The Itô formula and local times for stochastic processes of the Volterra type. (English. Ukrainian original) Zbl 0861.60067

Mishura, Yu. S.

Theory Probab. Math. Stat. 49, 123-136 (1994); translation from Teor. Jmovirn. Mat. Stat. 49, 173-192 (1993).

Summary: The paper contains a change of variables formula for stochastic processes of the Volterra type, that is, for processes that admit the representation \[ x_t=x_0+\int^t_0 a(t,s)ds+ \int^t_0 b(t,s)dw_s, \] where \(a(t,s)\) and \(b(t,s)\) are \({\mathcal F}_s\)-measurable functions for every \(t\geq s\geq 0\). The existence of a local time \({\mathcal L}^a_t\) is proved for such processes for every \(a\in {\mathbf R}\). Processes of the Volterra type generated by multiple stochastic integrals are also considered.

MSC:

60H05	Stochastic integrals
60G60	Random fields

Keywords:

stochastic processes of the Volterra type; local time; multiple stochastic integrals