Recurrence-transience criteria for storage processes
The paper considers storage processes governed by stochastic integral equations of the form \(X(t)=x- \int^t_0 r(X(s))ds+A(t)\), where \(r(x)\) is a function defined on \([0,\infty)\) with values in \([0,\infty)\), \(r(0)=0\), \(r(x)>0\) for \(x>0\) and left continuous with positive right limits on \((0,\infty)\), and \(A(t)\) is a subordinator with \[ E(e^{-\theta A(t)})= \exp \left\{\int^\infty_0 t(e^{-\theta y}-1) \nu(dy) \right\}. \] Storage processes are classified into three types: positive recurrent, null recurrent, and transient processes. The main results of the paper are: Theorem 3.2. If for some \(c>0\) \[ \int^\infty_c {\nu[x,\infty) \over r(x)}dx=\infty \quad\text{and} \quad\int^\infty_c {1\over r(x)}\exp \left\{-\int^x_c {\nu[y,\infty)\over r(y)} dy\right\} dx<\infty, \] then \(X(t)\) is transient. Theorem 3.3. Assume that \(\inf_{x\geq 1}r(x)>0\). If there is \(c>0\) such that \(\int^x_c{\nu(x-y,\infty) \over r(y)} dy\leq 1\) for all \(x\geq c\), then \(X(t)\) is recurrent.
scientific article; zbMATH DE number 1529334