Regularity and convergence of stochastic convolutions in duals of nuclear Fréchet spaces
\(\Phi\) is nuclear Fréchet space, \(\Phi'\) its dual, and \(D_ \infty(\Phi')\) the space (for an appropriate topology) of maps defined on \({\mathfrak R}_ +\), with values in \(\Phi'\), which are right-continuous, with left limits (càdlàg). \(M\) is a martingale with values in \(\Phi'\) and \(S(u,v)\) a family (depending on \((u,v)\)) of continuous linear operators which forms a reverse evolution system of contraction type. A stochastic convolution process is a process of the form \(Y_ t=\int_ 0^ t S'(t,u)dM_ u\), \(S'\) being the adjoint of \(S\). The paper provides results on three topics: weak convergence on \(D_ \infty(\Phi')\), càdlàg versions of \(Y\), convergence of \(Y^{(n)}\) to \(Y\) when \(M^{(n)}\) converges to \(M\). The technical requirements for the results to be true are too lengthy to be stated in a review.
scientific article; zbMATH DE number 148928
scientific article