Einige unstetige stochastische Prozesse

Summary

Stochastic processes of the following type are considered. At random time points, the variablex(t) jumps fromy tox, say. The heightsx−y of the jumps have a given distributionG ^*(x−y) that may depend ony ort. Between the jumps,x(t) is a solution to a given differential equationdx/dt=x(x, t). We look for the distributionF(x, t) ofx at timet>0,F(x, 0) being given. In the stationary case, stable distributions are investigated.

If there is a lower boundaryx ₀ and ifF(x ₀)>0, the problem is similar to the queueing problem. We solve it in the stationary case with integral equations of the Volterra type. Other problems can be transformed to differential equations for the moment generating functions. These equations are partial in the non stationary and ordinary in the stationary case.