On two-parameter non-degenerate Brownian martingales
type: Item
instance of: scholarly article
title: On two-parameter non-degenerate Brownian martingales
zbMATH Open document ID: 0910.60044
DOI: 10.1016/S0007-4497(98)80173-5
author: David Nualart
author: S. Tindel
published in: Bulletin des Sciences Mathématiques
publication date: 1999-04-19T00:00:00Z
review text: In the first part of the paper the authors study the existence and the properties of the density of the two-parameter Brownian martingale \(N(z)= \int_{[0,T]^2} G(\zeta) dW(\zeta)\), \(z \in [0,T]^2\), driven by a Brownian sheet \(\{W(z), z\in [0,T]^2\}\), and with a square-integrable adapted process \(G\) as integrand verifying, in addition to the condition \(G\geq \rho>0\), some boundedness assumption on the first and on the second order Malliavian derivatives. By using the Malliavin calculus relative to the Brownian sheet and an explicit formula for the density of a random variable \(F\) for which the derivative divided by the square of its Cameron-Martin space norm is Skorokhod integrable, the authors prove the existence of a continuous density of \(N(z)\), \(z\in (0,T]^2\), and give an estimate of its density. In the second part relatively independent of the first one, the authors study the hyperbolic stochastic partial differential equation (SPDE) \[ {\partial^2 \over \partial s\partial t} u(s,t)= f\bigl(s,t,u(s,t) \bigr)+ {\partial^2 \over \partial s\partial t} N(s,t),\;s,t\in [0,T], \quad u(s,0)= u(0,t)=a, \] where \(a\in R\) and \(f:[0,T]^2 \times R\to R\) is a bounded, measurable function which is nondecreasing in its second variable. The proof of existence and uniqueness adapts a monotonicity method exploited by \textit{R. Buckdahn} and \textit{E. Pardoux} [in: Diffusion processes and related problems in analysis, Vol. I: Diffusions in analysis and geometry. Prog. Probab. 22, 219-233 (1990; Zbl 0722.60061)]. The difficulties one meets here are: firstly, the comparison theorem does not hold for hyperbolic SPDEs [\textit{D. Nualart}, Stat. Probab. Lett. 5, 231-234 (1987; Zbl 0613.60054)]; and secondly, the Krylov type argument for the convergence of an approximating sequence \(u_n\) to the solution \(u\), used by the authors [Potential Anal. 7, No. 3, 661-680 (1997; Zbl 0886.60060)] in a special case, fails here because the change of probability is not a su…
Mathematics Subject Classification ID: 60H07
Mathematics Subject Classification ID: 60H15
zbMATH DE Number: 1202434
zbMATH Keywords: two-parameter Brownian martingale
zbMATH Keywords: density
zbMATH Keywords: occupation measure
zbMATH Keywords: Malliavin calculus
zbMATH Keywords: hyperbolic stochastic partial differential equations
reviewed by: Rainer Buckdahn
MaRDI profile type: Publication
cites work: Q3209951
cites work: Stochastic integrals in the plane
cites work: On nondegenerate quasilinear stochastic partial differential equations
cites work: On quasi-linear stochastic partial differential equations
cites work: Q4151471
cites work: Q3910361
cites work: Q4039796
cites work: Some remarks on a linear stochastic differential equation
cites work: Q4842684
cites work: Quasilinear stochastic hyperbolic differential equations with nondecreasing coefficient
Recommended article: Quasilinear stochastic hyperbolic differential equations with nondecreasing coefficient
Recommended article: Q3326546
Recommended article: Small perturbations in a hyperbolic stochastic partial differential equation
Recommended article: Hyperbolic Stochastic Partial Differential Equations with Additive Fractional Brownian Sheet
Recommended article: Q4840705
Recommended article: The law of the solution to a nonlinear hyperbolic SPDE
Recommended article: Q2712727
Recommended article: Two-parameter stochastic calculus and Malliavin's integration-by-parts formula on Wiener space
description: scientific article; zbMATH DE number 1202434