A stochastic model for a dam with non-additive input (Q2752175)

The content of a dam is described by a stochastic process \(X(t)\) determined by the integral equation NEWLINE\[NEWLINE X(t)=x+A(t)-\int_0^t r(X(s)) ds, NEWLINE\]NEWLINE where the initial content \(x\), the input process \(A(t)\), and the release rate function \(r(z)\) are given. The authors consider the cases where \(A(t)\) is a random piecewise linear function of a special structure and \(r(z)=cz\) (case I) or \(r(z)=cz^2\) (case II). They solve the integral equation explicitly in both cases and study the asymptotic behavior of the solution only in case I by means of an embedded Markov chain. They formulate conditions for the existence of a limiting distribution of that chain and derive properties of the limit.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00044].