A stability theorem for stochastic differential equations and application to stochastic control problems

For a sequence of stochastic differential equations of the the type: a stabilty theorem is presented under appropritate convergence mode of [d] and m application to stochastic control problems is also briefly discussed.     

A stability theorem of backward stochastic differential equations and its application

In this Note, we establish a stability theorem for backward stochastic differential equations, and we apply this theorem to study the homogenization of systems of semilinear parabolic partial differential equations.     

Backward stochastic partial differential equations with jumps and application to optimal control of random jump fields

We prove an existence and uniqueness result for a general class of backward stochastic partial differential equations (SPDE) with jumps. This is a type of equations, which appear as adjoint equations in the maximum principle approach to optimal control of systems described by SPDE driven by Lévy processes.     

Indefinite stochastic linear-quadratic optimal control problems with random jumps and related stochastic Riccati equations

We discuss the stochastic linear-quadratic (LQ) optimal control problem with Poisson processes under the indefinite case. Based on the wellposedness of the LQ problem, the main idea is expressed by the definition of relax compensator that extends the stochastic Hamiltonian system and stochastic Riccati equation with Poisson processes (SREP) from the positive definite case to the indefinite case. We mainly study the existence and uniqueness of the solution for the stochastic Hamiltonian system and obtain the optimal control with open-loop form. Then, we further investigate the existence and uniqueness of the solution for SREP in some special case and obtain the optimal control in close-loop form.     

Backward stochastic differential equations and applications to optimal control

We study the existence and uniqueness of the following kind of backward stochastic differential equation, $$x(t) + \int_t^T {f(x(s),y(s),s)ds + \int_t^T {y(s)dW(s) = X,} }$$ under local Lipschitz condition, where (Ω, ?,P, W(·), ?_t) is a standard Wiener process, for any given (x, y),f(x, y, ·) is an ?_t-adapted process, andX is ?_t-measurable. The problem is to look for an adapted pair (x(·),y(·)) that solves the above equation. A generalized matrix Riccati equation of that type is also investigated. A new form of stochastic maximum principle is obtained.     

A Lax equivalence theorem for stochastic differential equations

In this paper, a stochastic mean square version of Lax’s equivalence theorem for Hilbert space valued stochastic differential equations with additive and multiplicative noise is proved. Definitions for consistency, stability, and convergence in mean square of an approximation of a stochastic differential equation are given and it is shown that these notions imply similar results as those known for approximations of deterministic partial differential equations. Examples show that the assumptions made are met by standard approximations.     

An existence theorem for optimal control systems with state variable inC,and stochastic control problems

We consider an existence theorem for control systems whose state variables for everyt are inC, the set of continuous functions varying over a given setI. The dependence of the state variables upona I is induced by their dependence upon the initial state and the state equation governing the system. In contrast, the controlu=u(t) is taken as a measurable function oft alone. The usual space constraints and boundary conditions are also allowed to vary overaI, and the cost functional is now taken to be a continuous functional over a suitable class of continuous functions. We also discuss an application of these results to control systems with stochastic boundary conditions.This research was accomplished under Grant No. AF-AFOSR-942-65. The author is grateful to Dr. Lamberto Cesari for his suggestions and assistance in the preparation of this paper.     

A converse comparison theorem for backward stochastic differential equations with jumps

This paper establishes a converse comparison theorem for real-valued backward stochastic differential equations with jumps.     

Malliavin calculus applied to optimal control of stochastic partial differential equations with jumps

In this paper, we employ Malliavin calculus to derive a general stochastic maximum principle for stochastic partial differential equations with jumps under partial information. We apply this result to solve an optimal harvesting problem in the presence of partial information. Another application pertains to portfolio optimization under partial observation.     

Nonparametric estimation for stochastic differential equations with random effects

We consider N$N$ independent stochastic processes (_Xj(t),t∈[0,T])$(X_j\left(t\right),t\in [0,T])$, j=1,…,N$j=1,\dots ,N$, defined by a one-dimensional stochastic differential equation with coefficients depending on a random variable _?j${?}_j$ and study the nonparametric estimation of the density of the random effect _?j${?}_j$ in two kinds of mixed models. A multiplicative random effect and an additive random effect are successively considered. In each case, we build kernel and deconvolution estimators and study their L²$L^2$-risk. Asymptotic properties are evaluated as N$N$ tends to infinity for fixed T$T$ or for T=T(N)$T=T\left(N\right)$ tending to infinity with N$N$. For T(N)=N²$T\left(N\right)=N^2$, adaptive estimators are built. Estimators are implemented on simulated data for several examples.     

Rate of convergence of transport processes with an application to stochastic differential equations

Summary We obtain a rate of convergence of uniform transport processes to Brownian motion, which we apply to the Wong and Zakai approximation of stochastic integrals.The research of both authors was supported by a NSERC Canada Grant and by an EMR Canada Grant of M. Csörgö at Carleton University, Ottawa     

A new comparison theorem for solutions of stochastic differential equations

Let Z ₁(t) and Z ₂(t) be solutions of two stochastic differential equations. Then Z ₁(t)≦Z ₂(t) for all t?0 a.s. provided certain relations involving the coefficients and intial conditions of the equations hold. the diffusion coefficients are not required toi be the same for both equtions     

Existence of solutions and optimal control for reflecting stochastic differential equations with applications to population control theory *

For the d–dimensional reflecting stochastic differential equations (1) with non-smooth boundary and unbounded domain the existence of a strong solution, (weak solution) is obtained under the conditions that the coefficients are less than linear growth and they are non-Lipschitz, (and the diffusion coefficient is non-degenerate, the drift coefficient is bounded and measurable only). Moreover, the Girsanov theorem and the martingale representation theorem with respect to system (1) are also derived. Then by using the Ekeland lemma and the martingale method the existence, necessary and sufficient conditions for an optimal control and an optimal control are obtained. The results are then applied to solve an optimal control problem for a stochastic population model     

Existence theorems for optimal control problems involving functional differential equations

This paper extends some existence theorems of Cesari for optimal control problems to systems whose dynamics is described by functional differential equations of finitely-retarded type. We show that the proper choice of state space is the spaceE ¹×C[–, 0], where >0 represents the time-lag of the system, and that it is necessary to choose initial conditions from a compact set inC[–, 0] as well as to employ the usual growth condition.This research was accomplished in the frame of research project AFOSR-942-65 at the University of Michigan. In particular, the author would like to thank Professor L. Cesari (University of Michigan) and Professor N. Chafee (Brown University) for many helpful remarks during the preparation of the research, which forms part of the author's doctoral dissertation written at the University of Michigan.     

A comparison theorem and uniqueness theorem of backward doubly stochastic differential equations

In this paper, we deal with one dimensional backward doubly stochastic differential equations (BDSDEs). We obtain a comparison theorem and a uniqueness theorem for BDSDEs with continuous coefficients.     

Infinite-dimensional stochastic differential equations related to random matrices

We solve infinite-dimensional stochastic differential equations (ISDEs) describing an infinite number of Brownian particles interacting via two-dimensional Coulomb potentials. The equilibrium states of the associated unlabeled stochastic dynamics are the Ginibre random point field and Dyson’s measures, which appear in random matrix theory. To solve the ISDEs we establish an integration by parts formula for these measures. Because the long-range effect of two-dimensional Coulomb potentials is quite strong, the properties of Brownian particles interacting with two-dimensional Coulomb potentials are remarkably different from those of Brownian particles interacting with Ruelle’s class interaction potentials. As an example, we prove that the interacting Brownian particles associated with the Ginibre random point field satisfy plural ISDEs.     

Existence and Hyers-Ulam stability of random impulsive stochastic functional differential equations with finite delays

ABSTRACT

In this paper, we investigate the existence and Hyers-Ulam stability for random impulsive stochastic functional differential equations with finite delays. Firstly, we prove the existence of mild solutions to the equations by using Krasnoselskii's fixed point. Then, we investigate the Hyers-Ulam stability results under the Lipschitz condition on a bounded and closed interval. Finally, an example is given to illustrate our results.     

Exponential stability for nonlinear stochastic differential equations with respect to semimartingales

Consider a nonlinear stochastic differential equations with respect to semimartingales (1) dY(t) = F{Y(t),t)dti(t) + G(t)dM(t)+f(Y(t),t)dti(t)+g(Y(t),t)dM(t), which might be regarded as a stochastic perturbed system of (2) dX(t) = F(X(t),t)dfi(t) + G(t)dM(t). Suppose Eq. (2) is exponentially stable almost surely. Under what conditions is Eq. (1) still exponentially stable almost surely? In this paper we will give some sufficient conditions. As an application we also discuss the almost sure exponential stability for semilinear stochastic systems and small stochastic perturbed systems