Almost sure stability of linear stochastic differential equations with jumps (Q1849495)

Let \(W\) be a standard \(m\)-dimensional Wiener process and \(N\) a Poisson point process on \(\mathbb{R}_{+}\times Z\) with a finite characteristic measure \(\lambda\). The system of linear stochastic differential equations with jumps \[ dx(t) = Ax(t) dt + \sum^{m}_{i=1} B_{i}x(t) \circ dW_{i}(t) + \int_{Z} C(z)x(t-)N( dt, dz), \;x(0) = x_0,\tag{1} \] is considered. It is supposed that \(x_0\in\mathbb{R}^{d}\), \(x_0\neq 0\), \(A\), \(B_{i}\), \(C(z)\) are \(d\times d\)-matrices, \(C\in L^{d+1}(\lambda)\), and that the matrix \(I+C(z)\) has full rank \(\lambda\)-almost everywhere on \(Z\). By generalizing the procedure used in the diffusion case \(C\equiv 0\), the authors obtain a rather complete description of the long time behaviour of the solution \(x\) to (1), provided that a non-degeneracy condition on the diffusion part \(\sum^{m}_{i=1} \langle y,B_{i}x\rangle^2 \geq \kappa |x|^2|y|^2\) is satisfied for some \(\kappa>0\) and all \(x,y\in \mathbb{R}^{d}\). In particular, it is proved that the process \(\theta(t) = x(t)/|x(t)|\) on the unit sphere \(\mathbb{S}^{d-1}\) is strong Feller, its transition probability has a continuous density which is strictly positive for sufficiently large times, and there exists a unique invariant probability measure \(\mu\) for \(\theta\). Therefore, it is possible to obtain a formula of Furstenberg-Khas'minskii type showing that \(\lim_{t\to\infty}t^{-1}\log|x(t)|= \int_{\mathbb{S} ^{d-1}} Q(\varphi) d\mu(\varphi)\) for an explicitly given function \(Q\) on \(\mathbb{S}^{d-1}\) and all initial conditions \(x_0\). Furthermore, several interesting two-dimensional examples are discussed in detail.