Journal•ISSN: 0736-2994

# Stochastic Analysis and Applications

About: Stochastic Analysis and Applications is an academic journal. The journal publishes majorly in the area(s): Stochastic differential equation & Stochastic process. It has an ISSN identifier of 0736-2994. Over the lifetime, 1926 publication(s) have been published receiving 21548 citation(s).

Topics: Stochastic differential equation, Stochastic process, Stochastic partial differential equation, Differential equation, Markov chain

##### Papers published on a yearly basis

##### Papers

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TL;DR: In this article, a Monte-Carlo method is used to estimate the invariant probability law of a stochastic differential system by simulating a simple t,rajectory.

Abstract: Given the solution (Xt ) of a Stochastic Differential System, two situat,ions are considered: computat,ion of Ef(Xt ) by a Monte–Carlo method and, in the ergodic case, integration of a function f w.r.t. the invariant probability law of (Xt ) by simulating a simple t,rajectory. For each case it is proved the expansion of the global approximat,ion error—for a class of discret,isat,ion schemes and of funct,ions f—in powers of the discretisation step size, extending in the fist case a result of Gragg for deterministic O.D.E. Some nn~nerical examples are shown to illust,rate the applicat,ion of extrapolation methods, justified by the foregoing expansion, in order to improve the approximation accuracy

631 citations

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TL;DR: In this article, three characterizations of the minimal martingale measure [Pcirc] associated to a given d-dimensional semimartingale X are provided. And they extend the result of Ansel and Stricker on the Follmer-Schweizer decomposition to the case where X is continuous, but multidimensional.

Abstract: We provide three characterizations of the minimal martingale measure[Pcirc] associated to a given d-dimensional semimartingale X. In each case, [Pcirc] is shown to be the unique solution of an optimization problem where one minimizes a certain functional over a suitable class of signed local martingale measures for X. Furthermore, we extend a result of Ansel and Stricker on the Follmer-Schweizer decomposition to the case where X is continuous, but multidimensional.

349 citations

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TL;DR: This article proposes an adaptive algorithm to cope with the estimation of rare event probability that is asymptotically consistent, costs just a little bit more than classical multilevel splitting, and has the same efficiency in terms of asymPTotic variance.

Abstract: The estimation of rare event probability is a crucial issue in areas such as reliability, telecommunications, aircraft management. In complex systems, analytical study is out of question and one has to use Monte Carlo methods. When rare is really rare, which means a probability less than 10−9, naive Monte Carlo becomes unreasonable. A widespread technique consists in multilevel splitting, but this method requires enough knowledge about the system to decide where to put the levels at hand. This, unfortunately, is not always possible. In this article, we propose an adaptive algorithm to cope with this problem: The estimation is asymptotically consistent, costs just a little bit more than classical multilevel splitting, and has the same efficiency in terms of asymptotic variance. In the one-dimensional case, we rigorously prove the a.s. convergence and the asymptotic normality of our estimator, with the same variance as with other algorithms that use fixed crossing levels. In our proofs we mainly us...

257 citations

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TL;DR: In this paper, the Geral theory on the almost sure exponential stability and instability of the stochastically perturbed neural network is first established, and the theory is then applied to investigate stochastic stabilization and destabilization of the network.

Abstract: In this paper we shall discuss stochastic effects to the stability property of a neural network Suppose the stochastically perturbed neural network is described by an Ito equation [ILM002]The Geral theory on the almost sure exponential stability and instability of the stochastically perturbed neural network is first established. The theory is then applied to investigate the stochastic stabilization and destabilization of the neural network. Several interesting examples are also given for illustration

182 citations

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Fudan University

^{1}TL;DR: In this article, an adapted pair of process with values in H and K and respectively is defined, which solves a semilinear stochastic evolution equation of the backward form: where A is the infinitesimal generators of a C 0-semigroup {eAt } on H.

Abstract: Let K and H be two separable Hilbert spaces and be a cylindrical Wiener process with values in K defined on a probability space denote its natural filtration. Given , we look for an adapted pair of process with values in H and respectively is defined in §1),which solves a semilinear stochastic evolution equation of the backward form: where A is the infinitesimal generators of a C 0-semigroup {eAt } on H. The precise meaning of the equation is A linearized version of that equation appears in infinite-dimensional stochastic optimal control theory as the equation satisfied by the adjoint process. We also give our results to the following backward stochastic partial differential equation:

180 citations