On an estimate for the mean value of a nonnegative random process. (Russian) Zbl 0577.60021

Nakonechnyj, A. N.

Teor. Veroyatn. Mat. Stat. 29, 98-100 (1983).

The function \(I_{\epsilon}(t)=E(\exp (-\epsilon \xi (t)))\) with \(t\in [0,T]\), small parameter \(\epsilon\) and measurable, nonnegative, continuous random function \(\xi\) of the form \(\xi (t,\omega)=\int^{t}_{o}\psi (x,\omega)\) dx is estimated from above. Let \(a(t):=E(\xi (t))\), \(b_ j(t):=E((\xi (t)-a(t))^ j)\). The estimate has the form \(\exp (-\epsilon a(t))\cdot S_{k,\epsilon}\), where \(S_{k,\epsilon}\) is the sum of a power series in \(\epsilon\) with coefficients \(b_ j(t)/j!\) up to k and a term converging to 0 for \(k\to \infty\). An estimate for \(b_ j(t)\) is suggested.

Reviewer: C.Baldauf

MSC:

60E15	Inequalities; stochastic orderings
60G99	Stochastic processes

Keywords:

mean value of a nonnegative random process