Compensated Poisson Random Measure Research Articles

Impulse Control of Conditional McKean–Vlasov Jump Diffusions

In this paper, we consider impulse control problems involving conditional McKean–Vlasov jump diffusions, with the common noise coming from the σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma $$\\end{document}-algebra generated by the first components of a Brownian motion and an independent compensated Poisson random measure. We first study the well-posedness of the conditional McKean–Vlasov stochastic differential equations (SDEs) with jumps. Then, we prove the associated Fokker–Planck stochastic partial differential equation (SPDE) with jumps. Next, we establish a verification theorem for impulse control problems involving conditional McKean–Vlasov jump diffusions. We obtain a Markovian system by combining the state equation with the associated Fokker–Planck SPDE for the conditional law of the state. Then we derive sufficient variational inequalities for a function to be the value function of the impulse control problem, and for an impulse control to be the optimal control. We illustrate our results by applying them to the study of an optimal stream of dividends under transaction costs. We obtain the solution explicitly by finding a function and an associated impulse control, which satisfy the verification theorem.

An Averaging Principle for Caputo Fractional Stochastic Differential Equations with Compensated Poisson Random Measure

An Averaging Principle for Caputo Fractional Stochastic Differential Equations with Compensated Poisson Random Measure

Mean-field backward stochastic differential equations and applications

In this paper we study the linear mean-field backward stochastic differential equations (mean-field BSDE) of the form (0.1) d Y ( t ) = − [ α 1 ( t ) Y ( t ) + β 1 ( t ) Z ( t ) + ∫ R 0 η 1 ( t , ζ ) K ( t , ζ ) ν ( d ζ ) + α 2 ( t ) E [ Y ( t ) ] + β 2 ( t ) E [ Z ( t ) ] + ∫ R 0 η 2 ( t , ζ ) E [ K ( t , ζ ) ] ν ( d ζ ) + γ ( t ) ] d t + Z ( t ) d B ( t ) + ∫ R 0 K ( t , ζ ) N ̃ ( d t , d ζ ) , t ∈ 0 , T , Y ( T ) = ξ . where ( Y , Z , K ) is the unknown solution triplet, B is a Brownian motion, N ̃ is a compensated Poisson random measure, independent of B . We prove the existence and uniqueness of the solution triplet ( Y , Z , K ) of such systems. Then we give an explicit formula for the first component Y ( t ) by using partial Malliavin derivatives. To illustrate our result we apply them to study a mean-field recursive utility optimization problem in finance.

Stochastic Fubini Theorem for Jump Noises in Banach Spaces

We prove a general version of the stochastic Fubini theorem for stochastic integrals of Banach space valued processes with respect to compensated Poisson random measures under weak integrability assumptions, which extends this classical result from Hilbert space setting to Banach space setting.

Reflected backward stochastic partial differential equations with jumps in a convex domain

This paper is concerned with a class of multi-dimensional reflected backward stochastic partial differential equations (BSPDE for short), taking values in a convex domain in Rk, which are driven by an infinite dimensional Brownian motion and an independent compensated Poisson random measure. Existence and uniqueness of the solution to this class of reflected BSPDEs are proved. Penalization method plays an important role.

Pricing and hedging of general rating-sensitive claims in a jump-diffusion market model in the presence of stochastic factors

In this paper we solve a problem of finding a risk minimizing hedging strategy on a general Markovian market with ratings on which prices are influenced by additional factors and rating. The market is described by a system of SDEs driven by a Wiener process and a compensated Poisson random measure. Rating-sensitive claims are considered. We relate the problem of pricing and hedging the contracts described by a general cash-flow process to solving Cauchy/Dirichlet problems and subsequently to solving some linear system of equations. We illustrate our theory on two examples of different nature. The first is a general exponential Lévy model with stochastic volatility, and the second is a generalization of an exponential Lévy model with regime switching.

Pathwise uniqueness and non-explosion of SDEs driven by compensated Poisson random measures

In this note, we consider stochastic differential equations (SDEs) driven by compensated Poisson random measure. We showed that the solution of the SDEs admits non-explosion for a class of super-linear coefficients, and moreover, we proved that the pathwise uniqueness holds under certain non-Lipschitzian conditions on the coefficients.

Martingale solutions for the three-dimensional stochastic nonhomogeneous incompressible Navier–Stokes equations driven by Lévy processes

In this paper, the three-dimensional stochastic nonhomogeneous incompressible Navier–Stokes equations driven by Lévy processes consisting of the Brownian motion, the compensated Poisson random measure and the Poisson random measure are considered in a bounded domain. We obtain the existence of martingale solutions. The construction of the solution is based on the classical Galerkin approximation method, the stopping times, the stochastic compactness method and the Jakubowski–Skorokhod theorem.

Global Solutions to Stochastic Volterra Equations Driven by Lévy Noise

Abstract In this paper we investigate the existence and uniqueness of semilinear stochastic Volterra equations driven by multiplicative Lévy noise of pure jump type. In particular, we consider the equation $$\begin{array}{} \left\{ \begin{aligned} du(t) & = \left( A\int_0 ^t b(t-s) u(s)\,ds\right) \, dt + F(t,u(t))\,dt \\ & {} + \int_ZG(t,u(t), z) \tilde \eta(dz,dt) + \int_{Z_L}G_L(t,u(t), z) \eta_L(dz,dt),\, t\in (0,T],\\ u(0)&=u_0, \end{aligned} \right. \end{array} $$ where Z and ZL are Banach spaces, η̃ is a time-homogeneous compensated Poisson random measure on Z with intensity measure ν (capturing the small jumps), and ηL is a time-homogeneous Poisson random measure on ZL independent to η̃ with finite intensity measure νL (capturing the large jumps). Here, A is a selfadjoint operator on a Hilbert space H, b is a scalar memory function and F, G and GL are nonlinear mappings. We provide conditions on b, F G and GL under which a unique global solution exists. We also present an example from the theory of linear viscoelasticity where our result is applicable. The specific kernel b(t) = cρtρ−2, 1 < ρ < 2, corresponds to a fractional-in-time stochastic equation and the nonlinear maps F and G can include fractional powers of A.

Long time behavior for stochastic Burgers equations with jump noises

In this paper, we consider stochastic Burgers equations driven by compensated Poisson random measures. Under some appropriate conditions, we investigate the exponential stability on the solutions of the equations. Examples are presented to illustrate the applications of the results.

Finite element methods and their error analysis for SPDEs driven by Gaussian and non-Gaussian noises

In this paper, we investigate the mean square error of numerical methods for SPDEs driven by Gaussian and non-Gaussian noises. The Gaussian noise considered here is a Hilbert space valued Q-Wiener process and the non-Gaussian noise is defined through compensated Poisson random measure associated to a Lévy process. As the models consider the influences of Gaussian and non-Gaussian noises simultaneously, this makes the models more realistic when the models are also influenced by some randomly abrupt factors, but more complicated. As a consequence, the numerical analysis of the problems becomes more involved. We first study the regularity for the mild solution. Next, we propose a semidiscrete finite element scheme in space and a fully discrete linear implicit Euler scheme for the SPDEs, and rigorously obtain their error estimates. Both the regularity results of the mild solution and error estimates obtained in the paper are novel.

Linear Volterra backward stochastic integral equations

We present an explicit solution triplet (Y,Z,K) to the backward stochastic Volterra integral equation (BSVIE) of linear type, driven by a Brownian motion and a compensated Poisson random measure. The process Y is expressed by an integral whose kernel is explicitly given. The processes Z and K are expressed by Hida–Malliavin derivatives involving Y.

Second-Order Numerical Schemes for Decoupled Forward-Backward Stochastic Differential Equations with Jumps

We propose new numerical schemes for decoupled forward-backward stochastic differential equations (FBSDEs) with jumps, where the stochastic dynamics are driven by a $d$-dimensional Brownian motion and an independent compensated Poisson random measure. A semi-discrete scheme is developed for discrete time approximation, which is constituted by a classic scheme for the forward SDE [17, 25] and a novel scheme for the backward SDE. Under some reasonable regularity conditions, we prove that the semi-discrete scheme can achieve second-order convergence in approximating the FBSDEs of interest; and such convergence rate does not require jump-adapted temporal discretization. Next, to add in spatial discretization, a fully discrete scheme is developed by designing accurate quadrature rules for estimating the involved conditional mathematical expectations. Several numerical examples are given to illustrate the effectiveness and the high accuracy of the proposed schemes.

Itô formula for mild solutions of SPDEs with Gaussian and non-Gaussian noise and applications to stability properties

Abstract We use the Yosida approximation to find an Itô formula for mild solutions { X x ⁢ ( t ) , t ≥ 0 } {\{X^{x}(t),t\geq 0\}} of SPDEs with Gaussian and non-Gaussian colored noise, with the non-Gaussian noise being defined through a compensated Poisson random measure associated to a Lévy process. The functions to which we apply such Itô formula are in C 1 , 2 ⁢ ( [ 0 , T ] × H ) {C^{1,2}([0,T]\times H)} , as in the case considered for SDEs in [15]. Using this Itô formula, we prove exponential stability and exponential ultimate boundedness properties, in the mean square sense, for mild solutions. We also compare this Itô formula to an Itô formula for mild solutions introduced by Ichikawa in [12], and an Itô formula written in terms of the semigroup of the drift operator [5], which we extend to the non-Gaussian case.

Maximal inequalities for stochastic convolutions driven by compensated Poisson random measures in Banach spaces

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 Fr\'{e}chet 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)\xi(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 $(Z,\mathcal{Z})$ and $\xi:[0,\infty)\times\Omega\times Z\rightarrow E$ is an $\mathbb{F}\otimes \mathcal{Z}$-predictable function. We prove that there exists a c\`{a}dl\`{a}g modification a $\tilde{u}$ of the process $u$ which satisfies the following maximal inequality \begin{align*} \mathbb{E} \sup_{0\leq s\leq t} \|\tilde{u}(s)\|^{q^\prime}\leq C \mathbb{E} \left(\int_0^t\int_Z \|\xi(s,z) \|^{p}\,N(\mathrm{d} s,\mathrm{d} z)\right)^{\frac{q^\prime}{p}}, \end{align*} for all $ q^\prime \geq q$ and $1<p\leq 2$ with $C=C(q,p)$.

Long time stability of nonlocal stochastic Kuramoto–Sivashinsky equations with jump noises

In this paper, we consider a class of nonlocal stochastic Kuramoto–Sivashinsky equations driven by compensated Poisson random measures. Under appropriate conditions, we prove almost sure exponential stability and second moment stability of the solution.

Malliavin Differentiability of Solutions of SPDEs with Lévy White Noise

We consider a stochastic partial differential equation (SPDE) driven by a Lévy white noise, with Lipschitz multiplicative term σ. We prove that, under some conditions, this equation has a unique random field solution. These conditions are verified by the stochastic heat and wave equations. We introduce the basic elements of Malliavin calculus with respect to the compensated Poisson random measure associated with the Lévy white noise. If σ is affine, we prove that the solution is Malliavin differentiable and its Malliavin derivative satisfies a stochastic integral equation.

Prediction-Correction Scheme for Decoupled Forward Backward Stochastic Differential Equations with Jumps

AbstractBy introducing a new Gaussian process and a new compensated Poisson random measure, we propose an explicit prediction-correction scheme for solving decoupled forward backward stochastic differential equations with jumps (FBSDEJs). For this scheme, we first theoretically obtain a general error estimate result, which implies that the scheme is stable. Then using this result, we rigorously prove that the accuracy of the explicit scheme can be of second order. Finally, we carry out some numerical experiments to verify our theoretical results.

Mean-field SDEs with jumps and nonlocal integral-PDEs

Recently Buckdahn et al. (Mean-field stochastic differential equations and associated PDEs, arXiv:1407.1215 , 2014) studied a mean-field stochastic differential equation (SDE), whose coefficients depend on both the solution process and also its law, and whose solution process $${(X_s^{t,x,P_\xi},X_s^{t,\xi}=X_s^{t,x,P_\xi}|_{x=\xi})}$$ , $${s\in[t,T], (t,x)\in[0,T]\times \mathbb{R}^{d}, \xi\in L^2(\mathcal{F}_{t},\mathbb{R}^{d})}$$ , admits the flow property. This flow property is the key for the study of the associated nonlocal partial differential equation (PDE). In this work we extend these studies in a non-trivial manner to mean-field SDEs which, in addition to the driving Brownian motion, are governed by a compensated Poisson random measure. We show that under suitable regularity assumptions on the coefficients of the SDE, the solution $${X^{t,x,P_\xi}}$$ is twice differentiable with respect to x and its law. We establish the associated nonlocal integral-PDE, and we show that $${V(t,x,P_\xi)=E[\Phi(X_T^{t,x,P_\xi},P_{X_T^{t,\xi}})]}$$ is the unique classical solution $${V:[0,T]\times \mathbb{R}^{d}\times \mathcal{P}_{2}(\mathbb{R}^{d})\rightarrow\mathbb{R}}$$ of this nonlocal integral-PDE with terminal condition $${\Phi}$$ .

On some smoothening effects of the transition semigroup of a Lévy process

Let (Pt) be the transition semigroup of a Lévy process (Lt) taking values in a Hilbert space H. Let ν and N˜ respectively be the Lévy measure and compensated Poisson random measure of (Lt). It is shown that for any bounded and measurable function f,AqPtf(x)=1tE[f(Ltx)∫0t∫Hq(y)N˜(ds,dy)]for all t>0,x∈H, where Aq is some nonlocal operator. As a corollary,∫H|Ptf(x+y)−Ptf(x)|2ν(dy)≤1tPtf2(x)for all t>0,x∈H. As ν can be infinite this formula establishes some smoothening effect of the semigroup (Pt). In the paper some applications of the formula will be presented as well.