Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise
The article is devoted to the stochastic differential equation \[ dX(t)=A(t)X(t)dt+\sum _{k=1}^mB_kX(t)d\beta ^H_k(t).\tag{1} \] Here \(A(t), B_k, k=1,\dots,m\), are unbounded densely defined linear operators in the infinite-dimensional Hilbert space. The processes \(\beta ^H_k, k=1,\dots,m,\) are independent one-dimensional fractional Brownian motions with the Hurst parameter \(H.\) The authors propose the representation for the strong solution of (1). They suppose, in particular, that the operators \(B_k, k=1,\dots,m,\) generate mutually commuting strongly continuous groups \(S_k(s), k=1,\dots,m,\) and the family \(\tilde{A}(t)=A(t)-Ht^{2H-1}\sum _{k=1}^mB_k^2\) generates a strongly continuous evolution family of operators \(\{U(t,s); 0\leq s\leq t\leq T\}.\) Then the solution of (1) can be obtained by the following formula \[ X(t)=\prod _{k=1}^mS_k(\beta ^H_k(t))U(t,0)x_0. \] Such representation allows to study the stability of the solution using the law of the iterated logarithm for fractional Brownian motion and some kind of Lyapunov function despite of the absence of the Markov property of \(X.\) Examples, when \(\beta ^H\) can have the stabilizing influence, are also presented.
stochastic differential equations in Hilbert space
scientific article