Pathwise convergence rates for numerical solutions of Markovian switching stochastic differential equations

This work develops numerical approximation algorithms for solutions of stochastic differential equations with Markovian switching. The existing numerical algorithms all use a discrete-time Markov chain for the approximation of the continuous-time Markov chain. In contrast, we generate the continuous-time Markov chain directly, and then use its skeleton process in the approximation algorithm. Focusing on weak approximation, we take a re-embedding approach, and define the approximation and the solution to the switching stochastic differential equation on the same space. In our approximation, we use a sequence of independent and identically distributed (i.i.d.) random variables in lieu of the common practice of using Brownian increments. By virtue of the strong invariance principle, we ascertain rates of convergence in the pathwise sense for the weak approximation scheme.     

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 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     

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.     

Optimal approximation of stochastic differential equations by adaptive step-size control

We study the pathwise (strong) approximation of scalar stochastic differential equations with respect to the global error in the -norm. For equations with additive noise we establish a sharp lower error bound in the class of arbitrary methods that use a fixed number of observations of the driving Brownian motion. As a consequence, higher order methods do not exist if the global error is analyzed. We introduce an adaptive step-size control for the Euler scheme which performs asymptotically optimally. In particular, the new method is more efficient than an equidistant discretization. This superiority is confirmed in simulation experiments for equations with additive noise, as well as for general scalar equations.

     

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     

On the existence of weak solutions to stochastic differential equations with degenerate diffusion

For a certain class of stochastic differential equations with nonlinear drift and degenerate diffusion term existence of a weak solution is shown.     

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.      

Global existence of solutions for stochastic impulsive differential equations

In this paper we obtain some results on the global existence of solution to Itô stochastic impulsive differential equations in M([0,∞),? ⁿ ) which denotes the family of ? ⁿ -valued stochastic processes x satisfying sup_t∈[0,∞) \(\mathbb{E}\)|x(t)|² < ∞ under non-Lipschitz coefficients. The Schaefer fixed point theorem is employed to achieve the desired result. An example is provided to illustrate the obtained results.     

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.     

An optimization approach to weak approximation of stochastic differential equations with jumps

We propose an optimization approach to weak approximation of stochastic differential equations with jumps. A mathematical programming technique is employed to obtain numerically upper and lower bound estimates of the expectation of interest, where the optimization procedure ends up with a polynomial programming. A major advantage of our approach is that we do not need to simulate sample paths of jump processes, for which few practical simulation techniques exist. We provide numerical results of moment estimations for Doléans-Dade stochastic exponential, truncated stable Lévy processes and Ornstein-Uhlenbeck-type processes to illustrate that our method is able to capture very well the distributional characteristics of stochastic differential equations with jumps.     

Fundamental solutions of stochastic differential equations with drift

The paper deals with the pathwise uniqueness of solutions to one-dimensional time homogeneous stochastic differential equations with a diffusion coefficient σ satisfying the local time condition and measurable drift term b. We show that if the functions σ and b satisfy a non-degeneracy condition and fundamental solution to considered equation is unique in law, then pathwise uniqueness of solutions holds. Our result is in some sense negative, more precisely we give an example of an equation with Holder continuous diffusion coefficient and nondegenerate drift for which a fundamental solution is not unique in law and pathwise uniqueness of solutions does not hold.     

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.     

Existence and uniqueness of the solutions of stochastic differential equations

The standard existence and uniqueness theorem for stochastic differential equations requires Lipschitz condition of the coefficients. In this paper, we extend these results to the case in which the coefficients are not required to be Lipschitz continuous, instead they only satisfy a ‘weak’ type of Lipschitz condition.     

On almost periodic mild solutions for stochastic functional differential equations

In this paper, a class of stochastic functional differential equations given by

     

Strong convergence rate for multivalued stochastic differential equations via stochastic theta method

In this paper we study the stochastic theta method for multivalued stochastic differential equations driven by standard Brownian motions and obtain the strong convergence rate of this numerical scheme.     

An efficient approximation for stochastic differential equations on the partition ofsymmetricalirst

In certain applications of stochastic differential equations a numerical solution must be found corresponding to a particular sample path of the driving process. The order of convergence of approximations based on regular samples of the path is limited, and some approximations are asymptotically efficient in that they minimise the leading coefficient in the expansion of mean-square errors as power series in the sample step size. This paper considers approximations based on irregular samples taken at the passage times of the driving process through a series of thresholds. Such approximations can involve less computation than their regular sample counterparts, particularly for real-time applications. The orders of convergence of the Euler and Milshtein approximations are derived and a new approximation is defined which is asymptotically efficient with respect to the irregular samples. Its asymptotic mean-square error is less than half that of efficient approximations based on regular sample