Level-crossing properties of the risk process (Q2757551)

The classical risk process is described as NEWLINE\[NEWLINE R(t)=x_0+t-\sum_{i=1}^{N(t)}Y_i,\quad t\geq 0, NEWLINE\]NEWLINE where \(x_0\) is a nonnegative constant, \(N(t)\), \(t\geq 0\), is a homogeneous Poisson process with rate \(\lambda>0\) and \(Y_1, Y_2, \ldots\) are i.i.d. positive random variables, which are independent of \(N(\cdot)\). The associated level-crossing process \(C(x)=\left(L(x),\;(A_i(x),B_i(x)),\;1\leq i\leq L(x)\right)\) is considered. Here \(L(x)\) is the process counting the jumps from \((x,\infty)\) to \((-\infty,x]\), \(A_i(x)\) (\(B_i(x)\)) are the distances from \(x\) to \(R(t)\) after (before) the \(i\)th jump of this kind. It is known that \(C(\cdot)\) is a stationary Markov process; its transition mechanism is determined. As an application the covariance function \(E\left(L(x)L(x+y)\right)\) is calculated.