Solutions Of SDEs Research Articles

On approximation of solutions of multidimensional SDE's with reflecting boundary conditions

Let D be either a convex domain in R d or a domain satisfying the conditions (A) and (B) considered by Lions and Sznitman [7] and Saisho [11]. We estimate the rate of L p convergence for Euler and Euler–Peano schemes for stochastic differential equations in D with normal reflection at the boundary of the form X t=X 0+∫ t 0f(X s) dW s+∫ t 0g(X s) ds+K t,t∈ R + , where W is a d-dimensional Wiener process. As a consequence we give the rate of almost sure convergence for these schemes.

On weak solutions of one-dimensional SDEs with time-dependent coefficients

Let W be a standard Wiener process and let be a measurable function. We prove existence of weak solutions of the stochastic differential equation . As compared with earlier works we assume that f 2 is neither locally bounded above nor bounded below by a strictly positive constant

A generalized Haussmann's formula

In this paper, we establish a generalized Haussmann's formula on the stochastic integral representation of funtionals of solutions of SDEs of funtional type

A Deterministic Approach To Stochastic Optimal Control With Application To Anticipative Control

Using the decomposition of solution of SDE, we consider the stochastic optimal control problem with anticipative controls as a family of deterministic control problems parametrized by the paths of the driving Wiener process and of a newly introduced Lagrange multiplier stochastic process (nonanticipativity equality constraint). It is shown that the value function of these problems is the unique global solution of a robust equation (random partial differential equation) associated to a linear backward Hamilton-Jacobi-Bellman stochastic partial differential equation (HJB SPDE). This appears as limiting SPDE for a sequence of random HJB PDE's when linear interpolation approximation of the Wiener process is used. Our approach extends the Wong-Zakai type results [20] from SDE to the stochastic dynamic programming equation by showing how this arises as average of the limit of a sequence of deterministic dynamic programming equations. The stochastic characteristics method of Kunita [13] is used to represent the ...

Stability theorem for stochastic differential equations with jumps

Convergence in law of solutions of SDE having jumps is discussed assuming suitable convergence of the coefficients under a situation where the point process approaches a Poisson point process. As an application the asymptotic behavior of certain stochastic processes such as storage processes and random walks is also discussed.

Quasi sure stochastic flows

We prove that the solutions of SDE with smooth coefficients have ¥ ‐modifications and constitute quasi‐sure stochastic flows of C¥ ‐diffeomorphisms.

On solutions of stochastic differential equations with drift

We study stochastic differential equations of the formdX t=σ(X t)dMt+b(Xt)d t whereM is a continuous local martingale and stands for its quadratic variation process. The conditions introduced by Engelbert and Schmidt, which ensure the existence and uniqueness in law of solutions of SDE's driven by the Wiener process without drift (or with generalized drift) are shown to be no longer valid.

Comparison between solutions of SDEs and ODEs

A ratio-limit comparison between ξ t , the solution of an SDE driven by a semimartingale, and H t , the solution of an associated ODE, is proved on the set where lim t → ∞ ξ t =∞. Sufficient conditions in terms of the driving processes and the coefficients are obtained for lim t → ∞ ξ t to be ∞.

A comparison theorem for solutions of stochastic differential equations and its applications

A new kind of comparison theorem in which an SDE is compared with two deterministic ODEs is established by means of the generalized sample solutions of SDEs. Using this theorem, we can compare solutions of two SDEs with different diffusion coefficients and obtain some asymptotic estimations for the paths of diffusion processes.