Linear stochastic differential equations with cylindrical stochastic processes in normed spaces. (Russian) Zbl 0697.60064

Ponomarenko, A. I.

Teor. Veroyatn. Mat. Stat., Kiev 41, 105-113 (1989).

The author considers a stochastic differential equation of the n-th order of the form \[ d\Xi_ t^{(n-1)}=[A(t)+\sum^{n-1}_{k=0}\Xi_ t^{(k)}A_ k(t)]dt+dW_ tB(t),\quad d\Xi_ t^{(k)}=\Xi_ t^{(k+1)}dt \] where \(\Xi_ t\) is a process with values in \({\mathfrak L}(X^*,L^ 2(\Omega,{\mathcal U},P))\), the space of linear and continuous maps, \(B(t):\quad {\mathfrak L}(X^*,Y^*)\to {\mathbb{R}},\) X, Y are real normed spaces.
The Wiener process \(W_ t\) is \({\mathfrak L}(Y^*,L^ 2(\Omega,{\mathcal U},P))\)-valued with E (W\({}_ tW_ s)=\min (t,s)V\), where V is a positive symmetric operator from \({\mathfrak L}(Y^*,Y^{**})\). The coefficients are assumed to be \(L^ 1-A\), \(A_ i\), and \(L^ 2-B\), in the suitable spaces.
The main result (Theorem 4) gives the existence and uniqueness together with an explicit formula by means of the fundamental solution of the corresponding deterministic equation. The equations for the mean and correlation function of \(\Xi_ t\) are also derived.

Reviewer: M.Capinski

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
28C20	Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)

Keywords:

cylindrical Wiener process; stochastic differential equation