# Markovian lifting and optimal control for integral stochastic Volterra
equations with completely monotone kernels

Stefano Bonaccorsi
Department of Mathematics
Università degli Studi di Trento
via Sommarive 14, 38121 Trento, Italy
<EMAIL_ADDRESS>
& Fulvia Confortola
Department of Mathematics
Politecnico di Milano
via Bonardi 9, 20133 Milano, Italy
<EMAIL_ADDRESS>
Stefano Bonaccorsi and Fulvia Confortola are members of the group
Gnampa–Indam. This research is part of the activities of the PRIN Project
“Stochastic particle-based anomalous reaction-diffusion models with
heterogeneous interaction for radiation therapy”

###### Abstract

In this paper, we focus on solving the optimal control problem for integral
stochastic Volterra equations in a finite dimensional setting. In our setting,
the noise term is driven by a pure jump Lévy noise and the control acts on the
intensity of the jumps.

We use recent techniques proposed by Hamaguchi, where a crucial requirement is
that the convolution kernel should be a completely monotone function. This
allows us to use Bernstein’s representation and the machinery of Laplace
transform to obtain a Markovian lift.

It is natural that the Markovian lift, in whatever form constructed,
transforms the state equation into a stochastic differential equation in an
infinite-dimensional space. This space should be large enough to contain all
the information about the history of the process. Hence, although the original
equation is taken in a finite dimensional space, the resulting lift is always
infinite dimensional.

We solve the problem by using the forward-backward approach in the infinite-
dimensional setting and prove the existence of the optimal control for the
original problem. Under additional assumptions on the coefficients, we see
that a control in closed-loop form can be achieved.

##  1 Introduction

This paper aims to study an optimal control problem for a class of integral
stochastic Volterra equations in a finite-dimensional space $\mathbb{R}^{d}$,
where the equation is driven by a pure jump Lévy noise and the control acts on
the intensity of the jumps, a setting that resembles what already proposed in
[6].

Let us introduce the key ingredients of our construction. Let $\\{L_{t},\
t\geq 0\\}$ be a pure jump Lévy process on a filtered probability space
$(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\geq 0},\mathbb{P})$ where
$\\{\mathcal{F}_{t}\\}_{t\geq 0}$ is the completed, right-continuous
filtration generated by the Lévy process. Denote with $\tilde{\pi}({\rm
d}t,{\rm d}\xi):=\pi({\rm d}t,{\rm d}\xi)(\omega)-\,\nu({\rm d}\xi)\,{\rm d}t$
its compensated Poisson random measure under $\mathbb{P}$ with compensator
given by $\,\nu({\rm d}\xi)\,{\rm d}t$.

Under the measure $\mathbb{P}$, the problem that we will tackle here has the
form

$u(t)=x(t)+\int_{0}^{t}k(t-s)f(s,u(s))\,ds+\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}k(t-s)\sigma(s,\xi,u(s))\,\tilde{\pi}(ds,d\xi),$
(1)

where $k(t)$ is a completely monotone kernel with a suitable regularity at the
origin, and Lipschitz continuity conditions on the coefficients $f$ and
$\sigma$ are required, see Assumptions 2.1 and 3.1 below.

A crucial point in our construction is the requirement that the convolution
kernel is a completely monotone function, which allows us to use Bernstein’s
representation and the machinery of Laplace transform to obtain a Markovian
lift.

It is known that the non-Markovian nature of solutions to stochastic Volterra
equations introduces theoretical challenges, particularly when one addresses
the analysis of optimal control problems. Among various techniques aimed at
achieving a Markovian transformation of the problem, Carmona and Coutin [11]
proposed a noteworthy approach. This method involves utilizing Bernstein’s
measure of completely monotone kernels to derive an evolution equation in the
space of Laplace transforms. Subsequently, this technique has been applied in
recent works by Cuchiero and Teichmann [18], Abi Jaber and El Euch [1], Harms
and Stefanovits [24], with applications in mathematical finance, see also Abi
Jaber, Miller and Pham [2] for an application to a control problem. The
aforementioned approach has a weakness due to the proposed $L^{1}$ setting; a
more recent contribution by Hamaguchi [23] proves crucial in constructing a
Hilbert state space setting, with the application of advanced techniques such
as analytic semigroups and interpolation spaces. The broader context of this
construction finds its roots in the approach previously developed for integro-
differential equations by Desch and Miller [20]. This approach, originally
conceived for a deterministic framework, has been extended to the stochastic
problems by the authors in [9, 7, 16, 17, 6].

It seems natural that the Markovian lift, in whatever form constructed,
transform the state equation in a stochastic differential equation in an
infinite dimensional space, since this space shall be large enough to contain
all the information about the history of the process. This implies that,
although the original equation can be taken in a finite-dimensional space, the
resulting lift is always infinite-dimensional.

In our construction, the lifted stochastic equation has the form

$\left\\{\begin{aligned}
dY(t)&=[BY(t)+\tilde{F}(t,Y(t))]\,dt+\int_{\mathbb{R}^{n}\setminus\\{0\\}}\tilde{G}(t,\xi,Y(t))\,\tilde{\pi}(dt,d\xi)\\\
Y(0)&=y\end{aligned}\right.$ (2)

where $B$ is the infinitesimal generator of a strongly continuous, analytic
semigroup on $\mathcal{H}$, while $\tilde{F}$ and $\tilde{G}$ are unbounded
operators on $\mathcal{H}$ given in terms of $f$ and $\sigma$, respectively.
This construction’s details are given in section 3.

The main result we adapt from [22] is the correspondence between a mild
analytic solution for the evolution equation (2) and the integral equation
(1), see Theorem 3.12 for details. This allows the use of the machinery from
evolution equations to study the integral equation (1).

Next, we address an optimal control problem associated with the solution of
the integral stochastic Volterra equation in $\mathbb{R}^{d}$ driven by a Lévy
process as given in (1). We shall denote $T>0$ the time horizon of the control
problem. We define an admissible control process, or simply a control, as an
$\\{\mathcal{F}_{t}\\}$-predictable process $\gamma$ with values in a
measurable space $U$, called the action (or decision) space, endowed with a
$\sigma$-field $\mathcal{U}$. The set of admissible control processes is
denoted by $\mathcal{A}$. Further details will be provided in Section 4.

To model the effect of control, we suppose we have a function
$r:[0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{n}\times
U\rightarrow]0,\infty[$. A controller will act to modify the measure
$\mathbb{P}$ under which our system evolves, replacing it with the measure
$\mathbb{P}^{\gamma}$ that depends on the control $\gamma$ by the formula

$\frac{d\mathbb{P}^{\gamma}}{d\mathbb{P}}=\exp\left[\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}(r(s,u(s),\xi,\gamma(s))-1)\,[\pi(ds,d\xi)-\nu(d\xi)\,ds]\right.\\\
\left.-\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}(r(s,u(s),\xi,\gamma(s))-1-\ln(r(s,u(s),\xi,\gamma(s))))\,\pi(ds,d\xi)\right]$

Under suitable assumptions on $r$ (for instance if $r$ is strictly positive
and uniformly bounded), this defines a true probability measure
$\mathbb{P}^{\gamma}$ equivalent to $\mathbb{P}$. By Girsanov’s theorem the
compensator of $\pi$ under $\mathbb{P}^{\gamma}$ is given by
$\nu^{\gamma}({d}\xi,{d}t):=r(t,u(t),\xi,\gamma(t))\nu({d}\xi)\,{d}t$. This
means that our controller is effectively modifying the rates of jumps of
different sizes.

Then, if the control $\gamma$ is used, we see that $u$ has
$\mathbb{P}^{\gamma}$-dynamics

$\displaystyle u(t)=$ $\displaystyle x(t)+\int_{0}^{t}k(t-s)f(s,u(s))\,ds$ (3)
$\displaystyle+\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}k(t-s)\sigma(s,\xi,u(s))\,(r(s,u(s),\xi,\gamma(s))-1)\nu(d\xi)\,ds$
$\displaystyle+\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}k(t-s)\sigma(s,\xi,u(s))\,\tilde{\pi}^{\gamma}(ds,d\xi),$

where
$\tilde{\pi}^{\gamma}({d}t,{d}\xi):=\pi({d}t,{d}\xi)-r(t,u_{t},\xi,\gamma(t))\,\nu(d\xi)\,dt$
is the compensated Poisson random measure under $\mathbb{P}^{\gamma}$.
Therefore, our control can be seen as determining the jump rates of $u$ in a
weak sense, as $(u,\tilde{\pi}^{\gamma},\mathbb{P}^{\gamma})$ is a
probabilistic weak solution to (3).

The optimal control problem requires to minimize a cost functional
$\mathbb{J}$ defined through a running cost function $l$ and a terminal cost
function $g$ by

$\mathbb{J}(\gamma)=\mathbb{E}^{\gamma}\left[\int_{0}^{T}l(t,\gamma(t),u(t))\,dt+g(u(T))\right],$
(4)

where $\mathbb{E}^{\gamma}$ is the expectation under the measure
$\mathbb{P}^{\gamma}$ and $u$ is the (weak in the probabilistic sense)
solution of Equation (3) under $\mathbb{P}^{\gamma}$.

Due to the equivalence between (1) and (2), we can equivalently state the cost
functional in terms of the Markovian lift $Y$

$\mathbb{J}(\gamma)=\mathbb{E}^{\gamma}\left[\int_{0}^{T}l(t,\gamma(t),PY(t))\,dt+g(PY(T))\right]$
(5)

where now $Y$ is the solution of the controlled evolution equation

$\left\\{\begin{aligned} dY(t)=&[BY(t)+\tilde{F}(t,Y(t))\\\
&\phantom{[BY(t)}+\tilde{G}(t,\xi,Y(t))(r(t,PY(t),\xi,\gamma(t))-1)\nu({\rm
d}\xi)]\,dt\\\
&+\int_{\mathbb{R}^{n}\setminus\\{0\\}}\tilde{G}(t,\xi,Y(t))\,\tilde{\pi}^{\gamma}(dt,d\xi),\\\
Y(0)=&y\end{aligned}\right.$ (6)

The advantage of this expression is that we can proceed to find the solution
to the optimal control problem by using the forward-backward system approach,
well-known in the diffusive case (see, e.g., [21, 33]). Recently BSDEs driven
by random measures have been introduced to solve optimal control problems for
marked point processes [13, 14, 17, 15, 3].

The idea is to consider on the probability space
$(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\geq 0},\mathbb{P})$, the
forward-backward system

$\left\\{\begin{aligned}
&Y(t)=S(t)y+\int_{0}^{t}S(t-s)\tilde{F}(\sigma,Y(s))\,ds+\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}S(t-s)\tilde{G}(s,\xi,Y(s))\,\tilde{\pi}(ds,d\xi)\\\
&\Theta_{t}+\int_{t}^{T}\int_{\mathbb{R}^{n}\setminus\\{0\\}}Z_{s}(\xi)\,\tilde{\pi}({d}s,{d}\xi)=g(PY(T))+\int_{t}^{T}H(s,Y(s),Z_{s})\,ds.\end{aligned}\right.$

where the generator $H$ is the Hamiltonian function defined by:

$H(s,y,z)=\inf_{\chi\in
U}\left\\{l(s,Py,\chi)+\int_{\mathbb{R}^{n}\setminus\\{0\\}}z(\xi)\,(r(s,Py,\xi,\chi)-1)\,\nu({d}\xi)\right\\}.$
(7)

The solution to the first equation is a ${\mathcal{F}_{t}}$-predictable
process, which exists and is unique based on the results in Section 3.3
Assuming our conditions on $l$, $g$, and $r$, the Hamiltonian function is
Lipschitz continuous, and the solution $(\Theta,Z)$ to the backward stochastic
differential equation (the second equation in the system) is unique, as
demonstrated in Proposition 4.4.

Our main result, Theorem 4.5, proves that under appropriate assumptions the
optimal control problem has a solution and that the value function and the
optimal control can be represented using the solution to the backward
stochastic differential equation.

Finally, if we assume that the infimum in (7) is attained, we can prove that
there exists a feedback control, that is, a map
$\bar{\gamma}:[0,T]\times\mathcal{V}\rightarrow U$ such that the process
$\gamma^{*}_{t}=\bar{\gamma}(t,Y({t-}))$ is optimal among all predictable
controls. The optimal state is the probabilistic weak solution of the
controlled evolution equation (6) with $\gamma=\gamma^{*}_{t}$.

By taking the projection $u(t)=PY(t)$ we see that $\gamma^{*}$ is the optimal
control for the controlled Volterra problem (3) and the optimal solution is
given in closed-loop form.

### Motivation

The introduction of memory effects in the modeling of tumor growth and the
response of tumor cells to treatment over a finite period are area of
significant interest in the literature. The continuous analog of the
Richardson model [28] as described in the papers [29, 4, 26] has been applied
in population dynamics and tumor growth research. This approach allows for the
study of the effects of therapies on cancer growth, considering the non-
instantaneous response of tumor cells to treatment.

In a discrete setting, where a (planar or three-dimensional) object is
considered with a choice of $d$ possible directions of growth, the distance of
the object’s border from the origin in the $i$-th direction is denoted as
$R_{i}(t)$. The growth rate can be approximately described by the equation:

$\displaystyle\frac{\partial}{\partial
t}R_{i}(t)=\mu_{i}(t)+\int_{0}^{t}\int_{S}g_{i}(s,\xi)\,Z(ds,d\xi)$

where $\mu_{i}(t)$ represents the growth rate in the $i$-th direction and
$g_{i}$ is a suitable weight function determining the occurrence of certain
events in a neighborhood of the $i$-th direction. The integral term accounts
for the impact of events on growth over time and space. The model may be
modified to account for the effect of a control on the dosage and
effectiveness of therapy, particularly in the context of traditional cancer
treatment strategies, which often utilize the maximum tolerated dose (MTD).
However, the MTD strategy has drawbacks, such as the development of drug
resistance.

Numerous models addressing the impact of control on therapy dosage and
efficacy can be found in the medical literature; for a comprehensive overview,
we refer to the review [32]. Notably, the paper [30] focuses on modeling the
emergence of new metastatic tumor cells following drug treatment using a
stochastic differential equation driven by a jump process. The process
governing the dissemination of cells is characterized as a stochastic process
driven by a Poisson process. The pivotal parameter is the drug delivery rate,
directly influencing both the death rate of drug-sensitive tumor cells and,
consequently, the rate of metastatic cell dissemination.

##  2 Scalar kernels

We assume that the scalar kernel $k(t)$ is a completely monotone function,
with Bernstein representation

$k(t)=\int_{[0,\infty)}e^{-tx}\mu(dx).$ (8)

We shall require that the degree of singularity at 0 is small: more exactly,
we assume at least that

$\int_{0}^{1}t^{-1/2}k(t)\,dt<\infty.$ (9)

This assumption, in particular, implies that $k\in L^{2}_{\rm
loc}(\mathbb{R}_{+})$. To determine the exact order of degeneracy allowed, a
way to measure it needs to be introduced.

We know that the degree of degeneracy of $k(t)$ at 0 is (inversely)
proportional to the degeneracy at infinity of its Laplace transform $\hat{k}$.
In [8] the following index was used to study this characteristic

$\delta(k)=\inf\left\\{\rho>0\ :\
\int_{1}^{\infty}s^{-\rho}\hat{k}(s)\,ds<+\infty\right\\}.$ (10)

By using [8, Lemma 2.23], condition (9) can be stated equivalently by
requiring that

$\delta(k)<\frac{1}{2}.$ (11)

It is proved in [22] that for any $\alpha\in(0,1)$,
$\displaystyle\int_{0}^{1}t^{\alpha-1}k(t)\,dt<\infty$ if and only if
$\displaystyle\int_{[1,\infty)}x^{-\alpha}\,\mu(dx)<\infty$. We are ready to
provide the main assumption on the kernel $k$.

######  Assumption 2.1.

We assume that for some $\varepsilon>0$, the Bernstein’s measure associated to
the kernel satisfies

$\int_{[1,\infty)}x^{\varepsilon-1/2}\,\mu(dx)<\infty.$ (12)

In the notation of [8], the above assumption can be equivalently stated by
requiring that

$\displaystyle\delta(k)<\frac{1}{2}-\varepsilon.$

According to condition (9), we fix a weight function $\omega(x)=1\wedge
x^{-1/2}$ for every $x\in[0,\infty)$. Notice that the following bound holds:

$\int_{[0,\infty)}\left(\omega(x)+\frac{1}{(1+x)\omega(x)}\right)\,\mu(dx)<\infty.$
(13)

##  3 Stochastic Volterra equations in finite dimensions

In this section, we consider the problem

$u(t)=x(t)+\int_{0}^{t}k(t-s)f(s,u(s))\,ds+\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}k(t-s)\sigma(s,\xi,u(s))\,\tilde{\pi}(ds,d\xi),$
(1)

where $\tilde{\pi}(dt,d\xi)$ is a compensated Lévy measure having intensity
measure $dt\times\nu(d\xi)$.

###  3.1 Noise term

Let $S=\mathbb{R}^{n}\setminus\\{0\\}$ with the Borel $\sigma$-field
$\mathcal{S}=\mathcal{B}(\mathbb{R}^{n}\setminus\\{0\\})$. Let
$(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\},\mathbb{P})$ be a complete filtered
probability space. A Poisson random measure $\pi:\Omega\times S\to\mathbb{N}$,
with intensity measure $\nu(A)=\mathbb{E}[\pi(\omega,A)]$, is characterized by

  * (i)

$\omega\mapsto\pi(\omega,A)$ has Poisson distribution with parameter $\nu(A)$,
and

  * (ii)

for pairwise disjoint sets $\\{A_{1},\dots,A_{n}\\}$ the random variables
$\pi(A_{i})$ are independent.

The compensated Poisson random measure $\tilde{\pi}$ is defined by

$\displaystyle\tilde{\pi}(\omega,A)=\pi(\omega,A)-\nu(A).$

In this paper, we consider a time-homogeneous Poisson random measure
$\pi([0,t],A)$ on $(\mathbb{R}_{+}\times S)$ with corresponding intensity
measure $dt\times\nu(d\xi)$ and compensated measure $\tilde{\pi}$.

The random measure $\pi$ is associated to a pure jump Lévy process $L(t)$,
defined by

$\displaystyle
L(t)=\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}\xi\,\tilde{\pi}(ds,d\xi)$

with the Lévy coefficient given by

$\displaystyle\psi_{L}(x)=\int_{|y|<R}\left(1-e^{i\langle x,y\rangle}+i\langle
x,y\rangle\right)\,dy$

Let $E$ be a real separable Hilbert space. Let
$\sigma:\Omega\times[0,T]\times\mathbb{R}^{n}\to E$ be a progressively
measurable process, satisfying

$\displaystyle\mathbb{E}\int_{0}^{T}\int_{\mathbb{R}^{n}\setminus\\{0\\}}|\sigma(t,\xi)|^{2}\,dt\,\nu(d\xi)<\infty.$

Then we can define the Itô integral

$\displaystyle\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}\sigma(s,\xi)\tilde{\pi}(ds,d\xi)$

following [27]; Burkholder-Davis-Gundy inequality (see [10]) leads to the
following estimate

$\displaystyle\mathbb{E}\sup_{t\in[0,T]}\left|\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}\sigma(s,\xi)\tilde{\pi}(ds,d\xi)\right|^{2}\leq
c\,\mathbb{E}\int_{0}^{T}\int_{\mathbb{R}^{n}\setminus\\{0\\}}|\sigma(s,\xi)|^{2}\,\nu(d\xi)\,ds.$

To study stochastic convolutions, we can make the following remark (see for
instance [19]). Suppose we need to estimate

$\displaystyle\mathbb{E}\left|\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}k(t-s)\sigma(s,\xi)\tilde{\pi}(ds,d\xi)\right|^{2}$

Fix $t_{0}\in(0,T]$ and define $k(t_{0}-s)=0$ for $s>t_{0}$. Then Burkholder-
Davis-Gundy inequality implies

$\displaystyle\mathbb{E}\sup_{t\in[0,t_{0}]}\left|\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}k(t_{0}-s)\sigma(s,\xi)\tilde{\pi}(ds,d\xi)\right|^{2}\leq
c\,\mathbb{E}\int_{0}^{t_{0}}\int_{\mathbb{R}^{n}\setminus\\{0\\}}|k(t_{0}-s)\sigma(s,\xi)|^{2}\,\nu(d\xi)\,ds.$

In particular,

$\displaystyle\mathbb{E}\left|\int_{0}^{t_{0}}\int_{\mathbb{R}^{n}\setminus\\{0\\}}k(t_{0}-s)\sigma(s,\xi)\tilde{\pi}(ds,d\xi)\right|^{2}\leq
c\,\mathbb{E}\int_{0}^{t_{0}}\int_{\mathbb{R}^{n}\setminus\\{0\\}}|k(t_{0}-s)\sigma(s,\xi)|^{2}\,\nu(d\xi)\,ds;$

but since $t_{0}$ is arbitrary, we obtain

$\mathbb{E}\left|\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}k(t-s)\sigma(s,\xi)\tilde{\pi}(ds,d\xi)\right|^{2}\leq
c\,\mathbb{E}\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}|k(t-s)\sigma(s,\xi)|^{2}\,\nu(d\xi)\,ds.$
(14)

for every $t\in(0,T]$.

###  3.2 An abstract equation for the lift

We fix the following set of assumptions on the coefficients of problem (1).

######  Assumption 3.1.

The functions $f:[0,T]\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ and
$\sigma:[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{d}\rightarrow{\mathbb{R}^{d}}$
are jointly continuous and uniformly Lipschitz continuous in the last variable
uniformly with respect to the others.

We impose a structure assumption on the forcing term

$\displaystyle x(t)=\int_{[0,\infty)}e^{-xt}y(x)\,\mu(dx)$ (15)

for some function $y\in B_{b}([0,\infty);\mathbb{R}^{d})$ is a Borel
measurable and bounded function with values in $\mathbb{R}^{d}$.

Following [22, 23], by using (8) and an application of the (stochastic) Fubini
theorem Eq.(1) becomes

$\displaystyle
u(t)=\int_{[0,\infty)}\left(e^{-tx}y(x)+\int_{0}^{t}e^{-(t-s)x}f(s,u(s))\,ds+\int_{0}^{t}e^{-(t-s)x}\int_{\mathbb{R}^{n}\setminus\\{0\\}}\sigma(s,\xi,u(s))\,\tilde{\pi}(ds,d\xi)\right)\,\mu(dx)$

Thus, we obtain the representation

$u(t)=PY(t)=\int_{[0,\infty)}Y(t,x)\,\mu(dx)$ (16)

where $Y:\Omega\times[0,T]\times[0,\infty)\to\mathbb{R}^{d}$ is given by

$\displaystyle
Y(t,x)=e^{-tx}y(x)+\int_{0}^{t}e^{-(t-s)x}f(s,PY(s))\,ds+\int_{0}^{t}e^{-(t-s)x}\int_{\mathbb{R}^{n}\setminus\\{0\\}}\sigma(s,\xi,PY(s))\,\tilde{\pi}(ds,d\xi).$

We notice that the above expression can be equivalently given in differential
form

$\displaystyle dY(t,x)=$
$\displaystyle[-xY(t,x)+f(t,PY(t))]\,dt+\int_{\mathbb{R}^{n}\setminus\\{0\\}}\sigma(t,\xi,PY(t))\,{\tilde{\pi}(dt,d\xi)}$
$\displaystyle Y(0,x)=$ $\displaystyle y(x)$

To formalize the setting where to study the previous equation (that we will
call the Markovian lift equation) we define a Gelfand triple
$\mathcal{V}\subset\mathcal{H}\subset\mathcal{V}^{\star}$.

Let $y:[0,\infty)\to\mathbb{R}^{d}$ be Borel measurable; then we define the
norm

$\displaystyle\|y\|^{2}_{\mathcal{H}}=\int_{[0,\infty)}|y(x)|^{2}\,\omega(x)\,\mu(dx)$

and we let $\mathcal{H}$ be the Hilbert space of equivalent classes of
functions $y:[0,\infty)\to\mathbb{R}^{d}$ with $\|y\|_{\mathcal{H}}<\infty$.

Next, we introduce the norm

$\displaystyle\|y\|^{2}_{\mathcal{V}}=\int_{[0,\infty)}|y(x)|^{2}\,(1+x)\omega(x)\,\mu(dx)$

and we let $\mathcal{V}$ be the Hilbert space of equivalent classes of
functions $y:[0,\infty)\to\mathbb{R}^{d}$ with $\|y\|_{\mathcal{V}}<\infty$.
It is noted in [22] that $\mathcal{V}\subset\mathcal{H}$ continuously and
densely. Denote the continuous dual spaces of $\mathcal{H}$ and $\mathcal{V}$
by $\mathcal{H}^{\star}$ and $\mathcal{V}^{\star}$, respectively. Through the
Riesz isomorphism, we identify $\mathcal{H}$ with $\mathcal{H}^{\star}$, and
we see that $\mathcal{H}\subset\mathcal{V}^{\star}$ continuously and densely.

######  Lemma 3.1.

The operator $B:D(B)\subset\mathcal{H}\to\mathcal{H}$ given by $By(x)=-xy(x)$
with domain

$\displaystyle D(B)=\left\\{y\in\mathcal{H}\ :\
\int_{[0,\infty)}|y(x)|^{2}\,x^{2}\omega(x)\,\mu(dx)<\infty\right\\}$

generates a linear contraction semigroup $S(t)=e^{tB}$ on $\mathcal{H}$ given
by

$\displaystyle S(t)y(x)=e^{-tx}y(x).$

Moreover, $B$ can be extended to an operator
$B:\mathcal{V}\to\mathcal{V}^{\star}$ and

$\displaystyle{}_{\mathcal{V}^{\star}}\langle
By,y\rangle_{\mathcal{V}}\leq-\|y\|^{2}_{\mathcal{V}}+\|y\|^{2}_{\mathcal{H}}\leq
0$

for any $y\in\mathcal{V}$.

######  Lemma 3.2.

Let $i:\mathbb{R}^{d}\to\mathcal{H}$ be the immersion mapping $i(v)=1(x)v$,
where $1(x)=1$ for every $x$. Then $i$ is well defined and
$\|iv\|^{2}_{\mathcal{H}}\leq C\,\|v\|^{2}_{\mathbb{R}^{d}}$ for some finite
constant $C$.

###### Proof.

It holds

$\displaystyle\|iv\|^{2}_{\mathcal{H}}=\|v\|^{2}_{\mathbb{R}^{d}}\int_{[0,\infty)}\omega(x)\,\mu(dx)$

and the thesis follows since $\omega$ is an admissible weight function. ∎

We consider next the action of the semigroup $S(t)$ on $i$. We record the
following.

######  Lemma 3.3.

$S(t)i$ is a well defined mapping with values in $\mathcal{V}$, for every
$t>0$, and

$\displaystyle\int_{0}^{T}\|S(t)i\|_{L(\mathbb{R}^{d};\mathcal{V})}\,dt<\infty.$

###### Proof.

With a direct computation, we have

$\displaystyle\|S(t)i\|_{L(\mathbb{R}^{d};\mathcal{V})}^{2}=\int_{[0,\infty)}e^{-2tx}(1+x)\omega(x)\,\mu(dx)$

and since, for every $t>0$ there exists $C_{t}<\infty$ such that
$e^{-tx}(1+x)\leq C_{t}$, $x\geq 0$, we have

$\displaystyle\|S(t)i\|_{L(\mathbb{R}^{d};\mathcal{V})}^{2}\leq
C_{2t}\,\int_{[0,\infty)}\omega(x)\,\mu(dx)<\infty$

thanks to assumption (13).

Notice that $\sqrt{C_{t}}$ is an integrable function on $[0,T]$, being equal
to $\frac{1}{\sqrt{2t}}e^{t-1/2}\vee 1$, which leads to

$\displaystyle\int_{0}^{T}\|S(t)i\|_{L(\mathbb{R}^{d};\mathcal{V})}\,dt\leq
C(1+T)<\infty.$

∎

To obtain an estimate in $L^{2}$ in time, we need to take into account
Assumption 2.1.

######  Lemma 3.4.

Let $\varepsilon>0$ be given in Assumption 2.1. Then

$\displaystyle\int_{0}^{T}\|S(t)i\|^{2}_{L(\mathbb{R}^{d};\mathcal{V})}\,dt<\infty.$

###### Proof.

We can mimic the same construction as above, but we need the estimate

$\|S(t)i\|_{L(\mathbb{R}^{d};\mathcal{V})}^{2}=\int_{[0,1)}e^{-2tx}(1+x)\,\mu(dx)+\int_{[1,\infty)}e^{-2tx}(1+x)x^{-1/2}\,\mu(dx)\\\
\leq c_{1}+c_{2}\sup_{x\geq 1}e^{-2tx}(1+x)x^{-\varepsilon}\leq
c_{1}+c_{2,\varepsilon}t^{\varepsilon-1}.$

Since this function is integrable on $[0,T]$ for arbitrary $\varepsilon>0$,
the thesis follows. ∎

Next, we consider the operator $P$ that represents, in our construction, the
inverse of the Markovian lift. We have that $P$ is an unbounded operator on
$\mathcal{H}$ with domain (that contains) $\mathcal{V}$.

######  Lemma 3.5.

$P$ maps $\mathcal{V}$ into $\mathbb{R}^{d}$.

###### Proof.

We have

$\displaystyle\|PY\|^{2}_{\mathbb{R}^{d}}=$
$\displaystyle\left\|\int_{[0,\infty)}Y(x)\,\mu(dx)\right\|^{2}$
$\displaystyle\leq$
$\displaystyle\left(\int_{[0,\infty)}(1+x)^{-1}\omega^{-1}(x)\,\mu(dx)\right)\left(\int_{[0,\infty)}(1+x)\omega(x)\|Y(x)\|^{2}\,\mu(dx)\right)$
$\displaystyle\leq$ $\displaystyle C\,\|Y\|^{2}_{\mathcal{V}}$

since $\omega$ is an admissible weight function. ∎

Following the ideas in [22] we obtain the following result about the forcing
term.

######  Lemma 3.6.

Assume that $y\in B_{b}([0,\infty);\mathbb{R}^{d})$. Then the forcing term $x$
defined in (15) belongs to $L^{2}(0,T;\mathbb{R}^{d})$.

###### Proof.

By the inequality $e^{-x}\leq\frac{1}{1+x}$ we obtain

$\displaystyle\|e^{-\cdot t}y\|^{2}_{\mathcal{V}}$
$\displaystyle=\int_{[0,\infty)}e^{-2xt}|y(x)|^{2}(1+x)\omega(k)\,\mu(dx)$
$\displaystyle\leq\int_{[0,\infty)}\|y\|^{2}_{B_{b}(\mathbb{R}^{d})}\frac{1+x}{1+2xt}\omega(x)\,\mu(dx)\leq
C\,\left(\frac{1}{2t}\vee 1\right)\,\|y\|^{2}_{B_{b}(\mathbb{R}^{d})}.$

By an application of Fubini’s theorem, we have 111 Notice that
$\displaystyle\int_{0}^{T}e^{-2\theta t}\,dt\leq
T\qquad\text{and}\qquad\int_{0}^{T}e^{-2\theta t}\,dt\leq\frac{1}{2\theta}$
hence the actual bound is given by the minimum between the two values on the
right-hand side

$\displaystyle\int_{0}^{T}\|e^{-\cdot t}y\|^{2}_{\mathcal{V}}\,dt$
$\displaystyle=\int_{[0,\infty)}\int_{0}^{T}e^{-2xt}\,dt\,|y(x)|^{2}(1+x)\omega(x)\,\mu(dx)$
$\displaystyle\leq\int_{[0,\infty)}\left(\frac{1}{2x}\wedge
T\right)\,|y(x)|^{2}(1+x)\omega(x)\,\mu(dx)$
$\displaystyle\leq\left(\frac{1}{2}+T\right)\|y\|^{2}_{B_{b}(\mathbb{R}^{d})}\int_{[0,\infty)}\omega(x)\,\mu(dx).$

Since we can write (15) as $x(t)=P(e^{-\cdot t}y(t))$, by an application of
Lemma 3.5 we obtain that $x\in L^{2}(0,T;\mathbb{R}^{d})$ as required. ∎

The next results are quite similar to those proved in [22, Lemma 2.7] (we
present the proof of the stochastic part since in the quoted paper it is given
for a Wiener integral instead of a Lévy integral).

######  Proposition 3.7.

If $b$ is a predictable process such that $b\in L^{1}(0,T;\mathbb{R}^{d})$
almost surely, then the convolution

$\displaystyle B(t)=\int_{0}^{t}e^{-\cdot(t-s)}b(s)\,ds$

belongs to $\mathcal{V}$ almost surely for almost every $t\in[0,T]$ and the
estimate

$\int_{0}^{T}\|B(s)\|^{2}_{\mathcal{V}}\,ds\leq
C\left(\frac{1}{2}+T\right)\left(\int_{0}^{T}|b(s)|\,ds\right)^{2}$ (17)

holds almost surely.

######  Proposition 3.8.

If $\sigma$ is a predictable process with values in $\mathbb{R}^{d}$ such that

$\displaystyle\mathbb{E}\int_{0}^{T}\int_{\mathbb{R}^{n}\setminus\\{0\\}}|\sigma(t,\xi)|^{2}\,\nu(d\xi)\,ds<\infty$

then the stochastic convolution

$\displaystyle\Sigma(t)=\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}e^{(t-s)B}i\,\sigma(s,\xi)\,\pi(d\xi,ds)$

belongs to $\mathcal{V}$ almost surely for almost every $t\in[0,T]$ and
satisfies

$\displaystyle\mathbb{E}\int_{0}^{T}\|\Sigma(s)\|^{2}_{\mathcal{V}}\,ds\leq\mathbb{E}\int_{0}^{T}\int_{\mathbb{R}^{n}\setminus\\{0\\}}\left|\sigma(s,\xi)\right|^{2}\,\nu(d\xi)\,ds.$

###### Proof.

Let us start with the estimate

$\displaystyle\mathbb{E}\|\Sigma(t)\|_{\mathcal{V}}^{2}=\mathbb{E}\int_{[0,\infty)}\left|\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}(e^{(t-s)B}i)(x)\,\sigma(s,\xi)\,\pi(d\xi,ds)\right|^{2}\,(1+x)\omega(x)\,\mu(dx)$

and by applying Burkholder-Davis-Gundy inequality (14) we obtain

$\displaystyle\mathbb{E}\|\Sigma(t)\|_{\mathcal{V}}^{2}$ $\displaystyle\leq
c\,\mathbb{E}\int_{[0,\infty)}\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}\left|(e^{(t-s)B}i)(x)\,\sigma(s,\xi)\right|^{2}\,\nu(d\xi)\,ds\,(1+x)\omega(x)\,\mu(dx)$
$\displaystyle=c\,\mathbb{E}\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}\left\|e^{(t-s)B}i\right\|^{2}_{\mathcal{V}}\,\left|\sigma(s,\xi)\right|^{2}\,\nu(d\xi)\,ds.$

An application of Young’s inequality for convolutions implies

$\displaystyle\mathbb{E}\int_{0}^{T}\|\Sigma(t)\|_{\mathcal{V}}^{2}\,dt\leq
c\,\mathbb{E}\left(\int_{0}^{T}\int_{\mathbb{R}^{n}\setminus\\{0\\}}\left|\sigma(s,\xi)\right|^{2}\,\nu(d\xi)\,ds\right)\left(\int_{0}^{T}\left\|e^{tB}i\right\|^{2}_{\mathcal{V}}\,dt\right)$

The last quantity is bounded thanks to Lemma 3.4, and the thesis follows. ∎

###  3.3 Existence of the solution

We introduce the following SDE in the Hilbert space $\mathcal{H}$

$\left\\{\begin{aligned}
dY(t)&=[BY(t)+\tilde{F}(t,Y(t))]\,dt+\int_{\mathbb{R}^{n}\setminus\\{0\\}}\tilde{G}(t,\xi,Y(t))\,\tilde{\pi}(dt,d\xi)\\\
Y(0)&=y\end{aligned}\right.$ (2)

where we consider the nonlinear mappings
$\tilde{F}\colon[0,T]\times\mathcal{V}\to\mathcal{H}$, $\tilde{F}(t,Y)=i\cdot
f(t,PY)$, and
$\tilde{G}\colon[0,T]\times\mathbb{R}^{n}\times\mathcal{V}\to\mathcal{H}$,
${\tilde{G}(t,\xi,Y)=i\cdot\sigma(t,\xi,PY)}$.

######  Definition 3.9.

We define _mild analytical, strong probabilistic solution_ to (2) starting
from $y$ a predictable process $Y\in
D_{\mathcal{F}}([0,T];L^{2}(\Omega;\mathcal{V}))$, i.e.,
$Y\colon[0,T]\times\Omega\to\mathcal{V}$ is a predictable process with cádlág
trajectories and such that

$\displaystyle\sup_{t\in[0,T]}\mathbb{E}\|Y(t)\|^{2}_{\mathcal{V}}<\infty,$

which satisfies almost surely the integral equation

$Y(t)=S(t)y+\int_{0}^{t}S(t-r)\tilde{F}(r,Y(r))\,dr+\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}S(t-r)\tilde{G}(r,\xi,Y(r))\,\tilde{\pi}(dr,d\xi).$
(18)

######  Theorem 3.10.

Under Assumption 2.1 on the kernel and Assumption 3.1 on the coefficients of
the equation, there exists a unique mild analytical, strong probabilistic
solution $Y\in D_{\mathcal{F}}([0,T];L^{2}(\Omega;\mathcal{V}))$ for Eq.(2).

###### Proof.

Let us consider for some $\beta\in\mathbb{R}$ the space
$D_{\mathcal{F}}([0,T];L^{2}(\Omega;\mathcal{V}))$ endowed with the norm

$\displaystyle\|Y\|^{2}_{\beta,T}=\sup_{t\in[0,T]}e^{-\beta
t}\mathbb{E}[\|Y(t)\|^{2}_{\mathcal{V}}].$

We denote

$\displaystyle I_{F}(Y)(t)$
$\displaystyle=\int_{0}^{t}S(t-r)\tilde{F}(r,Y(r))\,dr$ $\displaystyle
I_{G}(Y)(t)$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}S(t-r)\tilde{G}(r,\xi,Y(r))\,\tilde{\pi}(dr,d\xi)$

They have a predictable version since they are adapted and stochastically
continuous processes for every $Y\in\mathcal{V}$. To apply Banach fixed point
theorem, it remains to prove that $I_{F}+I_{G}$ is a contraction on the space
$D_{\mathcal{F}}([0,T];L^{2}(\Omega;\mathcal{V}))$ for the norm
$\|\cdot\|^{2}_{\beta,T}$, for some $\beta$.

Fix $Y_{1},Y_{2}\in D_{\mathcal{F}}([0,T];L^{2}(\Omega;\mathcal{V}))$; then

$\displaystyle\|I_{F}(Y_{1})-I_{F}(Y_{2})\|^{2}_{\beta;T}$
$\displaystyle=\sup_{t\in[0,T]}e^{-\beta
t}\mathbb{E}\left\|\int_{0}^{t}S(t-r)[\tilde{F}(r,Y_{1}(r))-\tilde{F}(r,Y_{2}(r))]\,dr\right\|^{2}_{\mathcal{V}}$
$\displaystyle\leq\sup_{t\in[0,T]}e^{-\beta
t}\mathbb{E}\left|\int_{0}^{t}\|S(t-r)i\|_{\mathcal{V}}|f(r,PY_{1}(r))-f(r,PY_{2}(r))|\,dr\right|^{2}$
$\displaystyle\leq L_{f}^{2}\,\|P\|^{2}\,\sup_{t\in[0,T]}e^{-\beta
t}\left(\int_{0}^{t}\|S(t-r)i\|_{\mathcal{V}}\,dr\right)\,\mathbb{E}\int_{0}^{t}\|S(t-r)i\|_{\mathcal{V}}\|Y_{1}(r)-Y_{2}(r)\|^{2}_{\mathcal{V}}\,dr$

By an application of Lemma 3.3 we have

$\displaystyle\|I_{F}(Y_{1})-I_{F}(Y_{2})\|^{2}_{\beta;T}$ $\displaystyle\leq
C(1+T)\left(\int_{0}^{T}e^{-\beta
r}\|S(r)i\|_{\mathcal{V}}\,dr\right)\sup_{r\in[0,T]}e^{-\beta
r}\mathbb{E}\|Y_{1}(r)-Y_{2}(r)\|^{2}_{\mathcal{V}}$

Now the quantity in the bracket can be tackled with similar techniques to
those in Lemma 3.3:

$\displaystyle\int_{0}^{T}e^{-\beta r}$
$\displaystyle\left(\int_{[0,\infty)}e^{-2rx}(1+x)\omega(x)\,\mu(dx)\right)^{1/2}\,dr$
$\displaystyle\leq\left(\int_{0}^{T}e^{-\beta
r}\,dr\right)^{1/2}\left(\int_{0}^{T}e^{-\beta
r}\int_{[0,\infty)}e^{-2rx}(1+x)\omega(x)\,\mu(dx)\,dr\right)^{1/2}$
$\displaystyle\leq\frac{1}{\sqrt{\beta}}\left(\int_{[0,\infty)}\int_{0}^{T}e^{-\beta
r}e^{-2rx}\,dr\,(1+x)\omega(x)\,\mu(dx)\right)^{1/2}$
$\displaystyle\leq\frac{1}{\sqrt{\beta}}\left(\int_{[0,\infty)}\frac{1}{\beta+2x}\,(1+x)\omega(x)\,\mu(dx)\right)^{1/2}\leq
C\,\frac{1}{\sqrt{\beta}}\left(\frac{1}{\beta}\vee\frac{1}{2}\right)^{1/2}$

thanks again to assumption (13). Overall, we have obtained

$\displaystyle\|I_{F}(Y_{1})-I_{F}(Y_{2})\|^{2}_{\beta;T}$ $\displaystyle\leq
C\frac{1}{\sqrt{\beta}}(1+T)\|Y_{1}(r)-Y_{2}(r)\|^{2}_{\beta,T}$

and the constant on the right-hand side can be made arbitrarily small by
choosing $\beta$ large enough.

Let us turn to the stochastic term. We have

$\displaystyle\|I_{G}(Y_{1})-I_{G}(Y_{2})\|^{2}_{\beta;T}$
$\displaystyle=\sup_{t\in[0,T]}e^{-\beta
t}\mathbb{E}\left\|\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}S(t-r)i[\sigma(r,\xi,PY_{1}(r))-\sigma(r,\xi,PY_{2}(r))]\,\tilde{\pi}(dr,d\xi)\right\|^{2}_{\mathcal{V}}$
$\displaystyle\leq\sup_{t\in[0,T]}e^{-\beta
t}\mathbb{E}\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}\left\|S(t-r)i[\sigma(r,\xi,PY_{1}(r))-\sigma(r,\xi,PY_{2}(r))]\right\|^{2}_{\mathcal{V}}\,\nu(d\xi)\,dr$
$\displaystyle\leq
L_{\sigma}^{2}\|P\|^{2}\,\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}e^{-\beta(t-r)}\left\|S(t-r)i\right\|^{2}_{\mathcal{V}}\,\nu(d\xi)\,dr\,\|Y_{1}-Y_{2}\|^{2}_{\beta,T}$
$\displaystyle\leq
L_{\sigma}^{2}\|P\|^{2}\nu(\mathbb{R}^{n}\setminus\\{0\\})\,\left(\int_{0}^{T}e^{-\beta
r}\left\|S(r)i\right\|^{2}_{\mathcal{V}}\,dr\right)\,\|Y_{1}-Y_{2}\|^{2}_{\beta,T}$

the integral is bounded by

$\displaystyle\int_{0}^{T}e^{-\beta
r}\left\|S(r)i\right\|^{2}_{\mathcal{V}}\,dr=\int_{0}^{T}\int_{[0,\infty)}e^{-\beta
r}e^{-2rx}(1+x)\omega(x)\,\mu(dx)\,dr\leq\int_{[0,\infty)}\frac{1}{\beta+2x}(1+x)\omega(x)\,\mu(dx)$

Here we need assumption (12) and we obtain

$\displaystyle\int_{0}^{t}e^{-\beta
r}\left\|S(r)i\right\|^{2}_{\mathcal{V}}\,dr\leq
C\frac{1}{\beta^{\varepsilon}}\int_{[0,\infty)}(1+x)^{\varepsilon}\omega(x)\,\mu(dx)$

and we finally obtain

$\displaystyle\|I_{G}(Y_{1})-I_{G}(Y_{2})\|^{2}_{\beta;T}\leq
C\frac{1}{\beta^{\varepsilon}}\,\|Y_{1}-Y_{2}\|^{2}_{\beta,T}.$

Then, by taking $\beta$ large enough, the mapping $I_{F}+I_{G}$ above defines
a contraction in the space $D_{\mathcal{F}}([0,T];L^{2}(\Omega;\mathcal{V}))$
with respect to the norm $\|\cdot\|^{2}_{\beta,T}$, and the theorem is proved.
∎

###  3.4 Existence of the solution of the Volterra equation

For a solution to problem (1) we mean the following

######  Definition 3.11.

Given a stochastic basis
$(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\in[0,T]},\mathbb{P})$ and a Lévy
process $\\{L(t),\ t\in[0,T]\\}$ defined on it, a (probabilistic strong)
solution of (1) is a cádlág, square integrable, predictable process
$u:\Omega\times[0,T]\to\mathbb{R}^{d}$ that satisfies (1) almost surely.

The last step we need to take to prove the existence of the solution for
problem (1) is to mimic Hamaguchi [22] and prove that a solution of the SDE
(2) in $\mathcal{H}$ corresponds uniquely to a solution of the SVE (1) in
$\mathbb{R}^{d}$.

######  Theorem 3.12.

$Y$ is a solution of the the SDE (2) in $\mathcal{H}$ if and only if
$u(t)=PY(t)$ is solution of the SVE (1) in $\mathbb{R}^{d}$.

###### Proof.

Assume that $u$ is a solution of the Volterra equation (1) with initial
condition $x$ satisfying Assumption 3.1. Define

$Y(t,x)=e^{-tx}y(x)+\int_{0}^{t}e^{-x(t-s)}f(s,u(s))\,ds+\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}e^{-x(t-s)}\sigma(s,\xi,u(s))\,\tilde{\pi}(ds,d\xi).$
(19)

Since $u$ is a predictable process, then the stochastic integral is well-
defined; we obtain, from Lemma 3.6 and Propositions 3.7 and 3.8 that $Y(t)$ is
a $\mathcal{V}$-valued process almost surely for almost every $t$. Therefore,
we may apply the operator $P$ on both sides of (19) to obtain that

$\displaystyle
PY(t)=x(t)+\int_{0}^{t}k(t-s)f(s,u(s))\,ds+\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}k(t-s)\sigma(s,\xi,u(s))\,\tilde{\pi}(ds,d\xi)=u(t)$

holds almost surely, for almost every $t\in[0,T]$. Substituting this last
equality in formula (19) we obtain that $Y$ is a mild solution of (2).

Conversely, assume that $Y$ is a mild solution to (2). Then $Y(t)$ belongs to
$\mathcal{V}\subset D(P)$, hence we can define $u(t)=PY(t)$ and Lemma 3.5
implies that $u(t)\in\mathbb{R}^{d}$. We write the integral form of (2) in the
space $\mathcal{V}$ as

$Y(t,x)=e^{-tx}y(x)+\int_{0}^{t}e^{-(t-s)x}f(s,PY(s))\,ds+\int_{0}^{t}e^{-(t-s)x}\int_{\mathbb{R}^{n}\setminus\\{0\\}}\sigma(s,\xi,PY(s))\,\tilde{\pi}(ds,d\xi);$
(20)

by an application of the operator $P$ to both sides of (20) we obtain

$\displaystyle u(t)$
$\displaystyle=x(t)+\int_{0}^{t}k(t-s)f(s,PY(s))\,ds+\int_{0}^{t}k(t-s)\int_{\mathbb{R}^{n}\setminus\\{0\\}}\sigma(s,\xi,PY(s))\,\tilde{\pi}(ds,d\xi)$
$\displaystyle=x(t)+\int_{0}^{t}k(t-s)f(s,u(s))\,ds+\int_{0}^{t}k(t-s)\int_{\mathbb{R}^{n}\setminus\\{0\\}}\sigma(s,\xi,u(s))\,\tilde{\pi}(ds,d\xi)$

hence $u$ is a solution to (1). ∎

##  4 The control problem

In this section, we address the optimal control problem associated with the
solution $u$ of an integral stochastic Volterra equation in $\mathbb{R}^{d}$
driven by a Lévy process. In the reference filtered probability space
$(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\geq 0},\mathbb{P})$ we are
concerned with the equation

$u(t)=x(t)+\int_{0}^{t}k(t-s)f(s,u(s))\,ds+\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}k(t-s)\sigma(s,\xi,u(s))\,\tilde{\pi}(ds,d\xi).$
(1)

According to the results in Subsection 3.3, Equation (1) admits a unique
solution $u=\\{u(t),\ t\in[0,T]\\}$. We shall denote $T>0$ the time horizon of
the control problem. We define an admissible control process, or simply a
control, as an $\\{\mathcal{F}_{t}\\}$-predictable process $\gamma$ with
values in a measurable space $U$, called the action (or decision) space,
endowed with a $\sigma$-field $\mathcal{U}$. The set of admissible control
processes is denoted by $\mathcal{A}$.

The optimal control problem requires to minimize a cost functional
$\mathbb{J}$ defined through a running cost function $l$ and a terminal cost
function $g$. We will make the following assumptions.

######  Assumption 4.1.

  1. (i)

$(U,\mathcal{U})$ is a topological space which is the union of countably many
compact metrizable subsets of itself.

  2. (ii)

$r:[0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{n}\times U\rightarrow\mathbb{R}$
is
$\mathcal{B}([0,T])\otimes\mathcal{B}(\mathbb{R}^{d})\otimes\mathcal{B}(\mathbb{R}^{n})\otimes\mathcal{U}$-measurable
and there exists a constant $C_{r}>0$ such that

$0<r(t,x,\xi,\gamma)\leq C_{r},\qquad
t\in[0,T],\,x\in\mathbb{R}^{d},\,\xi\in\mathbb{R}^{n},\,\gamma\in U.$ (21)

  3. (iii)

$g:\mathbb{R}^{d}\rightarrow\mathbb{R}$ is
$\mathcal{B}(\mathbb{R}^{d})$-measurable and

$\mathbb{E}|g(u(T))|^{2}<\infty.$ (22)

  4. (iv)

$l:[0,T]\times\mathbb{R}^{d}\times U\rightarrow\mathbb{R}$ is
$\mathcal{B}([0,T])\otimes\mathcal{B}(\mathbb{R}^{d})\otimes\mathcal{U}$-measurable,
and there exists $\alpha>1$ such that for every $t\in[0,T]$,
$x\in\mathbb{R}^{d}$ and any admissible control $\gamma(\cdot)$ we have

$\inf_{\chi\in
U}l(t,x,\chi)>-\infty,\qquad\mathbb{E}\int_{0}^{T}|\inf_{\chi\in
U}l(s,u(s),\chi)|^{2}\,{d}s<\infty,$ (23)
$\mathbb{E}\left(\int_{0}^{T}|l(s,u(s),\gamma(s))|\,{d}s\right)^{\alpha}<\infty.$
(24)

######  Remark 4.1.

We note that the cost functions $g$ and $l$ need not be bounded. Clearly, (24)
follows from the other assumptions if we assume for instance that

$\displaystyle\mathbb{E}\int_{0}^{T}|\sup_{\chi\in
U}l(s,u(s),\chi)|\,{d}s<\infty.$

Using the function $r$, for each control process $\gamma$, we define the
measure $\mathbb{P}^{\gamma}$ through the Dolans-Dade exponential

$\frac{d\mathbb{P}^{\gamma}}{d\mathbb{P}}=\exp\left[\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}(r(s,u(s),\xi,\gamma(s))-1)\,[\pi({\rm
d}s,{\rm d}\xi)-\nu({\rm d}\xi)\,{\rm d}s]\right.\\\
\left.-\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}(r(s,u(s),\xi,\gamma(s))-1-\ln(r(s,u(s),\xi,\gamma(s))))\,\pi({\rm
d}s,{\rm d}\xi)\right].$

For ease of notation, we define

$\Lambda^{\gamma}_{t}:=\left.\frac{d\mathbb{P}^{\gamma}}{d\mathbb{P}}\right|_{\mathcal{F}_{t}}.$
(25)

It is a well-known result that $\Lambda^{\gamma}$ is a nonnegative
supermartingale relative to $\mathbb{P}$ and $\mathcal{F}_{t}$ (see [25,
Proposition 4.3] or [5]). Since the function $r$ is uniformly bounded, the
process $\Lambda^{\gamma}$ is a strictly positive martingale (relative to
$\mathbb{P}$ and $\mathcal{F}_{t}$) and $\mathbb{P}^{\gamma}$ is a true
probability measure equivalent to $\mathbb{P}$. By applying Girsanov’s theorem
we see that while the compensator of $\pi$ under $\mathbb{P}^{\gamma}$ is
given by $\nu^{\gamma}({d}\xi,{d}t):=r(t,\xi,\gamma(t))\nu({d}\xi)\,{d}t$.
This means that the control acts on the rates of jumps of different sizes.
Under such a control the process $u$ has the $\mathbb{P}^{\gamma}$-dynamics

$\displaystyle u(t)=$ $\displaystyle x(t)+\int_{0}^{t}k(t-s)f(s,u(s))\,ds$ (3)
$\displaystyle+\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}k(t-s)\sigma(s,\xi,u(s))\,(r(s,u(s),\xi,\gamma(s))-1)\nu(d\xi)\,ds$
$\displaystyle+\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}k(t-s)\sigma(s,\xi,u(s))\,\tilde{\pi}^{\gamma}(ds,d\xi).$

where
$\tilde{\pi}^{\gamma}({d}t,{d}\xi):=\pi({d}t,{d}\xi)-r(t,u_{t},\xi,\gamma(t))\,\nu(d\xi)\,dt$
is the compensated Poisson random measure under $\mathbb{P}^{\gamma}$,
associated to the process $L^{\gamma}$, which is a Lévy process relatively the
probability $\mathbb{P}^{\gamma}$.

The stochastic control problem we are considering aims to choose a control
$\gamma\in\mathcal{A}$ which minimizes the total expected cost
$\mathbb{J}(\gamma)$ that is given in the form

$\mathbb{J}(\gamma)=\mathbb{E}^{\gamma}\left[\int_{0}^{T}l(t,u(t),\gamma(t))\,dt+g(u(T))\right],$
(26)

where $\mathbb{E}^{\gamma}$ denotes the expectation with respect to
$\mathbb{P}^{\gamma}$. Taking into account assumptions (22), (24) and using
the Hölder inequality it is easily seen that the cost is finite for every
admissible control.

Proceeding as in Subsection 3.2, we associate to the stochastic Volterra
equation (3) the state equation in the Hilbert space $\mathcal{H}$

$\left\\{\begin{aligned} dY(t)=&[BY(t)+\tilde{F}(t,Y(t))\\\
&\phantom{[BY(t)}+\int_{\mathbb{R}^{n}\setminus\\{0\\}}\tilde{G}(t,\xi,Y(t))(r(t,PY(t),\xi,\gamma(t))-1)\nu({\rm
d}\xi)]\,dt\\\
&+\int_{\mathbb{R}^{n}\setminus\\{0\\}}\tilde{G}(t,\xi,Y(t))\,\tilde{\pi}^{\gamma}(dt,d\xi),\\\
Y(0)=&y\end{aligned}\right.$ (6)

######  Definition 4.2.

We say that Equation (6) admits a _mild analytical, weak probabilistic
solution_ if there exist a filtered probability space
$(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t},\mathbb{P}^{\gamma})$, a Lévy
process with compensated Poisson random measure $\pi^{\gamma}$ (relatively the
probability $\mathbb{P}^{\gamma}$) and predictable process $Y\in
D_{\mathcal{F}}([0,T];L^{2}(\Omega;\mathcal{V}))$, i.e.
$Y\colon[0,T]\times\Omega\to\mathcal{V}$ is a predictable process and with
cádlág trajectories such that

$\displaystyle\sup_{t\in[0,T]}\mathbb{E}^{\gamma}\|Y(t)\|^{2}_{\mathcal{V}}<\infty$

starting from $y$ which satisfies almost surely the integral equation

$\displaystyle Y(t)=$ $\displaystyle
S(t)y+\int_{0}^{t}S(t-s)\tilde{F}(s,Y(r))\,ds$ (18)
$\displaystyle+\int_{0}^{t}S(t-s)\tilde{G}(s,\xi,Y(s))(r(s,PY(s),\xi,\gamma(s))-1)\nu({d}\xi)\,ds$
$\displaystyle+\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}S(t-s)\tilde{G}(s,\xi,Y(s))\,\tilde{\pi}^{\gamma}(ds,d\xi).$

######  Theorem 4.3.

For any fixed control $\gamma\in\mathcal{A}$ the state equation (6) admits a
mild analytical, weak probabilistic solution.

###### Proof.

Let $Y$ be the mild analytical, strong probabilistic solution to the

$\displaystyle dY(t)=$
$\displaystyle[BY(t)+\tilde{F}(t,Y(t))]\,dt+\int_{\mathbb{R}^{n}\setminus\\{0\\}}\tilde{G}(t,\xi,Y(t))\,\tilde{\pi}(dt,d\xi),$
(27) $\displaystyle Y(0)=$ $\displaystyle y$

in the reference filtered probability space
$(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\geq 0},\mathbb{P})$ satisfying
almost surely the integral equation

$\displaystyle
Y(t)=S(t)y+\int_{0}^{t}S(t-s)\tilde{F}(s,Y(r))\,ds+\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}S(t-s)\tilde{G}(s,\xi,Y(s))\,\tilde{\pi}(ds,d\xi)$

This exists thanks to Theorem 3.10 under the conditions on $f$ and $\sigma$
given in Assumption 3.1. Now it is enough to pass to the probability
$\mathbb{P}^{\gamma}$ through the process $\Lambda^{\gamma}$ and observe that,
under the measure $\mathbb{P}^{\gamma}$, the process $Y$ satisfies the
equation

$\displaystyle Y(t)=$
$\displaystyle\,S(t)y+\int_{0}^{t}S(t-s)\tilde{F}(s,Y(s))\,ds$
$\displaystyle+\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}S(t-s)\tilde{G}(s,\xi,Y(s))\,[\pi(d\xi,ds)-\nu({d}\xi)\,{d}s]$
$\displaystyle=$
$\displaystyle\,S(t)y+\int_{0}^{t}S(t-s)\tilde{F}(s,Y(s))\,ds$
$\displaystyle\,+\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}S(t-s)\tilde{G}(s,\xi,Y(s))(r(s,PY(s),\xi,\gamma(s))-1)\,\nu({\rm
d}\xi)\,ds$
$\displaystyle\,+\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}S(t-s)\tilde{G}(s,\xi,Y(s))\,[\pi(d\xi,ds)-r(s,PY(s),\xi,\gamma(s))\,\nu({d}\xi)\,{d}s],$
$\displaystyle=$
$\displaystyle\,S(t)y+\int_{0}^{t}S(t-s)\tilde{F}(s,Y(s))\,ds$
$\displaystyle\,+\int_{0}^{t}S(t-s)\int_{\mathbb{R}^{n}\setminus\\{0\\}}\tilde{G}(s,\xi,Y(s))(r(s,PY(s),\xi,\gamma(s))-1)\,\nu({d}\xi)\,ds]$
$\displaystyle\,+\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}S(t-s)\tilde{G}(s,\xi,Y(s)\,\tilde{\pi}^{\gamma}(d\xi,ds).$

Therefore, $(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\geq
0},\mathbb{P}^{\gamma},L^{\gamma},Y)$ is a mild analytical, weak probabilistic
solution of (6). ∎

The original control problem can be rewritten equivalently in terms of the
Markovian lift $Y$: we have to minimize over all $\gamma\in\mathcal{A}$ the
cost functional

$\mathbb{J}(\gamma)=\mathbb{E}^{\gamma}\left[\int_{0}^{T}l(t,PY(t),\gamma(t))\,{d}t+g(PY(T))\right]$
(28)

where the state $Y$of the system evolves, under $\mathbb{P}^{\gamma}$,
according to the equation (6). We next provide the solution of the optimal
control problem formulated above.

We define classically the Hamiltonian function relative to the above problem:
for all $s\in[0,T],y\in\mathcal{V},z\in
L^{2}(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}),\nu({d}\xi))$

$H(s,y,z)=\inf_{\chi\in
U}\left\\{l(s,Py,\chi)+\int_{\mathbb{R}^{n}\setminus\\{0\\}}z(\xi)\,(r(s,Py,\xi,\chi)-1)\,\nu({d}\xi)\right\\}.$
(29)

The (possibly empty) set of minimizers will be denoted

$\Gamma(s,v,z)=\left\\{\chi\in U\ :\
H(s,y,z)=l(s,Py,\chi)+\int_{\mathbb{R}^{n}\setminus\\{0\\}}z(\xi)\,(r(s,Py,\xi,\chi)-1)\,\nu(d\xi)\right\\}.$
(30)

Let us consider, on the probability space
$(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\geq 0},\mathbb{P})$, the forward
equation

$Y(t)=S(t)y+\int_{0}^{t}S(t-s)\tilde{F}(s,Y(s))\,ds+\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}S(t-s)\tilde{G}(s,\xi,Y(s))\,\tilde{\pi}(d\xi,ds).$
(31)

The solution of (31) is a $\\{\mathcal{F}_{t}\\}$-predictable process, which
exists and is unique by the results in Subsection 3.3. Next, we consider the
associated backward equation associated with the Hamiltonian $H$, with the
final condition $g(PY(T))$ and driven by the random measure $\tilde{\pi}({\rm
d}s,{\rm d}\xi)$

$\Theta_{t}+\int_{t}^{T}\int_{\mathbb{R}^{n}\setminus\\{0\\}}Z_{s}(\xi)\,\tilde{\pi}(ds,d\xi)=g(PY(T))+\int_{t}^{T}H(s,Y(s),Z_{s})\,ds.$
(32)

BSDEs of this type can be considered as a case of BSDE driven by a general
Lévy processes (see [31] or [12, Chapter 19]).

######  Proposition 4.4.

In our assumptions, there exists a unique pair $(\Theta,Z)$ which solves
equation (32) such that $\Theta$ is real-valued, càdlàg and adapted,
$Z:\Omega\times[0,T]\times\mathbb{R}^{n}\rightarrow\mathbb{R}$ is
$\mathcal{P}\otimes\mathcal{B}(\mathbb{R}^{n})$-measurable and

$\displaystyle\mathbb{E}\int_{0}^{T}|\Theta_{s}|^{2}\,{d}s+\mathbb{E}\int_{0}^{T}\int_{\mathbb{R}^{n}\setminus\\{0\\}}|Z_{s}(\xi)|^{2}\nu({d}\xi)\,{d}s<\infty.$

Moreover there exist deterministic functions
$V:[0,T]\times\mathcal{V}\rightarrow\mathbb{R}$ and
$\tilde{V}:[0,T]\times\mathcal{V}\times\mathbb{R}^{n}\rightarrow\mathbb{R}$
such that

$\Theta_{s}=V(s,Y(s))\mbox{ and }Z_{s}(\xi)=\tilde{V}(s,Y(s);\xi).$ (33)

###### Proof.

We start by showing that, under our assumptions, the Hamiltonian function is
Lipschitz continuous in the last variable, i.e., there exists $L\geq 0$ such
that for every $s\in[0,T]$, $y\in\mathcal{V}$, $z,z^{\prime}\in
L^{2}(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}),\nu({d}\xi))$:

$|H(s,y,z)-H(s,y,z^{\prime})|\leq
L\left(\int_{\mathbb{R}^{n}\setminus\\{0\\}}|z(\xi)-z^{\prime}(\xi)|^{2}\,\nu({d}\xi)\right)^{1/2}.$
(34)

The boundedness assumption (21) implies that for every $s\in[0,T]$,
$y\in\mathcal{V}$, $z,z^{\prime}\in
L^{2}(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}),\nu({d}\xi))$, $\gamma\in U$,

$\displaystyle\int_{\mathbb{R}^{n}\setminus\\{0\\}}$ $\displaystyle
z(\xi)\,(r(s,Py,\xi,\gamma)-1)\,\nu(d\xi)$ $\displaystyle\leq$
$\displaystyle\int_{\mathbb{R}^{n}\setminus\\{0\\}}|z(\xi)-z^{\prime}(\xi)|\,(r(s,Py,\xi,\gamma)-1)\,\nu({\rm
d}\xi)$
$\displaystyle+\int_{\mathbb{R}^{n}\setminus\\{0\\}}z^{\prime}(\xi)\,(r(s,Py,\xi,\gamma)-1)\,\nu({d}\xi)$
$\displaystyle\leq$
$\displaystyle(C_{r}+1)\,\nu(\mathbb{R}^{n}\setminus\\{0\\})^{1/2}\left(\int_{\mathbb{R}^{n}\setminus\\{0\\}}|z(\xi)-z^{\prime}(\xi)|^{2}\,\nu({d}\xi)\right)^{1/2}$
$\displaystyle+\int_{\mathbb{R}^{n}\setminus\\{0\\}}z^{\prime}(\xi)\,(r(s,Py,\xi,\gamma)-1)\,\nu({d}\xi),$

so that adding $l(s,Py,\gamma)$ to both sides and taking the infimum over
$\gamma\in U$ it follows that

$\displaystyle H(s,Py,z)\leq
L\left(\int_{\mathbb{R}^{n}\setminus\\{0\\}}|z(\xi)-z^{\prime}(\xi)|^{2}\,\nu({\rm
d}\xi)\right)^{1/2}+H(s,Py,z^{\prime})$

where $L=(C_{r}+1)\,\nu(\mathbb{R}^{n}\setminus\\{0\\})^{1/2}<\infty$;
exchanging $z$ and $z^{\prime}$ we obtain (34).

We conclude the proof by applying Lemma 19.1.5 in [12], which can be applied
here since we are assuming that (22) and (23) hold and we have proved the
estimate in (34). Since the backward stochastic differential equation (32) is
Markovian in $Y$ the connection with a deterministic function is given by
Theorem 19.4.5 in [12]. ∎

Now, using the structure of BSDEs, we can then see that, if an optimal control
exists, then the optimal cost of the problem is given by $\Theta_{0}$ and an
optimal feedback control exists, that is, the optimal control depends only on
the current values of the state variables $(t,Y_{t})$.

######  Theorem 4.5.

In our assumptions, let $(\Theta,Z)$ be the unique solution to the BSDE (32).
For any admissible control $\gamma\in\mathcal{A}$ and for the corresponding
trajectory $Y$ starting at $Y(0)$, we have $\Theta_{0}\leq\mathbb{J}(\gamma)$
and the equality holds if and only if the following feedback law is verified
by $\gamma$ and $Y$, $\mathbb{P}$-a.s. for almost every $t\in[0,T]$:

$\gamma(t)\in\Gamma(t,Y({t-}),z),\qquad\text{where
}z(\xi)=\tilde{V}(t,Y({t-}),\xi),\quad\xi\in\mathbb{R}^{n}.$ (35)

Suppose in addition that the minimizer sets $\Gamma(t,y,z)$ introduced in (30)
are non empty, for $d\mathbb{P}\times{d}t$-almost all $(\omega,t)$ and all
$y\in\mathcal{V}$ and $z\in
L^{2}(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}),\nu(d\xi))$, i.e., there
exists $\gamma^{\prime}\in U$ such that

$\displaystyle H(t,y,z)=$ $\displaystyle
l(s,Py,\gamma^{\prime})+\int_{\mathbb{R}^{n}\setminus\\{0\\}}z(\xi)\,(r(s,Py,\xi,\gamma^{\prime})-1)\,\nu({\rm
d}\xi)$ (36) $\displaystyle=$ $\displaystyle\inf_{\gamma\in
U}\left\\{l(s,Py,\gamma)+\int_{\mathbb{R}^{n}\setminus\\{0\\}}z(\xi)\,(r(s,Py,\xi,\gamma)-1)\,\nu({\rm
d}\xi)\right\\}.$

Then there exists a feedback control, that is, a map
$\bar{\gamma}:[0,T]\times\mathcal{V}\rightarrow U$ such that the process
$\gamma^{*}_{t}=\bar{\gamma}(t,Y({t-}))$ is optimal among all predictable
controls.

Finally the closed-loop equation

$\left\\{\begin{aligned} dY(t)=&[BY(t)+\tilde{F}(t,Y(t))\\\
&\phantom{[BY(t)}+\int_{\mathbb{R}^{n}\setminus\\{0\\}}\tilde{G}(t,\xi,Y(t))(r(t,PY(t),\xi,\bar{\gamma}(t,Y(t)))-1)\nu({\rm
d}\xi)]\,dt\\\
&+\int_{\mathbb{R}^{n}\setminus\\{0\\}}\tilde{G}(t,\xi,Y(t))\,\tilde{\pi}^{\bar{\gamma}}(dt,d\xi),\\\
Y(0)=&y\\\ \end{aligned}\right.$ (37)

admits a mild analytical, weak probabilistic solution.

######  Remark 4.6.

The existence of an element $\gamma^{\prime}\in U$ that satisfies (36) is
crucial for the application of the theorem and the successful resolution of
the optimal control. It is possible to formulate general sufficient conditions
for the existence of $\gamma^{\prime}$. For instance, if
$r(s,x,\xi,\cdot),l(s,x,\cdot):U\rightarrow\mathbb{R}$ are continuous for
every $s\in[0,T],x\in H$ and $U$ is a compact metric space with its Borel
$\sigma$-algebra $\mathcal{U}$, then condition (36) is immediately satisfied.

Using the structure of the FBSDE system, we can then see that, if an optimal
control exists, then an optimal feedback control exists, that is, the optimal
control depends only on the current values of the state variables $(t,Y(t))$.

###### Proof.

Under the reference probability $\mathbb{P}$ the dynamic (6) of $Y$ can be
rewritten

$\left\\{\begin{aligned}
dY(t)=&[BY(t)+\tilde{F}(t,Y(t))]\,dt+\int_{\mathbb{R}^{n}\setminus\\{0\\}}\tilde{G}(t,\xi,Y(t))\,\tilde{\pi}(dt,d\xi),\\\
Y(0)=&y\end{aligned}\right.$ (38)

The process $Y$ is adapted to the filtration $\\{\mathcal{F}_{t}\\}$ generated
by $\tilde{\pi}$ and completed in the usual way. In the space
$(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\geq 0},\mathbb{P})$, we can
consider the system of forward-backward equations

$\left\\{\begin{aligned}
&Y(t)=S(t)y+\int_{0}^{t}S(t-s)\tilde{F}(s,Y(s))\,ds+\int_{0}^{t}\int_{\mathbb{R}^{n}\setminus\\{0\\}}S(t-s)\tilde{G}(s,Y(s))\,\tilde{\pi}(ds,d\xi)\\\
&\Theta_{t}+\int_{t}^{T}\int_{\mathbb{R}^{n}\setminus\\{0\\}}Z_{s}(\xi)\,\tilde{\pi}({d}s,{d}\xi)=g(PY(T))+\int_{t}^{T}H(s,Y(s),Z_{s})\,{d}s.\end{aligned}\right.$
(39)

where the generator $H$ is the Hamiltonian function associated with the
control problem as defined in (29). Writing the backward equation in (39) for
$t=0$ and with respect to the $\mathbb{P}^{\gamma}$ probability we obtain

$\Theta_{0}+\int_{0}^{T}\int_{\mathbb{R}^{n}\setminus\\{0\\}}Z_{s}(\xi)\,\tilde{\pi}^{\gamma}({d}s,{d}\xi)\\\
+\int_{0}^{T}\int_{\mathbb{R}^{n}\setminus\\{0\\}}Z_{s}(\xi)(r(s,\xi,PY(s),\gamma_{s})-1)\nu({d}\xi)\,{d}s=g(PY(T))+\int_{0}^{T}H(s,Y(s),Z_{s})\,{d}s.$
(40)

We observe that the stochastic integral has mean zero with respect to
$\mathbb{P}^{\gamma}$. It is sufficient to see that $Z_{s}$ is integrable with
respect to $\nu^{\gamma}({d}\xi)\,{d}t$ (recall that $\Lambda^{\gamma}_{t}$
were introduced in (25))

$\displaystyle\mathbb{E}^{\gamma}\int_{0}^{T}$
$\displaystyle\int_{\mathbb{R}^{n}\setminus\\{0\\}}|Z_{s}(\xi)|\,r(s,\xi,PY(s),\gamma(s))\nu({d}\xi)\,{d}s$
$\displaystyle\leq$ $\displaystyle
C_{r}\,\mathbb{E}^{\gamma}\int_{0}^{T}\int_{\mathbb{R}^{n}\setminus\\{0\\}}|Z_{s}(\xi)|\,\nu({d}\xi)\,{d}s$
$\displaystyle=$ $\displaystyle
C_{r}\,\mathbb{E}\left[\Lambda_{T}^{\gamma}\int_{0}^{T}\int_{\mathbb{R}^{n}\setminus\\{0\\}}|Z_{s}(\xi)|\,\nu({d}\xi)\,{d}s\right]$
$\displaystyle\leq$ $\displaystyle
C_{r}\left(\mathbb{E}|\Lambda_{T}^{\gamma}|^{2}\right)^{\frac{1}{2}}\left(\mathbb{E}\left|\int_{0}^{T}\int_{\mathbb{R}^{n}\setminus\\{0\\}}|Z_{s}(\xi)|\,\nu({d}\xi)\,{d}s\right|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle\displaystyle
C_{r}\left(\mathbb{E}|\Lambda_{T}^{\gamma}|^{2}\right)^{\frac{1}{2}}\left(T\nu(\mathbb{R}^{n}\setminus\\{0\\})\mathbb{E}\int_{0}^{T}\int_{\mathbb{R}^{n}\setminus\\{0\\}}|Z_{s}(\xi)|^{2}\,\nu({d}\xi)\,{d}s\right)^{\frac{1}{2}}\\!\\!.$

So, if we take in (40) the expectation with respect $\mathbb{P}^{\gamma}$,
adding and subtracting
$\mathbb{E}^{\gamma}\int_{0}^{T}l(s,PY(s),\gamma_{s})\,{d}s$ and recalling
(33), we conclude that

$\Theta_{0}=\mathbb{E}^{\gamma}J(Y(0),\gamma)+\mathbb{E}^{\gamma}\int_{0}^{T}H(s,Y(s),\tilde{V}(s,Y(s),\cdot))\,{d}s\\\
+\mathbb{E}^{\gamma}\int_{0}^{T}\left\\{-l(s,PY(s),\gamma_{s})-\int_{\mathbb{R}^{n}\setminus\\{0\\}}\tilde{V}(s,Y(s),\xi)(r(s,\xi,PY(s),\gamma_{s})-1)\,\nu({d}\xi)\right\\}\,{d}s.$
(41)

The above equality is sometimes called the fundamental relation and
immediately implies that $\Theta_{0}\leq J(\gamma)$ and that the equality
holds if and only if (35) holds.

Assume now that the set $\Gamma$ defined in (30) is not empty. Using
Filippov’s implicit function theorem (see Theorem 21.3.4 in [12]) and taking
into account (33) we see that there is a
$\mathcal{B}([0,T]\times\mathcal{V})$-measurable map $\bar{\gamma}(t,y)$ such
that

$l(t,Py,\bar{\gamma}(t,y))+\int_{\mathbb{R}^{n}\setminus\\{0\\}}\tilde{V}(t,Y(t),\xi)\,(r(t,\xi,Py,\bar{\gamma}(t,y))-1)\,\nu({d}\xi)\\\
=\inf_{\gamma}\\{l(t,Py,\gamma)+\int_{\mathbb{R}^{n}\setminus\\{0\\}}\tilde{V}(t,Y(t),\xi)\,(r(t,\xi,Py,\gamma)-1)\,\nu({d}\xi)$

for all $t\in[0,T]$ and $y\in\mathcal{V}$. Then the process
$\gamma^{*}_{t}=\bar{\gamma}(t,Y({t-}))$ satisfies (35), hence it is optimal.

Finally the existence of a mild analytical, weak probabilistic solution to
equation (37) is again a consequence of Girsanov’s theorem. Namely let $Y$ be
the mild solution of (31) and $\mathbb{P}^{\bar{\gamma}}$ be the probability
under which the compensator of the random measure $\tilde{\pi}$ is given by
$r(t,\xi,PY(t),\bar{\gamma}(t,Y(t)))\,\nu({\rm d}\xi)\,{\rm d}t$. Then $Y$ is
the mild solution of equation (37) relatively to the probability
$\mathbb{P}^{\bar{\gamma}}$ and the Lévy process $L^{\bar{\gamma}}$. ∎

######  Remark 4.7.

Assume further that

$\displaystyle y\mapsto r(t,y,\xi,\gamma)$

is Lipschitz continuous in $y$, uniformly with respect to $t$ and $\xi$ and
$\gamma$.

Suppose that $\bar{\gamma}$ is the process that defines the closed-loop
control. Then we can study directly Eq.(6), driven by the compensated random
measure $\tilde{\pi}^{\bar{\gamma}}$, in the stochastic basis
$(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\},\mathbb{P})$. Using the same
procedure as in Theorem 3.10 we see that $Y^{\bar{\gamma}}$ is a mild
analytical, strong probabilistic solution of Eq.(6) in the sense of Definition
3.9.

As stated in the introduction, the cost functional $\mathbb{J}$ is identical
for the original Volterra equation (3) and the lifted Markovian problem (6),
hence the optimal control presented in this section allows for simultaneous
solving of the control problem for both equations. In particular if, as stated
in the previous remark, we can prove the existence of a closed-loop solution
for the Markovian lift equation, then the same holds for the Volterra equation
(1).

## References

  * [1] E. Abi Jaber and O. El Euch.  Markovian structure of the Volterra Heston model.  Statist. Probab. Lett., 149:63–72, 2019.
  * [2] E. Abi Jaber, E. Miller, and H. Pham.  Linear-quadratic control for a class of stochastic Volterra equations: solvability and approximation.  Ann. Appl. Probab., 31(5):2244–2274, 2021.
  * [3] E. Bandini and F. Confortola.  Optimal control of semi-Markov processes with a backward stochastic differential equations approach.  Math. Control Signals Systems, 29(1):Art. 1, 35, 2017.
  * [4] O. E. Barndorff-Nielsen and J. Schmiegel.  Ambit processes: with applications to turbulence and tumour growth.  In Stochastic analysis and applications, volume 2 of Abel Symp., pages 93–124. Springer, Berlin, 2007.
  * [5] R. Boel, P. Varaiya, and E. Wong.  Martingales on jump processes. II. Applications.  SIAM J. Control, 13(5):1022–1061, 1975.
  * [6] S. Bonaccorsi and F. Confortola.  Optimal control for stochastic Volterra equations with multiplicative Lévy noise.  NoDEA Nonlinear Differential Equations Appl., 27(3):Paper No. 26, 26, 2020.
  * [7] S. Bonaccorsi, F. Confortola, and E. Mastrogiacomo.  Optimal control for stochastic Volterra equations with completely monotone kernels.  SIAM J. Control Optim., 50(2):748–789, 2012.
  * [8] S. Bonaccorsi and G. Desch.  Volterra equations in Banach spaces with completely monotone kernels.  NoDEA Nonlinear Differential Equations Appl., 20(3):557–594, 2013.
  * [9] S. Bonaccorsi and E. Mastrogiacomo.  An analytic approach to stochastic Volterra equations with completely monotone kernels.  J. Evol. Equ., 9(2):315–339, 2009.
  * [10] Z. Brzeźniak and E. Hausenblas.  Maximal regularity for stochastic convolutions driven by Lévy processes.  Probab. Theory Related Fields, 145(3-4):615–637, 2009.
  * [11] P. Carmona and L. Coutin.  Fractional Brownian motion and the Markov property.  Electron. Comm. Probab., 3:95–107, 1998.
  * [12] S. N. Cohen and R. J. Elliott.  Stochastic calculus and applications.  Probability and its Applications. Springer, Cham, second edition, 2015.
  * [13] F. Confortola and M. Fuhrman.  Filtering of continuous-time Markov chains with noise-free observation and applications.  Stochastics, 85(2):216–251, 2013.
  * [14] F. Confortola and M. Fuhrman.  Backward stochastic differential equations associated to jump Markov processes and applications.  Stochastic Process. Appl., 124(1):289–316, 2014.
  * [15] F. Confortola, M. Fuhrman, and J. Jacod.  Backward stochastic differential equation driven by a marked point process: an elementary approach with an application to optimal control.  Ann. Appl. Probab., 26(3):1743–1773, 2016.
  * [16] F. Confortola and E. Mastrogiacomo.  Optimal control for stochastic heat equation with memory.  Evol. Equ. Control Theory, 3(1):35–58, 2014.
  * [17] F. Confortola and E. Mastrogiacomo.  Feedback optimal control for stochastic Volterra equations with completely monotone kernels.  Math. Control Relat. Fields, 5(2):191–235, 2015.
  * [18] C. Cuchiero and J. Teichmann.  Generalized Feller processes and Markovian lifts of stochastic Volterra processes: the affine case.  J. Evol. Equ., 20(4):1301–1348, 2020.
  * [19] G. Desch and S. O. Londen.  Regularity of stochastic integral equations driven by poisson random measures.  Journal of Evolution Equations, 17(1):263–274, 2017.
  * [20] W. Desch and R. K. Miller.  Exponential stabilization of Volterra integro-differential equations in Hilbert space.  J. Differential Equations, 70(3):366–389, 1987.
  * [21] N. El Karoui and S.-J. Huang.  A general result of existence and uniqueness of backward stochastic differential equations.  In Backward stochastic differential equations (Paris, 1995–1996), volume 364 of Pitman Res. Notes Math. Ser., pages 27–36. Longman, Harlow, 1997.
  * [22] Y. Hamaguchi.  Markovian lifting and asymptotic log-Harnack inequality for stochastic Volterra integral equations, arXiv:2304.06683, 2023\.
  * [23] Y. Hamaguchi.  Weak well-posedness of stochastic Volterra equations with completely monotone kernels and non-degenerate noise, arXiv:2310.16030, 2023.
  * [24] P. Harms and D. Stefanovits.  Affine representations of fractional processes with applications in mathematical finance.  Stochastic Process. Appl., 129(4):1185–1228, 2019.
  * [25] J. Jacod.  Multivariate point processes: predictable projection, radon-nikodym derivatives, representation of martingales.  Z. Wahrsch. Verw. Gebiete, 31:235–253, 1974/75.
  * [26] K. Y. Jónsdóttir, J. Schmiegel, and E. B. Vedel Jensen.  Lévy-based growth models.  Bernoulli, 14(1):62–90, 2008.
  * [27] S. Peszat and J. Zabczyk.  Stochastic partial differential equations with Lévy noise, volume 113 of Encyclopedia of Mathematics and its Applications.  Cambridge University Press, Cambridge, 2007.  An evolution equation approach.
  * [28] D. Richardson.  Random growth in a tessellation.  Proc. Cambridge Philos. Soc., 74:515–528, 1973.
  * [29] J. Schmiegel.  Self-scaling tumor growth.  Physica A: Statistical Mechanics and its Applications, 367:509–524, 2006.
  * [30] X. Sun, J. Bao, and Y. Shao.  Mathematical modeling of therapy-induced cancer drug resistance: Connecting cancer mechanisms to population survival rates.  Scientific Reports, 6(1):22498, 2016.
  * [31] S. Tang and X. Li.  Necessary conditions for optimal control of systems with random jumps.  SIAM J. Control Optim, 32:1447–1475, 1994.
  * [32] A. Yin, D. J. A. R. Moes, J. G. C. van Hasselt, J. J. Swen, and H.-J. Guchelaar.  A review of mathematical models for tumor dynamics and treatment resistance evolution of solid tumors.  CPT Pharmacometrics Syst Pharmacol, 8(10):720–737, Aug. 2019.
  * [33] J. Yong and X. Y. Zhou.  Stochastic controls, volume 43 of Applications of Mathematics (New York).  Springer-Verlag, New York, 1999.  Hamiltonian systems and HJB equations.


# Gate-tunable giant superconducting nonreciprocal transport in few-layer
$T_{d}$-MoTe2

T. Wakamura,1 M. Hashisaka,1 S. Hoshino2, M. Bard1, S. Okazaki,3 T. Sasagawa,3
T. Taniguchi,4 K. Watanabe,5 K. Muraki,1 N. Kumada1 1NTT Basic Research
Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi, 243-0198,
Japan  2Department of Physics, Saitama University, Shimo-Okubo, Saitama
338-8570, Japan  3Laboratory for Materials and Structures, Tokyo Institute of
Technology, Nagatsuta, 226-8503, Japan  4International Center for Materials
Nanoarchitectronics, National Institute for Materials and Science, 1-1 Namiki,
Tsukuba, 305-0044, Japan  5Research Center for Functional Materials, National
Institute for Materials and Science, 1-1 Namiki, Tsukuba, 305-0044, Japan

###### Abstract

We demonstrate gate-tunable giant field-dependent nonreciprocal transport
(magnetochiral anisotropy) in a noncentrosymmetric superconductor $T_{\rm
d}$-MoTe2 in the thin limit. Giant magnetochiral anisotropy (MCA) with a
rectification coefficient $\gamma$ = $3.1\times 10^{6}$ T-1 A-1, is observed
at 230 mK, below the superconducting transition temperature ($T_{c}$). This is
one of the largest values reported so far and is likely attributed to the
reduced symmetry of the crystal structure. The temperature dependence of
$\gamma$ indicates that the ratchet-like motion of magnetic vortices is the
origin of the MCA, as supported by our theoretical model. For bilayer $T_{\rm
d}$-MoTe2, we successfully perform gate control of the MCA and realize
threefold modulation of $\gamma$. Our experimental results provide a new route
to realizing electrically controllable superconducting rectification devices
in a single material.

††preprint: APS/123-QED

Recent intensive studies on nonreciprocal transport have revealed the
potential of using noncentrosymmetric materials or inversion-symmetry-breaking
multilayer structures to develop novel rectification devices [1, 2, 3]. In
systems with broken inversion and time-reversal symmetries, Onsager’s
reciprocal theorem allows the electrical resistance to be different for
opposite current directions. This is called magnetochiral anisotropy (MCA),
and it leads to the rectification effect [1].

Broken inversion symmetry is more profitable in superconductors. Rectification
via ratchet-like motion of magnetic vortices was reported more than a decade
ago for superconductors with asymmetric artificial magnetic nanostructures or
with asymmetric antidots as an asymmetric pinning potential, and it was found
that as the asymmetry becomes stronger, rectification is more efficient [4, 5,
6, 7, 8, 9]. Recent studies have pointed out that such ratchet-like motion of
magnetic vortices is also possible in noncentrosymmetric superconductors and
provides large MCA as for other mechanisms such as paraconductivity [10, 11,
12, 13, 14]. In such systems, the asymmetry of the crystal structure
intrinsically induces asymmetric pinning potential for magnetic vortices.
Taking the analogy with previous findings on superconductors with artificial
asymmetric pinning potentials, the symmetry of the crystal should play a
crucial role for the efficiency of the rectification. However, previous
reports on MCA in noncentrosymmetric superconductors has been limited to those
with trigonal symmetry [15, 16, 11, 12, 17]. Therefore, it is particularly
called for to explore MCA in noncentrosymmetric superconductors with different
symmetries, especially with lower symmetry than trigonal symmetry for further
enhancement of the efficiency.

Figure 1: (a) Top view of the crystal structure, in which broken inversion
symmetry is evident. For thin layers, only one mirror plane is present. (b)
Optical microscope image of a 4 ML device. A thin $T_{\rm d}$-MoTe2 flake is
deposited on metallic contacts prepared in advance. (c) Temperature dependence
of the resistance of 4 ML and bilayer samples. (d) Comparison of $R_{2\omega}$
when current is parallel to the $a$-axis (red) and $b$-axis (blue) taken of
the 4 ML sample.  Figure 2: (a) Experimental data from the 4 ML sample. Top:
Nonreciprocal resistance ($R_{\rm 2\omega}$ = $V_{\rm 2\omega}/I_{\rm\omega}$)
measured at different temperatures. Bottom: $R_{\omega}$ signals measured
simultaneously with $R_{2\omega}$. (b) Temperature dependence of $\gamma$
taken from the 4 ML sample. The orange curve shows the fit based on equation
(3). Inset: Experimental data of $\gamma$ as a function of temperature
obtained from the bilayer sample with the fit. (c) Left: Schematic
illustration of the motion of magnetic vortices with the velocity $\mathbf{v}$
driven by an external current $\mathbf{J}$, which generates an electric field
$\mathbf{E}=\mathbf{B}\times\mathbf{v}$. Right: Image of sawtooth potential
assumed as the ratchet pinning potential in (3).

In this Letter, we demonstrate gate-tunable giant MCA in a noncentrosymmetric
superconductor $T_{\rm d}$-MoTe2 in the thin limit. $T_{\rm d}$-MoTe2 lacks
inversion symmetry and, for thin layers, has only one mirror plane normal to
the $b$-axis as shown in Fig. 1(a). This reduced symmetry of the crystal
structure may make the pinning potential for magnetic vortices highly
asymmetric, which can generate a large MCA. We exploit few-layer $T_{\rm
d}$-MoTe2 for our measurements. Below $T_{c}$, we observe MCA under a
perpendicular magnetic field. The rectification coefficient $\gamma$, i.e.,
the ratio of the nonreciprocal (second harmonic) resistance to the linear
resistance, reaches 3.1 $\times$ 106 T-1A-1 at 230 mK, one of the largest
values reported so far. The monotonic increase in $\gamma$ with decreasing
temperature indicates that the giant MCA is due to the ratchet-like motion of
magnetic vortices in the mixed state of the type-II superconductor [10].
Interestingly, despite that $T_{\rm d}$-MoTe2 is a semimetal, in the bilayer
sample we can successfully modulate the MCA via an external gate voltage and
demonstrate threefold modulation of $\gamma$. This ability to produce a large
variation in the nonreciprocal resistance by changing the gate voltage may
provide key insights into the mechanisms behind the giant MCA by associating
it with modulation of the superconducting properties.

Thin semimetallic MoTe2 samples are prepared from high-quality $T_{\rm
d}$-MoTe2 crystals with a residual-resistivity ratio (RRR) $\sim$ 1000 grown
via the flux growth method. Mechanically exfoliated flakes are transferred
onto prepatterned electrodes on a Si/SiO2 chip or hexagonal boron-nitride
(h-BN) deposited on a Si/SiO2 chip by a typical dry transfer technique with
polycarbonate (PC) and poli-dimethylpolysiloxane (PDMS) [18] in an argon-
filled glovebox with a low concentration of O2 and H2O ($<$ 0.5 ppm).
Electrical measurements are performed with a lock-in amplifier and 3He low
temperature measurement system. More details of the fabrication process and
measurement setup are in the Supplemental Material [19].

In materials under broken inversion and time-reversal symmetries, Onsager’s
reciprocal theorem allows the linear longitudinal resistance to be different
for opposite current directions [1]. Rikken et al. heuristically found a
general formula for the nonreciprocal transport, also called the MCA,
expressed as [20]

$R=R_{0}(1+\gamma BI),$ (1)

where $\gamma$ is the rectification coefficient, which quantifies the
efficiency of generating the nonreciprocal resistance. $R_{0}$, $B$ and $I$
are the linear resistance, magnetic field and excitation current,
respectively. Substituting equation (1) into Ohm’s law $V=RI$ leads to

$V=R_{0}I+\gamma BR_{0}I^{2}.$ (2)

The first term is the typical linear voltage response to the current and the
second term is related to the nonreciprocal transport. Thus, the nonreciprocal
response is obtained as a second harmonic signal for the ac excitation current
$I_{\omega}\propto\sin(\omega t)$.

First, we show the temperature dependence of the resistance for the four
monolayer (ML) and bilayer (2 ML) samples in Fig. 1(c). While $T_{c}$ is low
($\sim$ 100 mK) for bulk $T_{\rm d}$-MoTe2 [21], that for the 4 ML and bilayer
samples is 750 mK and 2.2 K, respectively. This large enhancement in $T_{c}$
for thin layers is consistent with previous studies [22, 23]. Note that here
$T_{c}$ is defined as the temperature where the resistance becomes half of
that in the normal state.

Now let us focus on measuring the nonreciprocal transport in the
superconducting state. Figure 1(d) shows the second-harmonic longitudinal
resistance $R_{2\omega}$ for $I_{\omega}\parallel b$ and for
$I_{\omega}\parallel a$ at 230 mK. $a$ and $b$ are the crystal axes as defined
in Fig. 1(a). A clear peak and dip are observed in $R_{2\omega}$ for
$I_{\omega}\parallel b$. The field-asymmetric $R_{2\omega}$ signals are in
agreement with the MCA in (1) and are consistent with previous experimental
results [15, 16, 24, 11]. Note that the nonlinearity of the resistance due to
the transition between the normal and superconducting state is symmetric in
$B$, so it is excluded as the origin of $R_{2\omega}$. In contrast to the case
for $I_{\omega}\parallel b$, $R_{2\omega}$ for $I_{\omega}\parallel a$ is
dramatically suppressed. This is also consistent with the geometry of MCA,
where the symmetry plane, the directions of the magnetic field and generated
second-harmonic voltage are all perpendicular to each other [1]. Note that the
finite signal for $I_{\omega}\parallel a$ is due to misalignment of the
electrodes to the crystal axis (see Fig. 1(b)) [19]. Below we focus on the
geometry where $I_{\omega}\parallel b$.

The top part of Fig. 2(a) displays $R_{2\omega}$ as a function of
perpendicular magnetic field measured at different temperatures. The amplitude
of the signals monotonically decreases with increasing temperature. Above
$T_{c}$, $R_{2\omega}$ is completely suppressed, indicating that the effect is
related to superconductivity. The bottom part of Fig. 2(a) displays the
$R_{\omega}$ signals measured simultaneously with the $R_{2\omega}$ signals.

Now that we have obtained $R_{2\omega}$ and $R_{\omega}$, we can estimate the
value of the rectification coefficient
$\gamma=2R_{2\omega}/(R_{\omega}BI_{\omega})$ [1, 15, 10]. To obtain $\gamma$,
we use the values of $R_{\omega}$ and $R_{2\omega}$ at $B$ where $R_{2\omega}$
is at a peak. Figure 2(b) shows that $\gamma$ continues to increase with
decreasing temperature and reaches $\gamma$ = 3.1 $\times$ 106 T-1 A-1 at 230
mK, the lowest measurement temperature. This value is two to three orders of
magnitude larger than that of other two-dimensional superconductors, such as
MoS2 and NbSe2 as we will discuss later. In the inset of Fig. 2(b) we also
plot the temperature dependence of $\gamma$ for the bilayer sample, which
shows the similar trend with slightly smaller amplitudes than those of the 4
ML sample.

So far, several mechanisms have been proposed to explain the MCA in the
superconducting state [14, 10]. The temperature dependence of the signals and
the direction of the applied magnetic field are clues with which to determine
the mechanism. For example, paraconductivity is one of the mechanisms proposed
as an origin of MCA under an in-plane magnetic field. Since it is relevant to
thermal fluctuations of the superconducting order parameter, the nonreciprocal
signal is slightly enhanced above $T_{c}$ and suppressed much below $T_{c}$.
On the other hand, the ratchet-like motion of the magnetic vortices enhances
the MCA below $T_{c}$ under a perpendicular magnetic field [10]. In the mixed
state of type-II superconductors, magnetic fluxes penetrate the
superconductor, and they are usually trapped by pinning potentials induced by
disorder. External current can drive the magnetic fluxes through the Lorenz
force as schematically shown in the left image of Fig. 2(c), if it is large
enough to overcome the pinning potential [25, 26]. In superconductors with
broken inversion symmetry, the asymmetry of the crystal structure locally
affects the shape of the pinning potentials, making them asymmetric [20, 27].
In this case, the magnetic vortices can exhibit ratchet-like motion, where the
leftward and rightward motion of the vortex is not equivalent [4, 5]. This
asymmetry provides a source for nonreciprocal transport [10, 12, 11, 16, 4, 5,
8, 6, 7]. Increasing $\gamma$ with decreasing temperature is consistent with
the ratchet-like motion of the magnetic vortices as the origin of the MCA
[10]. Such a temperature dependence is because thermal fluctuations of the
magnetic vortices inside the pinning potential, which disturb the ratchet
motion, are suppressed with decreasing temperature, and also the coherence
length, which determines the diameter of the vortex, becomes smaller, making
the vortex more sensitive to the pinning potentials.

Figure 3: (a) Gate voltage ($V_{g}$) dependence of $T_{c}$ for the bilayer
sample taken at 230 mK. (b) $R_{2\omega}$ as a function of $B$ at different
$V_{g}$ at 230 mK. (c) $\gamma$ as a function of $V_{g}$. Inset: Gate voltage
dependence of $R_{2\omega}$.

In Fig. 2(b), we plot the theoretical fit to the experimental data based on
the following theoretical expression assuming the ratchet-like motion of
magnetic vortices as the origin of the MCA [10, 17, 19]:

$\gamma=\frac{\phi_{0}^{*}\beta\ell}{WB}\,\frac{g_{2}(\beta U)}{g_{1}(\beta
U)},$ (3)

where $W$ is the width of the sample, $\phi_{0}^{*}=h/2|e|$ is the flux
quantum and $\beta=1/k_{B}T$ is the inverse temperature. $\ell$ and $U$ are
the mean periodicity and the height of the pinning potential for a vortex,
respectively. We take the simple potential shape shown in the right figure of
Fig. 2(c), where the dimensionless parameter $f$ controls the asymmetry of the
potential. $g_{1}$ and $g_{2}$ are dimensionless functions determined from the
linear- and second-order responses. The ratio is given by $\frac{g_{2}(\beta
U)}{g_{1}(\beta U)}\sim\frac{f(\beta U)^{3}}{180}$ for a moving vortex regime
with a small ratchet potential. The fits follow the experimental data
qualitatively as shown in Fig. 2(b), which supports the ratchet-like motion of
magnetic vortices as the dominant mechanism for the giant MCA in this system.
There are two fitting parameters, and we emphasize that the fits shown in Fig.
2(b) are obtained by using the same fitting parameters both for the 4 ML and
bilayer sample, which also corroborates the validity of our theoretical model
(see SM for more details [19]). Note that the vortex picture may be justified
specifically in the intermediate temperature range below $T_{c}$. At higher
temperatures, the vortex mechanism should be replaced by superconducting
fluctuation and normal contributions [15]. The origin of the deviation at
lower temperatures can be explained by quantum effects of the ratchet-like
motion of vortices [28], where quantum tunneling through the ratchet potential
suppresses the MCA [11]. Nonetheless, the monotonic enhancement of $\gamma$
with decreasing temperature represents the advantage of $T_{\rm d}$-MoTe2
compared to other high-$\gamma$ noncentrosymmetric superconductors with
paraconductivity-based nonreciprocal transport, because even larger $\gamma$
is expected at lower temperatures, and the temperature range for large
$\gamma$ is much broader [24].

We next discuss the amplitude of $\gamma$. Superconductors with trigonal
symmetry are often used in MCA measurements, and $\gamma$ = 8.0 $\times$ 103
T-1 A-1 and 3.4 $\times$ 104 T-1 A-1 have been obtained for MoS2 and NbSe2,
respectively [15, 16]. These values are two or three orders of magnitude
smaller than the value of $\gamma$ = 3.1 $\times$ 106 T-1 A-1 obtained in our
study. The only value comparable to ours reported in the previous studies is
$\gamma=$ 3.2 $\times$ 106 T-1 A-1 from a SrTiO3 Rashba superconductor under
an in-plane magnetic field [24]. As for nonsuperconducting material with
broken inversion symmetry, (Bi1-xSbx)2Te3 (BST) topological nanowires provide
$\gamma\sim 1.0\times 10^{5}$ T-1 A-1 [3]. Therefore, the value obtained in
our study is one of the largest reported so far [15, 16, 11, 12, 17]. We
attribute this gigantic enhancement in $\gamma$ to the reduced symmetry of
$T_{\rm d}$-MoTe2 compared with other noncentrosymmetric superconductors
employed in the previous studies. In comparison with other two-dimensional
superconductors with trigonal symmetry, $T_{\rm d}$-MoTe2 has reduced symmetry
with only one mirror plane for thin layers. This reduced symmetry affects the
asymmetry of the pinning potential. Since the symmetry of the pinning
potential is crucial for the vortex dynamics, as reported previously [8, 9,
29, 30], the lower symmetry in the pinning potentials may generate larger
nonreciprocal signals.

Finally, we demonstrate the gate modulation of the MCA for the bilayer $T_{\rm
d}$-MoTe2. While gate control of the MCA in the normal state has been studied
in a BST topological nanowire [3] and at the LaTiO3/SrTiO3 interface [31], it
has not been reported yet in superconductors. The primary reason for this is
that the concentration of charge carriers in a superconductor is typically
high, making it challenging to employ a conventional solid gate to regulate
superconducting characteristics due to the electric field screening on the
nanometer scale within the material. We can overcome this problem by thinning
down $T_{\rm d}$-MoTe2 to a thickness comparable to the screening length [32,
33]. Figure 3(a) displays the gate dependence of $T_{c}$ obtained from the
bilayer sample. Here the gate voltage ($V_{g}$) is applied through a h-BN (34
nm in thickness) as a gate insulator. $T_{c}$ is successfully modulated by
$V_{g}$, and at $V_{g}$ = 8 V it is larger by around 20 $\%$ compared with at
$V_{g}$ = $-$8 V. In addition to the variation of $T_{c}$, the MCA signals are
also largely modulated by $V_{g}$ (Fig. 3(b)). Figure 3(c) plots $\gamma$ as a
function of $V_{g}$. Here, $\gamma$ varies with $V_{g}$, and $\gamma$ at
$V_{g}$ = $-$8 V is almost three times larger than at $V_{g}$ = 8 V. Note that
this large variation is enabled by modulation of not only $R_{2\omega}$, but
also $R_{\omega}$ and $B$.

The gate voltage can modulate some parameters relevant to superconductivity,
such as $T_{c}$, $B_{c2}$, the magnetic penetration length $\lambda$ and the
coherence length $\xi$. Interestingly, we found that superconductivity becomes
more robust as the gate voltage is made more positive. This means that
$B_{c2}$ becomes larger and $\xi$ smaller for more positive $V_{g}$. This
trend is counterintuitive when we consider the variation in $R_{2\omega}$ with
$V_{g}$, because a larger $B_{c2}$ provides larger $U_{0}$, and a smaller
$\xi$ should be more advantageous for the ratchet-like motion. By contrast,
$\lambda$, which quantifies the scale for the vortex-vortex interaction,
increases as $V_{g}$ decreases, concomitantly with the decline in
superconductivity. Since the importance of the vortex-vortex interaction for
the ratchet-like motion has already been discussed in the context of the
artificial ratchet potentials [29, 34], it may also play a role in intrinsic
ratchet potentials in noncentrosymmetric superconductors. Although further
theoretical studies are required to fully understand our experimental data,
the demonstration of gate control of nonreciprocal transport illustrates the
rich functionality of superconducting nonreciprocal devices for future
applications and also provides key insights into exploring detailed mechanisms
behind the ratchet-like motion of magnetic vortices.

In conclusion, we have shown giant superconducting nonreciprocal transport
(MCA) in thin samples of the noncentrosymmetric superconductor $T_{\rm
d}$-MoTe2. We obtain 3.1 $\times$ 106 T-1 A-1 at 230 mK, one of the largest
values of $\gamma$ recorded so far. The temperature dependence of $\gamma$
supports the ratchet-like motion of magnetic vortices as the origin of the
nonreciprocal transport. The giant nonreciprocal signal is likely due to the
reduced symmetry of the crystal structure of $T_{\rm d}$-MoTe2. We have also
demonstrated gate modulation of the MCA in the superconducting state. In
bilayer $T_{\rm d}$-MoTe2, we obtain a threefold modulation of $\gamma$ using
a typical solid gate. Simultaneous demonstration of the gigantic MCA and its
gate modulation in the superconducting state reveals that $T_{\rm d}$-MoTe2 is
a promising candidate for realizing an electrically-tunable efficient
superconducting rectification devices.

We gratefully acknowledge M. Imai, S. Sasaki, H. Murofushi and S. Wang for
their support in the experiments. This project is financially supported in
part by the JPSJ KAKENHI (grant no. 21H01022, 21H04652, 21K18181, 21H05236,
20H00354 and 19H05790).

## References

  * Tokura and Nagaosa [2018] Y. Tokura and N. Nagaosa, Nonreciprocal responses from non-centrosymmetric quantum materials, Nature Communications 9, 3740 (2018).
  * Ideue  _et al._ [2017] T. Ideue, K. Hamamoto, S. Koshikawa, M. Ezawa, S. Shimizu, Y. Kaneko, Y. Tokura, N. Nagaosa, and Y. Iwasa, Bulk rectification effect in a polar semiconductor, Nature Physics 13, 578 (2017).
  * Legg  _et al._ [2022] H. F. Legg, M. Rossler, F. Munning, D. Fan, O. Breunig, A. Bliesener, G. Lippertz, A. Uday, A. A. Taskin, D. Loss, J. Klinovaja, and Y. Ando, Giant magnetochiral anisotropy from quantum-confined surface states of topological insulator nanowires, Nature Nanotechnology 17, 696 (2022).
  * Villegas  _et al._ [2003a] J. E. Villegas, S. Savel’ev, F. Nori, E. M. Gonzalez, J. V. Anguita, R. García, and J. L. Vicent, A superconducting reversible rectifier that controls the motion of magnetic flux quanta, Science 302, 1188 (2003a).
  * Lee  _et al._ [1999] C.-S. Lee, B. Jankó, I. Derényi, and A.-L. Barabási, Reducing vortex density in superconductors using the ‘ratchet effect’, Nature 400, 337 (1999).
  * de Souza Silva  _et al._ [2006] C. C. de Souza Silva, J. Van de Vondel, B. Y. Zhu, M. Morelle, and V. V. Moshchalkov, Vortex ratchet effects in films with a periodic array of antidots, Phys. Rev. B 73, 014507 (2006).
  * Villegas  _et al._ [2003b] J. E. Villegas, E. M. Gonzalez, M. I. Montero, I. K. Schuller, and J. L. Vicent, Directional vortex motion guided by artificially induced mesoscopic potentials, Phys. Rev. B 68, 224504 (2003b).
  * Zhu  _et al._ [2001] B. Y. Zhu, L. V. Look, V. V. Moshchalkov, B. R. Zhao, and Z. X. Zhao, Vortex dynamics in regular arrays of asymmetric pinning centers, Phys. Rev. B 64, 012504 (2001).
  * Villegas  _et al._ [2005] J. E. Villegas, E. M. Gonzalez, M. P. Gonzalez, J. V. Anguita, and J. L. Vicent, Experimental ratchet effect in superconducting films with periodic arrays of asymmetric potentials, Phys. Rev. B 71, 024519 (2005).
  * Hoshino  _et al._ [2018] S. Hoshino, R. Wakatsuki, K. Hamamoto, and N. Nagaosa, Nonreciprocal charge transport in two-dimensional noncentrosymmetric superconductors, Phys. Rev. B 98, 054510 (2018).
  * Itahashi  _et al._ [2020a] Y. M. Itahashi, Y. Saito, T. Ideue, T. Nojima, and Y. Iwasa, Quantum and classical ratchet motions of vortices in a two-dimensional trigonal superconductor, Phys. Rev. Research 2, 023127 (2020a).
  * Ideue  _et al._ [2020] T. Ideue, S. Koshikawa, H. Namiki, T. Sasagawa, and Y. Iwasa, Giant nonreciprocal magnetotransport in bulk trigonal superconductor $\mathrm{PbTa}{\mathrm{se}}_{2}$, Phys. Rev. Research 2, 042046 (2020).
  * Daido  _et al._ [2022] A. Daido, Y. Ikeda, and Y. Yanase, Intrinsic superconducting diode effect, Phys. Rev. Lett. 128, 037001 (2022).
  * Wakatsuki and Nagaosa [2018] R. Wakatsuki and N. Nagaosa, Nonreciprocal current in noncentrosymmetric $\mathrm{R}$ashba superconductors, Phys. Rev. Lett. 121, 026601 (2018).
  * Wakatsuki  _et al._ [2017] R. Wakatsuki, Y. Saito, S. Hoshino, Y. M. Itahashi, T. Ideue, M. Ezawa, Y. Iwasa, and N. Nagaosa, Nonreciprocal charge transport in noncentrosymmetric superconductors, Science Advances 3, e1602390 (2017).
  * Zhang  _et al._ [2020] E. Zhang, X. Xu, Y.-C. Zou, L. Ai, X. Dong, C. Huang, P. Leng, S. Liu, Y. Zhang, Z. Jia, X. Peng, M. Zhao, Y. Yang, Z. Li, H. Guo, S. J. Haigh, N. Nagaosa, J. Shen, and F. Xiu, Nonreciprocal superconducting $\mathrm{NbSe}_{2}$ antenna, Nature Communications 11, 5634 (2020).
  * Itahashi  _et al._ [2022] Y. M. Itahashi, T. Ideue, S. Hoshino, C. Goto, H. Namiki, T. Sasagawa, and Y. Iwasa, Giant second harmonic transport under time-reversal symmetry in a trigonal superconductor, Nature Communications 13, 1659 (2022).
  * Zomer  _et al._ [2014] P. J. Zomer, M. H. D. Guimaraes, J. C. Brant, N. Tombros, and B. J. van Wees, Fast pick up technique for high quality heterostructures of bilayer graphene and hexagonal boron nitride, Applied Physics Letters 105, 013101 (2014).
  * [19] Supplemental material, .
  * Rikken  _et al._ [2001] G. L. J. A. Rikken, J. Folling, and P. Wyder, Electrical magnetochiral anisotropy, Phys. Rev. Lett. 87, 236602 (2001).
  * Qi  _et al._ [2016] Y. Qi, P. G. Naumov, M. N. Ali, C. R. Rajamathi, W. Schnelle, O. Barkalov, M. Hanfland, S.-C. Wu, C. Shekhar, Y. Sun, V. Suss, M. Schmidt, U. Schwarz, E. Pippel, P. Werner, R. Hillebrand, T. Forster, E. Kampert, S. Parkin, R. J. Cava, C. Felser, B. Yan, and S. A. Medvedev, Superconductivity in $\mathrm{W}$eyl semimetal candidate $\mathrm{MoTe}_{2}$, Nature Communications 7, 11038 (2016).
  * Rhodes  _et al._ [2021] D. A. Rhodes, A. Jindal, N. F. Q. Yuan, Y. Jung, A. Antony, H. Wang, B. Kim, Y.-c. Chiu, T. Taniguchi, K. Watanabe, K. Barmak, L. Balicas, C. R. Dean, X. Qian, L. Fu, A. N. Pasupathy, and J. Hone, Enhanced superconductivity in monolayer $\mathit{T}_{\mathrm{d}}$-$\mathrm{MoTe}_{2}$, Nano Letters 21, 2505 (2021).
  * Gan  _et al._ [2019] Y. Gan, C.-W. Cho, A. Li, J. Lyu, X. Du, J.-S. Wen, and L.-Y. Zhang, Giant enhancement of superconductivity in few layers $\mathrm{MoTe}_{2}$, Chinese Physics B 28, 117401 (2019).
  * Itahashi  _et al._ [2020b] Y. M. Itahashi, T. Ideue, Y. Saito, S. Shimizu, T. Ouchi, T. Nojima, and Y. Iwasa, Nonreciprocal transport in gate-induced polar superconductor $\mathrm{SrTiO}_{3}$, Science Advances 6, eaay9120 (2020b).
  * Tinkham [2004] M. Tinkham,  _Introduction to Superconductivity: 2nd Ed._ (Dover Publications, 2004).
  * Kwok  _et al._ [2016] W.-K. Kwok, U. Welp, A. Glatz, A. E. Koshelev, K. J. Kihlstrom, and G. W. Crabtree, Vortices in high-performance high-temperature superconductors, Reports on Progress in Physics 79, 116501 (2016).
  * Fente  _et al._ [2018] A. Fente, W. R. Meier, T. Kong, V. G. Kogan, S. L. Bud’ko, P. C. Canfield, I. Guillamón, and H. Suderow, Influence of multiband sign-changing superconductivity on vortex cores and vortex pinning in stoichiometric high-$\mathit{T}_{c}\ \ \mathrm{CaKFe}_{4}\mathrm{As}_{4}$, Phys. Rev. B 97, 134501 (2018).
  * Hamamoto  _et al._ [2019] K. Hamamoto, T. Park, H. Ishizuka, and N. Nagaosa, Scaling theory of a quantum ratchet, Phys. Rev. B 99, 064307 (2019).
  * Palau  _et al._ [2012] A. Palau, C. Monton, V. Rouco, X. Obradors, and T. Puig, Guided vortex motion in $\mathrm{YBa}_{2}\mathrm{Cu}_{3}\mathrm{O}_{7}$ thin films with collective ratchet pinning potentials, Phys. Rev. B 85, 012502 (2012).
  * Morgan and Ketterson [1998] D. J. Morgan and J. B. Ketterson, Asymmetric flux pinning in a regular array of magnetic dipoles, Phys. Rev. Lett. 80, 3614 (1998).
  * Choe  _et al._ [2019] D. Choe, M.-J. Jin, S.-I. Kim, H.-J. Choi, J. Jo, I. Oh, J. Park, H. Jin, H. C. Koo, B.-C. Min, S. Hong, H.-W. Lee, S.-H. Baek, and J.-W. Yoo, Gate-tunable giant nonreciprocal charge transport in noncentrosymmetric oxide interfaces, Nature Communications 10, 4510 (2019).
  * Ma  _et al._ [2019] Q. Ma, S.-Y. Xu, H. Shen, D. MacNeill, V. Fatemi, T.-R. Chang, A. M. Mier Valdivia, S. Wu, Z. Du, C.-H. Hsu, S. Fang, Q. D. Gibson, K. Watanabe, T. Taniguchi, R. J. Cava, E. Kaxiras, H.-Z. Lu, H. Lin, L. Fu, N. Gedik, and P. Jarillo-Herrero, Observation of the nonlinear $\mathrm{H}$all effect under time-reversal-symmetric conditions, Nature 565, 337 (2019).
  * Fei  _et al._ [2018] Z. Fei, W. Zhao, T. A. Palomaki, B. Sun, M. K. Miller, Z. Zhao, J. Yan, X. Xu, and D. H. Cobden, Ferroelectric switching of a two-dimensional metal, Nature 560, 336 (2018).
  * Olson  _et al._ [2001] C. J. Olson, C. Reichhardt, B. Jankó, and F. Nori, Collective interaction-driven ratchet for transporting flux quanta, Phys. Rev. Lett. 87, 177002 (2001).