Diffusion equation for multivalued stochastic differential equations

Deterministic oscillations with bilinear hysteresis are governed by a multivalued differential equation of the type ξ′ + kξ?b(ξ) + g, where k is maximal monotonic and b is Lipschitzian. An existence and uniqueness result is proven for corresponding stochastic equation. The diffusion equation satisfied by the laws of ξ(t) is established. In the particular case k = 0, this equation is equivalent to the Fokker-Planck equation.     

On existence of solutions of multivalued stochastic differential equations with discontinuous coefficients

The subject of the paper is to find existence conditions of weak solutions to multivalued stochastic differential equations with discontinuous coefficients. First we prove that a non-exploding solution exists when the drift coefficient b satisfies linear growth and the diffusion coefficient σ is uniformly elliptic. On this basis, we continue to obtain a solution (up to the explosion time) in the weak sense under certain local integrability, improving the result of Rozkosz and S?omiński.     

Generalized solutions of a stochastic partial differential equation

We discuss the Cauchy problem of a certain stochastic parabolic partial differential equation arising in the nonlinear filtering theory, where the initial data and the nonhomogeneous noise term of the equation are given by Schwartz distributions. The generalized (distributional) solution is represented by a partial (conditional) generalized expectation ofT(t)° _0,t ^–1 , whereT(t) is a stochastic process with values in distributions and _s,t is a stochastic flow generated by a certain stochastic differential equation. The representation is used for getting estimates of the solution with respect to Sobolev norms.Further, by applying the partial Malliavin calculus of Kusuoka-Stroock, we show that any generalized solution is aC -function under a condition similar to Hörmander's hypoellipticity condition.     

On Wiener-Poisson type multivalued stochastic differential equations with non-Lipschitz coefficients

In this paper, we prove local uniqueness for multivalued stochastic differential equations with Poisson jumps. Then existence and uniqueness of global solutions is obtained under the conditions that the coefficients satisfy locally Lipschitz continuity and one-sided linear growth of b. Moreover, we also prove the Markov property of the solution and the existence of invariant measures for the corresponding transition semigroup.     

On approximations of the Euler-Peano scheme for multivalued stochastic differential equations

It is known that a unique strong solution exists for multivalued stochastic differential equations under the Lipschitz continuity and linear growth conditions. In this paper we apply the Euler-Peano scheme to show that existence of weak solution and pathwise uniqueness still hold when the coefficients are random and satisfy one-sided locally Lipschitz continuous and an integral condition (i.e. Krylov's conditions put forward in On Kolmogorov's equations for finite-dimensional diffusions, Stochastic PDE's and Kolmogorov Equations in Infinite Dimensions (Cetraro, 1998), Lecture Notes in Math., 1715, Springer, Berlin, 1999, pp. 1–63). When the coefficients are nonrandom and possibly discontinuous but only satisfy some integral conditions, the sequence of solutions of the Euler-Peano scheme converges weakly, and the limit is a weak solution of the corresponding MSDE. As a particular case, we obtain a global semi-flow for stochastic differential equations reflected in closed, convex domains.     

Exponential ergodicity of non-Lipschitz multivalued stochastic differential equations

Under the conditions of coefficients being non-Lipschitz and the diffusion coefficient being elliptic, we study the strong Feller property and irreducibility for the transition probability of solutions to general multivalued stochastic differential equations by using the coupling method, Girsanov's theorem and a stopping argument. Thus we can establish the exponential ergodicity and the spectral gap.     

On non-explosion for the solution of a stochastic differential equation

In the present work we give general sufficient conditions in terms of Lyapunov functions which ensure existence of a weak solution for a stochastic differential equation for which conditions of existence hold in every bounded region.     

Continuous dependence properties on solutions of backward stochastic differential equation

The existence theorem and continuous dependence property in ”L²” sense for solutions of backward stochastic differential equation (shortly BSDE) with Lipschitz coefficients were respectively established by Pardoux-Peng and Peng in [1,2], Mao and Cao generalized the Pardoux-Peng’s existence and uniqueness theorem to BSDE with non-Lipschitz coefficients in [3,4]. The present paper generalizes the Peng’s continuous dependence property in ”L²” sense to BSDE with Mao and Cao’s conditions. Furthermore, this paper investigates the continuous dependence property in “almost surely” sense for BSDE with Mao and Cao’s conditions, based on the comparison with the classical mathematical expectation.     

Numerical schemes for multivalued backward stochastic differential systems

We define approximation schemes for generalized backward stochastic differential systems, considered in the Markovian framework. More precisely, we propose a mixed approximation scheme for the following backward stochastic variational inequality:

A transfer principle for multivalued stochastic differential equations

In this paper we prove a transfer principle for multivalued stochastic differential equations.     

On a stochastic fractional partial differential equation with a fractional noise

We will prove the existence, uniqueness and regularity of the solution for a stochastic fractional partial differential equation driven by an additive fractional space–time white noise. Moreover, the absolute continuity of the solution is also obtained.     

On a stochastic differential equation arising in a price impact model

We provide sufficient conditions for the existence and uniqueness of solutions to a stochastic differential equation which arises in the price impact model developed by Bank and Kramkov (2011)  and . These conditions are stated as smoothness and boundedness requirements on utility functions or Malliavin differentiability of payoffs and endowments.     

On solutions to Ito stochastic differential equations

In this paper we obtain general conditions under which stochastic differential equations possess a strong solution representable in an explicit form as a functional of the Wiener process. Particular interest bears the problem of determining conditions that guarantee non-explosion of the solution. The necessary as well as sufficient condition is derived.