Filtration shrinkage by level-crossings of a diffusion (Q2371954)

Let \(x_1<\cdots<x_N\) be a finite collection of points in \(\mathbb{R}\) and \(R\) the indicator function \[ R(x)=\begin{cases} k & \text{if } x\in(x_k, x_{k+1}],\;k\in\{1, \dots,N-1\},\\ 0 & \text{if } x\leq x_1,\\ N & \text{if }x>x_N.\end{cases} \] Let \(X\) be a non singular diffusion with state space an interval in \(\mathbb{R}\) that contains \(x_1,\dots,x_N\) and with generator \({\mathcal A}\). The author establishes a stochastic basis for the problem and proves that, to each point \(x_i\), it is possible to associate a random measure \(\mu_i\), with compensator \(\nu_i\), such that the filtration generated by \(R(X_t)\) has the martingale representation property with respect to the random measures \((\mu_i-\nu_i,i=1,\dots,N)\). A (corollary is that the graph of all inaccessible stopping times is contained in the set \(D\) of times where \(X\) reaches the levels \(x_1,\dots,x_n\). The compensator \(\nu_i\) is explicitly computed: its expression involves the local time of \(X\) at \(x_i\) and the Lévy measure of the inverse of the local time, which is related to the eigenvectors of \({\mathcal A}\).