On a class of stochastic differential equations arising from the stochastic approximation theory

The paper dealt with generalized stochastic approximation procedures of Robbins-Monro type. We consider these procedures as strong solutions of some stochastic differential equations with respect to semimartingales and investigate their almost sure convergence and mean square convergence     

On Wong-Zakai approximation of stochastic differential equations

An approximation theorem of stochastic differential equations driven by semimartingales is proved, based on approximation of semimartingales by a sequence of processes with piecewise monotonic sample functions.     

On the approximation of stochastic differential equations

The stability properties of stochastic differential equations with respetct to the perturbation of the coefficients and of the driving processes are investigated in the topology of uniform convergence in probability     

Optimal control of stochastic differential equations with dynamical boundary conditions

In this paper we investigate the optimal control problem for a class of stochastic Cauchy evolution problems with nonstandard boundary dynamic and control. The model is composed by an infinite dimensional dynamical system coupled with a finite dimensional dynamics, which describes the boundary conditions of the internal system. In other terms, we are concerned with nonstandard boundary conditions, as the value at the boundary is governed by a different stochastic differential equation.     

Optimal control of stochastic functional differential equations with a bounded memory

This paper treats a finite time horizon optimal control problem in which the controlled state dynamics are governed by a general system of stochastic functional differential equations with a bounded memory. An infinite dimensional Hamilton–Jacobi–Bellman (HJB) equation is derived using a Bellman-type dynamic programming principle. It is shown that the value function is the unique viscosity solution of the HJB equation.     

The dynamic programming equation for stochastic optical control in hilbert spaces: a variational approach

The global existence of a point wise solution to the Hamilton-Jacobi equation for totally observed controlled diffusions in Hilbert spaces is proved by studying the corresponding control problem. The optimality principle for the control problem leads to local results, whilst an a priori bound is achieved by introducing a secondary minimization problem.     

On the approximation of stochastic partial differential equations i

The stability of abstract stochastic partial differential equations with respect to the simultaneous perturbation of the driving processes and of the differential operators is investigated. The results obtained here will be applied to concrete stochastic partial differential equations in the continuation of this paper     

approximation for the solutions of stochastic differential equations. iii. jointly weak convergence

We give sufficient conditions for a family Z, e > 0 of continuous finite variation processes to converge weakly to a diffusion process Z. Then we consider the integral equation dXE(t) = (l)(Xe(t))dZE{t) and the stochastic equation dX{i) = (j)(X{t))dZ{t) and denote by X(t,x,w respectively X{t,x,(jo), the solution starting at x. We prove that PoX~l, e>0 converge weakly to Pol     

A step size control algorithm for the weak approximation of stochastic differential equations

A vriable step size control algorithm for the weak approximation of stochastic differential equations is introduced. The algorithm is based on embedded Runge–Kutta methods which yield two approximations of different orders with a negligible additional computational effort. The difference of these two approximations is used as an estimator for the local error of the less precise approximation. Some numerical results are presented to illustrate the effectiveness of the introduced step size control method.      

Optimal control for infinite dimensional stochastic differential equations with infinite Markov jumps and multiplicative noise

In this paper we solve an infinite-horizon linear quadratic control problem for a class of differential equations with countably infinite Markov jumps and multiplicative noise. The global solvability of the associated differential Riccati-type equations is studied under detectability hypotheses. A nonstochastic, operatorial approach is used. Some properties of the linear stochastic systems, such as stability, stabilizability and detectability, are also discussed on the basis of a new solution representation result. A generalized Ito's formula which applies to infinite dimensional stochastic differential equations with countably infinite Markov jumps is also provided.     

Random attractors of stochastic partial differential equations: A smooth approximation approach

We use the method of smooth approximation to examine the random attractor for two classes of stochastic partial differential equations (SPDEs). Roughly speaking, we perturb the SPDEs by a Wong-Zakai scheme using smooth colored noise approximation rather than the usual polygonal approximation. After establishing the existence of the random attractor of the perturbed system, we prove that when the colored noise tends to the white noise, the random attractor of the perturbed system with colored noise converges to that of the original SPDEs by invoking some continuity results on attractors in random dynamical systems.     

Adaptive stepsize based on control theory for stochastic differential equations

The numerical solution of stochastic differential equations (SDEs) has been focussed recently on the development of numerical methods with good stability and order properties. These numerical implementations have been made with fixed stepsize, but there are many situations when a fixed stepsize is not appropriate. In the numerical solution of ordinary differential equations, much work has been carried out on developing robust implementation techniques using variable stepsize. It has been necessary, in the deterministic case, to consider the “best” choice for an initial stepsize, as well as developing effective strategies for stepsize control—the same, of course, must be carried out in the stochastic case.

In this paper, proportional integral (PI) control is applied to a variable stepsize implementation of an embedded pair of stochastic Runge–Kutta methods used to obtain numerical solutions of nonstiff SDEs. For stiff SDEs, the embedded pair of the balanced Milstein and balanced implicit method is implemented in variable stepsize mode using a predictive controller for the stepsize change. The extension of these stepsize controllers from a digital filter theory point of view via PI with derivative (PID) control will also be implemented. The implementations show the improvement in efficiency that can be attained when using these control theory approaches compared with the regular stepsize change strategy.     

An analytic approximation of solutions of stochastic differential delay equations with Markovian switching

In this paper, we are concerned with the stochastic differential delay equations with Markovian switching (SDDEwMSs). As stochastic differential equations with Markovian switching (SDEwMSs), most SDDEwMSs cannot be solved explicitly. Therefore, numerical solutions, such as EM method, stochastic Theta method, Split-Step Backward Euler method and Caratheodory’s approximations, have become an important issue in the study of SDDEwMSs. The key contribution of this paper is to investigate the strong convergence between the true solutions and the numerical solutions to SDDEwMSs in the sense of the L^p-norm when the drift and diffusion coefficients are Taylor approximations.     

Optimal non-proportional reinsurance control and stochastic differential games

We study stochastic differential games between two insurance companies who employ reinsurance to reduce risk exposure. We consider competition between two companies and construct a single payoff function of two companies’ surplus processes. One company chooses a dynamic reinsurance strategy in order to maximize the payoff function while its opponent is simultaneously choosing a dynamic reinsurance strategy so as to minimize the same quantity. We describe the Nash equilibrium of the game and prove a verification theorem for a general payoff function. For the payoff function being the probability that the difference between two surplus reaches an upper bound before it reaches a lower bound, the game is solved explicitly.     

A numerical approximation of parabolic stochastic partial differential equations driven by a Poisson random measure

The paper deals with SPDEs driven by Poisson random measures in Banach spaces and its numerical approximation. We investigate the accuracy of space and time approximation. As the space approximation we consider spectral methods and as time approximation the implicit Euler scheme and the explicit Euler scheme. AMS subject classification (2000) 60H15, 35R30