On two-parameter non-degenerate Brownian martingales
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 suitable tool here. So the authors use an equi-absolute continuity property of the occupation measures of \(u_n\), following an argument due to \textit{I. Gyöngy} [in: Probability theory and mathematical statistics with applications, 87-96 (1988; Zbl 0665.60060)].
hyperbolic stochastic partial differential equations
scientific article; zbMATH DE number 1202434
scientific article