A weak stochastic integral in Banach space with application to a linear stochastic differential equation
In this readable paper the authors study in detail cylindrical stochastic processes in separable Banach spaces. They start with the notion of cylindrical Wiener process \(W_ t\), which may be represented as a weakly convergent sum of independent, real Wiener processes. Similar constructions may be found in \textit{A. Bensoussan}, Filtrage optimal des systèmes linéaires (1971; Zbl 0231.93022), \textit{A. V. Balakrishnan}, Applied functional analysis (1976; Zbl 0333.93051) and \textit{L. Gross}, Proc. 5th Berkeley Sympos. math. Statist. Probab., Univ. Calif. 1965/1966, 2, part 1, 31-42 (1967; Zbl 0187.409). Since the covariance may be a general bounded symmetric operator, the distribution of \(W_ t\) is only a cylinder set measure. They then define an adequate notion of stochastic integral with respect to \(W_ t\) and give a theorem on existence and uniqueness for certain stochastic differential equations with linear drift and state-independent diffusion.
cylindrical stochastic process in Banach space
cylindrical Wiener process in Banach space
linear stochastic differential equations in Banach space
scientific article; zbMATH DE number 3848338
scientific article