,
$U_{m,0}\left(\frac{k}{10}\right)$, …, $U_{m,m}\left(\frac{k}{10}\right)$
coming exactly from the trajectory depicted on Figure 2. In order to estimate

$\displaystyle\widehat{\Phi}_{1}\left(\frac{k}{10},X_{m}\left(\frac{k}{10}\right),Y_{m}\left(\frac{k}{10}\right),U_{m,0}\left(\frac{k}{10}\right),...,U_{m,m}\left(\frac{k}{10}\right)\right),$
$\displaystyle\widehat{\Phi}_{2}\left(\frac{k}{10},X_{m}\left(\frac{k}{10}\right),Y_{m}\left(\frac{k}{10}\right),U_{m,0}\left(\frac{k}{10}\right),...,U_{m,m}\left(\frac{k}{10}\right)\right),$

100000 simulations were used for each $k=0,1,...,9$.

Figure 3: Hedging strategy estimated for the path from Fig. 2 for $m=10$ (red)
and $m=30$ (blue). The corresponding partition is $k/10$, $k=0,1,...,10$.
Simulating the red line took 63231 seconds whereas the blue line – 73410
seconds.

######  Example 5.2.

(Rough kernel) Consider the SVV model

$\displaystyle X(t)$ $\displaystyle=X(0)+\int_{0}^{t}Y(s)X(s)\left(\rho
dB_{1}(s)+\sqrt{1-\rho^{2}}dB_{2}(s)\right),$ $\displaystyle Y(t)$
$\displaystyle=Y(0)+\int_{0}^{t}b(s,Y(s))ds+\int_{0}^{t}\mathcal{K}(t,s)dB_{1}(s),$

with $X(0)=5$, $Y(0)=1$, $\rho=0.5$,
$b(t,y)=\frac{1}{(y-0.01)^{4}}-\frac{1}{(5-y)^{4}}$ and
$\mathcal{K}(t,s)=\mathcal{K}(t-s)=\frac{1}{\Gamma(0.8)(t-s)^{0.2}}$. The
approximation $(X_{m},Y_{m},U_{m,1},...,U_{m,m})$ is constructed by
approximating the Volterra noise
$Z(t):=\frac{1}{\Gamma(0.8)}\int_{0}^{t}\frac{1}{(t-s)^{0.2}}dB_{1}(s)$ by a
linear combination of standard Ornstein-Uhlenbeck processes as described in
Subsection 4.2.

(a) Kernels $\mathcal{K}$ and $\mathcal{K}_{m}$

(b) Volterra noises $Z$ and $Z_{m}$

(c) Volatility processes $Y$ and $Y_{m}$

(d) Price processes $X$ and $X_{m}$

Figure 4: Approximation of the SVV model with rough fractional kernel

Just as in Example 5.1, Figure 4 illustrates approximation of the SVV model;
original kernel and generated sample paths of the corresponding processes are
depicted in black whereas approximations are depicted in red ($m=10$
summands), green ($m=100$ summands) and blue ($m=1000$ summands). Note that
rough volatility requires much higher values of $m$ in order to ensure a
decent level of approximation in comparison to Example 5.1.

Figure 5 contains a path of the process

$\displaystyle\widehat{u}_{\pi}(t)$
$\displaystyle:=\sum_{k=0}^{9}\frac{\widehat{\Phi}_{1}\left(\frac{k}{10},X_{m}\left(\frac{k}{10}\right),Y_{m}\left(\frac{k}{10}\right),U_{m,0}\left(\frac{k}{10}\right),...,U_{m,m}\left(\frac{k}{10}\right)\right)}{\widehat{\Phi}_{2}\left(\frac{k}{10},X_{m}\left(\frac{k}{10}\right),Y_{m}\left(\frac{k}{10}\right),U_{m,0}\left(\frac{k}{10}\right),...,U_{m,m}\left(\frac{k}{10}\right)\right)}\mathbbm{1}_{\left(\frac{k}{10},\frac{k+1}{10}\right]}(t),$
(5.2)

$t\in[0,1]$, where the payoff function $F(x):=\max\\{x-4,0\\}$ and the values
$X_{m}\left(\frac{k}{10}\right)$, $Y_{m}\left(\frac{k}{10}\right)$,
$U_{m,1}\left(\frac{k}{10}\right)$, …, $U_{m,m}\left(\frac{k}{10}\right)$ are
exactly the ones from the trajectory depicted on Figure 4. In order to
estimate

$\displaystyle\widehat{\Phi}_{1}\left(\frac{k}{10},X_{m}\left(\frac{k}{10}\right),Y_{m}\left(\frac{k}{10}\right),U_{m,0}\left(\frac{k}{10}\right),...,U_{m,m}\left(\frac{k}{10}\right)\right),$
$\displaystyle\widehat{\Phi}_{2}\left(\frac{k}{10},X_{m}\left(\frac{k}{10}\right),Y_{m}\left(\frac{k}{10}\right),U_{m,0}\left(\frac{k}{10}\right),...,U_{m,m}\left(\frac{k}{10}\right)\right),$

100000 simulations were used for each $k=0,1,...,9$.

Figure 5: Hedging strategy estimated for the path from Fig. 4 for $m=10$
(red), $m=100$ (green), $m=1000$ (blue) and $m=2000$ (black). The
corresponding partition is $k/10$, $k=0,1,...,10$. The figure also illustrates
slower rate of convergence in comparison to the Hölder kernel case: the black
and blue lines are close to each other but the red and green lines
(corresponding to relatively low values of $m$) differ substantially.
Computation time: 24162 seconds for $m=10$, 25348 seconds for $m=100$, 38530
seconds for $m=1000$ and 42431 seconds for $m=2000$.

###  5.2 Least squares Monte Carlo method

Despite its simplicity and clear theoretical justification, the nested Monte
Carlo approach has a substantial disadvantage: it takes long to compute and
thus requires powerful computational resources in order to be used in
practice. In order to overcome this issue, one can use the Least Squares Monte
Carlo (LSMC) method instead:

  * 1)

simulate $N$ independent realizations

$(X^{(i)}_{m},Y^{(i)}_{m},U^{(i)}_{m,1},...,U^{(i)}_{m,m})=\left\\{\big{(}X^{(i)}_{m}(t),Y^{(i)}_{m}(t),U^{(i)}_{m,1}(t),...,U^{(i)}_{m,m}(t)\big{)},~{}t\in[0,T]\right\\},$

$i=1,...,N$;

  * 2)

for each path, evaluate

$\big{(}X^{(i)}_{m}(t_{k}),Y^{(i)}_{m}(t_{k}),U^{(i)}_{m,1}(t_{k}),...,U^{(i)}_{m,m}(t_{k})\big{)},$
$F(X^{(i)}_{m}(T))(X^{(i)}_{m}(t_{k+1})-X^{(i)}_{m}(t_{k}))$

and

$(X^{(i)}_{m}(t_{k+1})-X^{(i)}_{m}(t_{k}))^{2},$

$i=1,...,N$;

  * 3)

apply an appropriate non-parametric regression (with a mean squared error loss
function) to the generated “dataset” treating

$\big{(}X^{(i)}_{m}(t_{k}),Y^{(i)}_{m}(t_{k}),U^{(i)}_{m,1}(t_{k}),...,U^{(i)}_{m,m}(t_{k})\big{)},\quad
i=1,...,N,$

as “input” and

$F(X^{(i)}_{m}(T))(X^{(i)}_{m}(t_{k+1})-X^{(i)}_{m}(t_{k})),\quad(X^{(i)}_{m}(t_{k+1})-X^{(i)}_{m}(t_{k}))^{2},\quad
i=1,...,N,$

as “output” variables. The resulting estimates
$\widehat{\Phi}_{1}(t_{k},\cdot)$, $\widehat{\Phi}_{2}(t_{k},\cdot)$ of the
regression functions are then used to calculate $\widehat{u}_{\pi}$ on the
interval $(t_{k},t_{k+1}]$.

######  Remark 5.3.

The Least Squares Monte Carlo method described above can be regarded as a
simplified version of the approach suggested in [11, 33, 38] for pricing
American options or in [30, 31, 32] for modeling of life insurance companies.

######  Example 5.4.

Fig. 6 contains approximations of the optimal hedging strategy constructed for
the path of $X_{m}$ from Example 5.1 that corresponds to $m=10$. In order to
obtain the dark green line, we simulated the “dataset” containing the “input”
variables

$\big{(}X^{(i)}_{m}(t_{k}),Y^{(i)}_{m}(t_{k}),U^{(i)}_{m,0}(t_{k}),...,U^{(i)}_{m,m}(t_{k})\big{)},\quad
i=1,...,N,$

together with the corresponding “output” variables

$F(X^{(i)}_{m}(T))(X^{(i)}_{m}(t_{k+1})-X^{(i)}_{m}(t_{k})),\quad(X^{(i)}_{m}(t_{k+1})-X^{(i)}_{m}(t_{k}))^{2},\quad
i=1,...,N,$

with $N=1000000$ for each $k=0,1,...,9$. In order to compute
$\widehat{\Phi}_{1}(t_{k},\cdot)$, $\widehat{\Phi}_{2}(t_{k},\cdot)$, we used
the idea from [31] and utilized a neural network with 3 hidden layers and 20
nods in each layer. The red line is exactly the path of $\hat{u}_{\pi}$ for
$m=10$ from Fig. 3 and is treated as a reference.

Figure 6: Approximation of the optimal hedge using the Nested Monte Carlo
method for $m=10$ from Fig. 3 (red) and using the Least Squares Monte Carlo
method (dark green).

Note that the Nested approach described in Example 5.1 required 63231 seconds
to simulate a path of $\hat{u}_{\pi}$ corresponding to a single realization of
$(X_{m},Y_{m},U_{m,0},...U_{m,m})$ whereas the Least Squares modification took
roughly 6 days to simulate the initial dataset and about 5 hours to fit the
neural network. Once ready, the actual computations were conducted almost
instantly.

######  Remark 5.5.

The neural network architecture in Example 5.4 is not optimal. It is fairly
clear from Fig. 6: even though the LSMC Monte Carlo estimate preserves in
general the shape of the reference optimal hedge simulated using the NMC
approach, the difference between the two is still quite substantial. We
emphasize that performance of the NMC method heavily depends on the chosen
non-parametric regression method, and therefore a separate investigation on
performance of different regression approaches is required.

######  Remark 5.6.

Similar method can theoretically be applied for the SVV model with a rough
fractional kernel. However, as noted in Example 5.2, one may need a very high
dimensionality of the “input” variables vector (over 1000) which requires a
dataset with possibly unrealistically huge number of observations.

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