On $L_p$-Solvability of Stochastic Integro-Differential Equations

Itô’s formula for jump processes in Lp-spaces

“…Our main theorems on Itô's formula, Theorem 2.1 and Theorem 2.2 generalise Lemma 5.1 and Theorem 2.1, respectively, from [10]. We use them to prove an existence and uniqueness theorem for a class of stochastic integro-differential equations in [7]. In [7] we need an Itô's formula for d| u t | p Lp , where u t = ( M i=1 |u i t | 2 ) 1/2 and (u i t ) t∈[0,T ] is a W 1 p -valued process having a stochastic differential of the type (1.4) for each i = 1, 2, ..., M .…”

“…We use them to prove an existence and uniqueness theorem for a class of stochastic integro-differential equations in [7]. In [7] we need an Itô's formula for d| u t | p Lp , where u t = ( M i=1 |u i t | 2 ) 1/2 and (u i t ) t∈[0,T ] is a W 1 p -valued process having a stochastic differential of the type (1.4) for each i = 1, 2, ..., M . Therefore in Theorem 2.1 of the present paper we consider a system of stochastic differentials instead of a single one.…”