Searcharxiv⌕ Search

arXiv subjects

Tusheng Zhang

Publications and source records attributed to Tusheng Zhang.

At least 73 records · Page 4Linked to original sources
arXiv2017

A stochastic HJB equation for optimal control of forward-backward SDEs

We study optimal stochastic control problems of general coupled systems of forward-backward stochastic differential equations with jumps. By means of the Itô-Ventzell formula the system is transformed to a controlled backward stochastic partial differential equation (BSPDE) with jumps. Using a comparison principle for such BSPDEs we obtain a general stochastic Hamilton-Jacobi- Bellman (HJB) equation for such control problems. In the classical Markovian case with optimal control of jump diffusions, the equation reduces to the classical HJB equation. The results are applied to study risk minimization in financial markets.

math.OC↗
arXiv2016

Backward Doubly SDEs and Semilinear Stochastic PDEs in a convex domain

This paper presents existence and uniqueness results for reflected backward doubly stochastic differential equations (in short RBDDSEs) in a convex domain D. Moreover, using a stochastic flow approach a probabilistic interpretation for a class of reflected SPDE's in a domain is given via such RBDSDEs. The solution is expressed as a pair (u,ν) where u is a predictable continuous process which takes values in a Sobolev space and m is a random regular measure. The bounded variation process K, component of the solution of the reflected BDSDE, controls the set when u reaches the boundary of D. This bounded variation process determines the measure m from a particular relation by using the inverse of the flow associated to the the diffusion operator.

math.PR↗
arXiv2016

Smooth densities of the laws of perturbed diffusion processes

Under some regularity conditions on $b$, $σ$ and $α$, we prove that the following perturbed stochastic differential equation \begin{equation} X_t=x+\int_0^t b(X_s)ds+\int_0^t σ(X_s) dB_s+α\sup_{0 \le s \le t} X_s, \ \ \ α<1 \end{equation} admits smooth densities for all $0 \le t \le t_0$, where $t_0>0$ is some finite number.

math.PR↗
arXiv2015

Lattice Approximations of Reflected Stochastic Partial Differential Equations Driven by Space-Time White Noise

We introduce a discretization/approximation scheme for reflected stochastic partial differential equations driven by space-time white noise through systems of reflecting stochastic differential equations. To establish the convergence of the scheme, we study the existence and uniqueness of solutions of Skorohod-type deterministic systems on time-dependent domains. We also need to establish the convergence of an approximation scheme for deterministic parabolic obstacle problems. Both are of independent interest on their own.

math.PR↗
arXiv2014

A Note on the Malliavin Differentiability of One-Dimensional Reflected Stochastic Differential Equations with Discontinuous Drift

We consider a one-dimensional Stochastic Differential Equation with reflection where we allow the drift to be merely bounded and measurable. It is already known that such equations have a unique strong solution. Recently, it has been shown that non-reflected SDE's with discontinuous drift possess more regularity than one could expect, namely they are Malliavin differentiable and weakly differentiable w.r.t. the initial value. In this paper we show that similar results hold for one-dimensional SDE's with reflection. We then apply the results to get a Bismut-Elworthy-Li formula for the corresponding Kolmogorov equation.

math.PR↗
arXiv2014

A comparison theorem for backward SPDEs with jumps

In this paper we obtain a comparison theorem for backward stochastic partial differential equation (SPDEs) with jumps. We apply it to introduce space-dependent convex risk measures as a model for risk in large systems of interacting components.

math.PR↗
arXiv2014

Approximations of Stochastic Partial Differential Equations

In this paper we show that solutions of stochastic partial differential equations driven by Brownian motion can be approximated by stochastic partial differential equations forced by pure jump noise/random kicks. Applications to stochastic Burgers equations are discussed.

math.PR↗
arXiv2013

Stochastic Generalized Porous Media Equations with Reflection

A non-negative Markovian solution is constructed for a class of stochastic generalized porous media equations with reflection. To this end, some regularity properties and a comparison theorem are proved for stochastic generalized porous media equations, which are interesting by themselves. Invariant probability measures and ergodicity of the solution are also investigated.

math.PR↗