Bounded in the \(p\)-order mean solutions of differential equations in a Banach space. (Ukrainian, English) Zbl 1052.34067

Gorodnij, M. F.

Teor. Jmovirn. Mat. Stat. 67, 20-25 (2002); translation in Theory Probab. Math. Stat. 67, 23-28 (2003).

The author deals with equations of the type \(X'(t)=AX(t)+\xi(t)\), \(t\in \mathbb{R},\) in a complex separable Banach space \(X\) equipped with the norm \(\| \cdot\| \). Here, \(A\) is a sectorial operator and \(\xi\) is a \(p\)-mean continuous \(X\)-valued random process, \(p \in [1,\infty)\). He investigates conditions for existence and uniqueness of solutions of such equations. Conditions which provide the continuous differentiability (with probability 1) of the trajectories of solutions are obtained too.

Reviewer: N. M. Zinchenko (Kyïv)

MSC:

34G10	Linear differential equations in abstract spaces
60H99	Stochastic analysis
34F05	Ordinary differential equations and systems with randomness

Keywords:

Banach space; stochastic differential equation; continuous differentiability