Stochastic differential equations with time-dependent reflecting barriers

We study existence, uniqueness and stability of solutions of stochastic differential equations with time-dependent reflecting barriers in the general case where compensating reflection processes are not necessarily of bounded variations and solutions need not be semimartingales. Applications to models of stock prices with natural boundaries of Bollinger bands type are given.     

Stochastic differential equations with jump reflection at the boundary

A unique solution is shown to exist of a stochastic differential equation where the solution is subject to random kicks whenever it reaches or crosses a pre-selected boundary. The driving terms are arbitrary semimartingales, and the solutions are shown not to have explosions, even with essentially no restrictions on the stochastic kicking process.     

Stochastic partial differential equations in bounded domains with Dirichlet boundary conditions

In this paper we prove the existence of a unique solution for a class of stochastic parabolic partial differential equations in bounded domains, with Dirichlet boundary conditions. The main tool is an equivalence result, provided by the stochastic characteristics method, between the stochastic equations under investigation and a class of deterministic parabolic equations with moving boundaries, depending on random coefficients. We show the existence of the solution to this last problem, thus providing a solution to the former.     

Penalization methods for reflecting stochastic differential equations with jumps

The problem of approximation of a solution to a reflecting stochastic differential equation (SDE) with jumps by a sequence of solutions to SDEs with penalization terms is considered. The approximating sequence is not relatively compact in the Skorokhod topology J ¹ and so the methods of approximation based on the J ¹-topology break down. In the paper, we prove our convergence results in the S-topology on the Skorokhod space D(R⁺,?R ^d ) introduced recently by Jakubowski. The S-topology is weaker than J ¹ but stronger than the Meyer-Zheng topology and shares many useful properties with J ¹.     

Stochastic differential equations

When a system is acted upon by exterior disturbances, its time-development can often be described by a system of ordinary differential equations, provided that the disturbances are smooth functions. But for sound reasons physicists and engineers usually want the theory to apply when the noises belong to a larger class, including for example “white noise.” If the integrals in the system derived for smooth noises are reinterpreted as Itô integrals, the equations make sense; but in nonlinear cases they often fail to describe the time-development of the system. In this paper (extending previous work of the author) a calculus is set up for stochastic systems that extends to a theory of differential equations. When the equations are known that describe the development of the system when noises are smooth, an extension to the larger class of noises is proposed that in many cases gives results consistent with the smooth-noise case and also has “robust” solutions, that change by small amounts when the noises undergo small changes. This is called the “canonical” extension.Nevertheless, there are certain systems in which the canonical equations are inappropriate. A criterion is suggested that may allow us to distinguish when the canonical equations are the right choice and when they are not.     

Stochastic differential equations with fractal noise

Stochastic differential equations in ?ⁿ with random coefficients are considered where one continuous driving process admits a generalized quadratic variation process. The latter and the other driving processes are assumed to possess sample paths in the fractional Sobolev space W^β₂ for some β > 1/2. The stochastic integrals are determined as anticipating forward integrals. A pathwise solution procedure is developed which combines the stochastic Itô calculus with fractional calculus via norm estimates of associated integral operators in W^α ₂ for 0 < α < 1. Linear equations are considered as a special case. This approach leads to fast computer algorithms basing on Picard's iteration method. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stochastic differential equations for Dirichlet processes

We consider the stochastic differential equation dX _t = a(X _t )dW _t + b(X _t )dt, where W is a one-dimensional Brownian motion. We formulate the notion of solution and prove strong existence and pathwise uniqueness results when a is in C ^1/2 and b is only a generalized function, for example,the distributional derivative of a H?lder function or of a function of bounded variation. When b = aa′, that is, when the generator of the SDE is the divergence form operator ℒ = , a result on non-existence of a strong solution and non-pathwise uniqueness is given as well as a result which characterizes when a solution is a semimartingale or not. We also consider extensions of the notion of Stratonovich integral. Received: 23 February 2000 / Revised version: 22 January 2001 / Published online: 23 August 2001     

正向多维带跳随机微分方程比较定理

本文研究了高维及矩阵值带跳随机微分方程, 给出其比较定理成立的一个充分必要条件.     

Stochastic set differential equations

In this paper we present the existence and uniqueness of solutions to the stochastic set differential equations driven by a Wiener process. In the paper we also study their continuous dependence on initial conditions and a stability property.     

Stochastic functional differential equations with infinite delay

The stability and boundedness of the solution for stochastic functional differential equation with finite delay have been studied by several authors, but there is almost no work on the stability of the solutions for stochastic functional differential equations with infinite delay. The main aim of this paper is to close this gap. We establish criteria of pth moment ψγ(t)-bounded for neutral stochastic functional differential equations with infinite delay and exponentially stable criteria for stochastic functional differential equations with infinite delay, and we also illustrate the result with an example.     

Interior and boundary difference equations for hyperbolic differential equations

Interior and boundary difference equations are derived for several hyperbolic partial differential equations by means of an integral method. The method is applied to a simple transport equation, to waves in a compressible, isentropic fluid, and to surface waves in shallow water. Boundary conditions treated are (a) a perfectly reflecting boundary, (b) an open boundary with outgoing waves and a specified incoming wave, and (c) a partially reflecting boundary. For open boundaries, the major assumption for the algorithms to be valid is that outgoing waves can be defined, an assumption equivalent to the most general statement of Sommerfeld's radiation condition. The difference equations obtained are conservative, second-order accurate, two time-level, explicit, and stable (for one-dimensional, time-dependent problems) for cΔt/Δx ? 1 where c is the wave speed, Δt is the temporal grid size, and Δx is the spatial grid size. Numerical calculations demonstrate the excellent accuracy of the procedure.     

Linear stochastic differential equations with boundary conditions

Summary We study linear stochastic differential equations with affine boundary conditions. The equation is linear in the sense that both the drift and the diffusion coefficient are affine functions of the solution. The solution is not adapted to the driving Brownian motion, and we use the extended stochastic calculus of Nualart and Pardoux [16] to analyse them. We give analytical necessary and sufficient conditions for existence and uniqueness of a solution, we establish sufficient conditions for the existence of probability densities using both the Malliavin calculus and the co-aera formula, and give sufficient conditions that the solution be either a Markov process or a Markov field.Supported in part by NSF Grant No. MCS-8301880The research was carried out while this author was visiting the Institute for Advanced Study, Princeton NJ, and was supported by a grant from the RCA corporation     

Fractional partial differential equations with boundary conditions

We identify the stochastic processes associated with one-sided fractional partial differential equations on a bounded domain with various boundary conditions. This is essential for modelling using spatial fractional derivatives. We show well-posedness of the associated Cauchy problems in $C_0(\mathrm{\Omega })$ and $L_1(\mathrm{\Omega })$. In order to do so we develop a new method of embedding finite state Markov processes into Feller processes on bounded domains and then show convergence of the respective Feller processes. This also gives a numerical approximation of the solution. The proof of well-posedness closes a gap in many numerical algorithm articles approximating solutions to fractional differential equations that use the Lax–Richtmyer Equivalence Theorem to prove convergence without checking well-posedness.     

Advanced differential equations with nonlinear boundary conditions

We investigate the existence of solutions for advanced differential equations with nonlinear boundary conditions. Sufficient conditions when the problem has extremal solutions or a unique solution are formulated. Linear advanced differential inequalities are also discussed.