Construction of the resolvent of a process with independent increments breaking at the moment of leaving the interval. (Russian) Zbl 0566.60072

Bratijchuk, N. S.

Teor. Veroyatn. Mat. Stat. 30, 25-34 (1984).

Let \(\xi\) (t), \(t\geq 0\), be a stochastically continuous and homogeneous stochastic process with independent increments and right continuous paths. Let \(\zeta\) be the moment of leaving the interval (0,T), \(x\in (0,T)\) and \(E_ X(\cdot)=E(\cdot | \xi (0)=x)\). The resolvent of the process \(\xi^ T(t)=\xi (t)\), \(0\leq t\leq \zeta\) is defined by \(R^ T_{\lambda}f(x)=E_ x\int^{\zeta}_{0}e^{-\lambda t}f(\xi (t))dt.\)
In this paper functions \(B_{\lambda}(\cdot)\), \(\psi\) (\(\cdot,f)\) and a constant \(c_ 0(f)\) are introduced such that the resolvent \(R^ T_{\lambda}\) has the form \[ R^ T_{\lambda}f(x)=c_ 0(f)B_{\lambda}(x)+\int^{x}_{0}\psi (y,f)B_{\lambda}(x-y)dy- \int^{x}_{0}f(y)B_{\lambda}(x-y)dy \] under suitable conditions.

Reviewer: G.Oprişan

MSC:

60J35	Transition functions, generators and resolvents
60J99	Markov processes
60J45	Probabilistic potential theory

Keywords:

resolvent