Monotone operators in stochastic set-valued equations

✍ Scribed by J. Motyl

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 294 KB
• Volume: 47
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/s0362-546x(01)00155-9

✦ Synopsis

Let (F) and (G) be set-valued functions. Under certain measurability conditions there exist set-valued stochastic integrals (\int_{s}^{t} F d \tau), and (\int_{s}^{t} G d W_{\tau}) defined as Aumann's type integrals. Given such integrals we study a stochastic inclusion of the form:

[
x_{t}-x_{s} \in \int_{s}^{t} F(x){\tau} d \tau+\int{s}^{t} G(x){\tau} d W{\tau}
]

We find sufficient conditions for the existence of strong solutions to the inclusion which differ both from Lipschitz and Pardoux "monotone" conditions. Secondary, the viability property for such a type inclusion will be discussed.