Finite Explosion for One-dimensional Lévy Driven SDEs and its Application to SPDEs

Source abstract

This paper addresses finite-time explosion of one-dimensional stochastic differential equations (SDEs) driven by pure-jump Lévy processes \[ X_{t}^{x}=x-\int_{0}^{t}b(X_{s}^{x})\,\dd s+L_{t}, \] where the drift coefficient $b$ is locally Lipschitz continuous, and $(L_t)_{t\ge0}$ is a pure jump Lévy process. We formulate three sets of conditions: a right-tail return condition; a nondegeneracy condition together with a left-tail Osgood bound; and a right-tail Osgood bound. The first two imply almost-sure explosion to $-\infty$ from every finite initial state, whereas the latter two imply a uniform bound on the mean explosion time. As an application, we give explicit conditions on the nonlinearity and the Lévy measure under which every local weak solution of a semilinear parabolic stochastic partial differential equation (SPDE) with additive Lévy space--time white noise and homogeneous Dirichlet boundary conditions has an almost surely finite lifetime.