In an M-type 2 Banach space, firstly we explore some properties of the set-valued stochastic integral associated with the stationary Poisson point process. By using the Hahn decomposition theorem and bounded linear functional, we obtain the main result: the integral of a set-valued stochastic process with respect to the compensated Poisson measure is a set-valued submartingale but not a martingale unless the integrand degenerates into a single-valued process. Secondly we study the strong solution to the set-valued stochastic integral equation, which includes a set-valued drift, a single-valued diffusion driven by a Brownian motion and the set-valued jump driven by a Poisson process.
We prove new upper and lower bounds for Banach space-valued stochastic integrals with respect to a compensated Poisson random measure. Our estimates apply to Banach spaces with non-trivial martingale (co)type and extend various results in the literature. We also develop a Malliavin framework to interpret Poisson stochastic integrals as vector-valued Skorohod integrals, and prove a Clark-Ocone representation formula.
Let $(E, \| \cdot\|)$ be a Banach space such that, for some $q\geq 2$, the function $x\mapsto \|x\|^q$ is of $C^2$ class and its first and second Fréchet derivatives are bounded by some constant multiples of $(q-1)$-th power of the norm and $(q-2)$-th power of the norm and let $S$ be a $C_0$-semigroup of contraction type on $(E, \| \cdot\|)$. We consider the following stochastic convolution process \begin{align*} u(t)=\int_0^t\int_ZS(t-s)ξ(s,z)\,\tilde{N}(\mathrm{d} s,\mathrm{d} z), \;\;\; t\geq 0, \end{align*} where $\tilde{N}$ is a compensated Poisson random measure on a measurable space $(Z…
We prove maximal inequalities for $L_q$-valued martingales obtained by stochastic integration with respect to compensated random measures. A version of these estimates for integrals with respect to compensated Poisson random measures were first obtained by Dirksen (arXiv:1208:3885) using arguments based on inequalities for sums of independent Banach-space-valued random variables, geometric properties of Banach spaces, and decoupling inequalities. Our proofs are completely different and rely almost exclusively on classical stochastic analysis for real semimartingales.
In this note we consider a class of neutral stochastic functional differential equations with finite delay driven simultaneously by a fractional Brownian motion and a Poisson point processes in a Hilbert space. We prove an existence and uniqueness result and we establish some conditions ensuring the exponential decay to zero in mean square for the mild solution by means of the Banach fixed point principle.
Let $G$ be a locally compact group and $π$ a representation of $G$ by weakly^* continuous isometries acting in a dual Banach space $E$. Given a probability measure $μ$ on $G$ we study the Choquet-Deny equation $π(μ)x=x$, $x\in E$. We prove that the solutions of this equation form the range of a projection of norm 1 and can be represented by means of a ``Poisson formula'' on the same boundary space that is used to represent the bounded harmonic functions of the random walk of law $μ$. The relation between the space of solutions of the Choquet-Deny equation in $E$ and the space of bounded harmon…
We study measures in Banach space which arise as the skew convolution product of two other measures where the convolution is deformed by a skew map. This is the structure that underlies both the theory of Mehler semigroups and operator self-decomposable measures. We show how that given such a set-up the skew map can be lifted to an operator that acts at the level of function spaces and demonstrate that this is an example of the well known functorial procedure of second quantisation. We give particular emphasis to the case where the product measure is infinitely divisible and study the second q…
In this paper, we employ Markov process theory to prove asymptotic results for a class of stochastic processes which arise as solutions of a stochastic evolution inclusion and are given by the representation formula \begin{align*} \mathbb{X}_{x}(t)=\sum \limits_{m=0}\limits^{\infty}T((t-α_{m})_{+})(x_{x,m})1\hspace{-0,9ex}1_{[α_{m},α_{m+1})}(t), \end{align*} where $(T(t))_{t \geq 0}$ is a (nonlinear) time-continuous, contractive semigroup acting on a separable Banach space $(V,||\cdot||_{V})$, $(α_{m})_{m \in \mathbb{N}}$ is the sequence of arrival times of a homogeneous Poisson process, $x$ i…
In this paper we prove Burkholder-Davis-Gundy inequalities for a general martingale $M$ with values in a UMD Banach space $X$. Assuming that $M_0=0$, we show that the following two-sided inequality holds for all $1\leq p<\infty$: \begin{align}\label{eq:main}\tag{$\star$} \mathbb E \sup_{0\leq s\leq t} \|M_s\|^p \eqsim_{p, X} \mathbb E γ([\![M]\!]_t)^p ,\;\;\; t\geq 0. \end{align} Here $ γ([\![M]\!]_t) $ is the $L^2$-norm of the unique Gaussian measure on $X$ having $[\![M]\!]_t(x^*,y^*):= [\langle M,x^*\rangle, \langle M,y^*\rangle]_t$ as its covariance bilinear form. This extends to general U…
In this article we investigate the existence and uniqueness of the stochastic Volterra equation driven by a \levy noise of pure jump type. In particular, we consider the following type of equation $ du(t) = ( A\int_0 ^t b(t-s) u(s)\,ds) \, dt + F(t,u(t))\,dt+ \int_ZG(t,u(t), z) \tilde η(dz,dt) + \int_{Z_L}G_L(t,u(t), z) η_L(dz,dt) ;\, t\in (0,T], $, $u(0)=u_0$, where $Z$ and $Z_L$ are Banach spaces, $\tilde η$ is a time-homogeneous compensated Poisson random measure on $Z$ with \levy measure $ν$ capturing the small jumps, and $η_L$ is a time-homogeneous Poisson random measure on $Z_L$ with fin…
We will give a proof of the following fact. If $\mathfrak{A}_1$ and $\mathfrak{A}_2$, $\tilde η_1$ and $\tilde η_2$, $ξ_1$ and $ξ_2$ are two examples of filtered probability spaces, time homogeneous compensated Poisson random measures, and progressively measurable Banach space valued processes such that the laws on $L^p([0,T],{L}^{p}(Z,ν;E))\times \CM_I([0,T]\times Z)$ of the pairs $(ξ_1,η_1)$ and $(ξ_2,η_2)$ %, $i=1,2$, are equal, and $u_1$ and $u_2$ are the corresponding stochastic convolution processes, then the laws on $ (\DD([0,T];X)\cap L^p([0,T];B)) \times L^p([0,T],{L}^{p}(Z,ν;E))\time…
On explicit solutions to a class of quadratic BSDEJs driven by affine Volterra processes with jumps and applications
Laboratoire de Probabilités, Statistique et Modélisation, UMR 8001, Sorbonne Université and Université Paris Cité（概率、统计与建模实验室，UMR 8001，索邦大学与巴黎西岱大学）；BPCE SA（BPCE股份公司）
The dynamics of an infinite system of point particles in $\mathbb{R}^d$, which hop and interact with each other, is described at both micro- and mesoscopic levels. The states of the system are probability measures on the space of configurations of particles. For a bounded time interval $[0,T)$, the evolution of states $μ_0 \mapsto μ_t$ is shown to hold in a space of sub-Poissonian measures. This result is obtained by: (a) solving equations for correlation functions, which yields the evolution $k_0 \mapsto k_t$, $t\in [0,T)$, in a scale of Banach spaces; (b) proving that each $k_t$ is a correla…
