Stratonovich-type integral with respect to a general stochastic measure

Abstract

Let µ be a general stochastic measure, where we assume for µ only σ-additivity in probability and continuity of paths. We prove that the symmetric integral [0,T ] f (µ t , t) • dµ t is well defined. For stochastic equations with this integral, we obtain the existence and uniqueness of a solution.

Transport equation driven by a stochastic measure

“…The symmetric integral of random functions with respect to stochastic measures was considered in [17]. We review the basic facts and definitions concerning this integral.…”

“…where F (z, v) = z 0 f (y, v) dy. Some other properties and equations with the symmetric integral are considered in [17], [18], [19].…”

“…Stochastic integral with respect to µ is defined as symmetric integral. This Stratonovich-type integral was studied in [17], we recall it's definition and basic properties in subsection 2.2. SMs include many important classes of processes, but we can prove existence of the integral only for integrands of the form f (µ t , t) where f ∈ C 1,1 (R × [0, T ]).…”

Averaging principle for equation driven by a stochastic measure

“…The symmetric integral of random functions with respect to stochastic measures was considered in [16].…”

“…Theorem 2.6. (Theorem 5.3 [16]) Let A1 and A4 hold, X 0 be an arbitrary random variable. Then equation (2.4) has a unique solution X t = F (µ t , Y t ), where Y t is the solution of the random equation…”