Logarithmic estimates for the density of an anticipating stochastic differential equation
type: Item
instance of: scholarly article
title: Logarithmic estimates for the density of an anticipating stochastic differential equation
zbMATH Open document ID: 0962.60044
DOI: 10.1016/S0304-4149(98)00092-1
author: Marta Sanz-Solé
author: Mònica Sarrà
published in: Stochastic Processes and their Applications
publication date: 2001-01-17T00:00:00Z
review text: Let \(\{X^\varepsilon _t\), \(t\in [0,1]\}\) be a family of \(R^d\)-valued stochastic processes indexed by the parameter \(\varepsilon \in (0,1)\), which satisfy the anticipating stochastic differential equation \[ X^\varepsilon _t= x_0^\varepsilon +\sqrt \varepsilon \int _0^t\sum _{j=1}^k \sigma _j(X^\varepsilon _s)\circ dW^j_s +\int _0^t \sigma _0 (X^\varepsilon _s) ds \] where \(W=\{W^j_t\), \(1\leq j\leq k\), \(t\in [0,1]\}\) is an \(R^k\)-valued standard Brownian motion and \(\sigma _j:R^d\to R^d\), \(j=0,\dots{},k\), and \(\{X^\varepsilon _0\}\) are random vectors not necessarily adapted to the filtration associated with the Wiener process. The stochastic integral is defined in the Stratonovich sense. Varadhan's type estimates for the density of the probability law of \(X^\varepsilon _t\) as \(\varepsilon \downarrow 0\) are proved.
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60F10
zbMATH DE Number: 1554287
zbMATH Keywords: anticipating calculus
zbMATH Keywords: large deviations
zbMATH Keywords: Malliavin calculus
MaRDI profile type: Publication
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full work available at URL: s0304-4149(98)00092-1
OpenAlex ID: W1510927666
description: scientific article; zbMATH DE number 1554287