Convergence of numerical solutions to neutral stochastic delay differential equations with Markovian switching

Recently, numerical solutions of stochastic differential equations have received a great deal of attention. It is surprising that there are not any numerical methods established for neutral stochastic delay differential equations yet. In the paper, the Euler–Maruyama method for neutral stochastic delay differential equations is developed. The key aim is to show that the numerical solutions will converge to the true solutions under the local Lipschitz condition.     

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.     

Weak solutions to gamma-driven stochastic differential equations

We study a stochastic differential equation driven by a gamma process, for which we give results on the existence of weak solutions under conditions on the volatility function. To that end we provide results on the density process between the laws of solutions with different volatility functions.     

Convergence of numerical solutions to stochastic age-dependent population equations

In this paper, stochastic age-dependent population equations, one of the important classes of hybrid systems, are studied. In general, most of stochastic age-dependent population equations do not have explicit solutions, thus numerical approximation schemes are invaluable tools for exploring their properties. The main purpose of this paper is to develop a numerical scheme and show the convergence of the numerical approximation solution to the true solution.     

Convergence of solutions of backward stochastic equations

We establish conditions for the weak convergence of solutions of backward stochastic equations in the case of the weak convergence of coefficients.     

Markov solutions of stochastic differential equations

Probability Theory and Related Fields -     

Numerical solutions to time-fractional stochastic partial differential equations

Numerical Algorithms - This study is concerned with numerical approximations of time-fractional stochastic heat-type equations driven by multiplicative noise, which can be used to model the...     

Simplified existence for solutions to stochastic differential equations

Nonstandard methods are used to give a simple construction of a solution to SDEs of the form , where are required only to be measurable, with, bounded. By working with an internal Brownian motion the proof avoids the complicated lifting and approximation arguments needed in previous existence proofs.     

Convergence of solutions of nonlinear differential equations

Summary Solutions of are said to converge if every pair of solutions x(t), y(t) satisfy x(t) − y(t) →0 as t → ∞. An invariance principle of LaSalle is used to determine conditions under which the solutions of converge. In certain cases the approach used does not require boundedness of solutions as has been required in most previous results on convergence of solutions. The results of this investigation are applied to a number of nonlinear second order differential equations. Sufficient conditions are also found for the convergence of solutions of certain functional differential equations. Entrata in Redazione il 10 febbraio 1976.     

Successive approximations of solutions to stochastic functional differential equations

In the present paper, by means of the successive approximations method, the local or global existence and uniqueness theorems for a stochastic functional differential equation of the Ito type are proved.     

Malliavin regularity of solutions to mixed stochastic differential equations

For a mixed stochastic differential equation driven by independent fractional Brownian motions and Wiener processes, the existence and integrability of the Malliavin derivative of the solution are established. It is also proved that the solution possesses exponential moments.     

Almost sure explosion of solutions to stochastic differential equations

This paper is concerned with the problem of explosive solutions for a class of stochastic differential equations. Our main results are presented as two theorems. Theorem 1 is concerned with the existence of explosive solutions with positive probability under certain sufficient conditions. With some additional mild conditions, it is shown in Theorem 2 that the explosion will occur almost surely. The methods of auxiliary functions and cycles are used in the proofs. Several remarks about their applications are given.     

Lp solutions of backward stochastic differential equations

In this paper, we are interested in solving backward stochastic differential equations (BSDEs for short) under weak assumptions on the data. The first part of the paper is devoted to the development of some new technical aspects of stochastic calculus related to BSDEs. Then we derive a priori estimates and prove existence and uniqueness of solutions in L^p p>1, extending the results of El Karoui et al. (Math. Finance 7(1) (1997) 1) to the case where the monotonicity conditions of Pardoux (Nonlinear Analysis; Differential Equations and Control (Montreal, QC, 1998), Kluwer Academic Publishers, Dordrecht, pp. 503–549) are satisfied. We consider both a fixed and a random time interval. In the last section, we obtain, under an additional assumption, an existence and uniqueness result for BSDEs on a fixed time interval, when the data are only in L¹.     

Periodic solutions of affine stochastic differential equations

We discuss the problem of the existence of periodic and stationary solutions of affine stochastic differential equations. We prove that under a controllability condition the system has a periodic solution if and only if the linear part is eyponentially stable in mean square.

It is also shown that the controllability assumption is necessary for the existence of a “unique” weakly periodic solution with nondegenerate covariance.     

Convergence of numerical solutions to stochastic pantograph equations with Markovian switching

In this paper, a class of stochastic pantograph equations with Markovian switching is considered. The main purpose is to investigate the convergence of the Euler method of the equations. It is proved that the Euler approximation solution converge to the analytic solution in probability under weaker conditions. An example is provided to illustrate our theory.     

Convergence of solutions of perturbed nonlinear differential equations

Summary We use the nonlinear variation of parameters formula to investigate the convergence of the solutions of nonlinear perturbed systems of differential equations. This research was supported in part by the National Science Foundation under grant GP-11543. Entrata in Redazione il 9 ottobre 1971.