Consider a bivariate Lévy-driven risk model in which the loss process of an insurance company and the investment return process are two independent Lévy processes. Under the assumptions that the loss process has a Lévy measure of consistent variation and the return process fulfills a certain condition, we investigate the asymptotic behavior of the finite-time ruin probability. Further, we derive two asymptotic formulas for the finite-time and infinite-time ruin probabilities in a single Lévy-driven risk model, in which the loss process is still a Lévy process, whereas the investment return process reduces to a deterministic linear function. In such a special model, we relax the loss process with jumps whose common distribution is long tailed and of dominated variation.

This paper is concerned with a class of high-order fractional stochastic partial differential equations driven by fractional noise. We firstly prove the existence and uniqueness of the mild solution and then study the Hölder continuity of the solution with respect to space and time variables. In addition, we also prove the existence and Gaussian-type estimates for the density of the solution by using the techniques of Malliavin calculus.

In this paper, we study the following nonlinear fractional stochastic partial differential equation $\frac{\mathrm{\partial }}{\mathrm{\partial }t}u(t,x)=\mathfrak{D}(x,D)u(t,x)+g(u(t,x))\frac{{\mathrm{\partial }}^2{w}_{\rho }}{\mathrm{\partial }t\mathrm{\partial }x}(t,x),t\ge 0,x\in \mathbb{R},$ where $\mathfrak{D}(x,D)$ denotes the Markovian generator of a stable-like Feller process with variable order and $g(\cdot ):\mathbb{R}\to \mathbb{R}$ is a measurable function. The forcing noise denoted by $\frac{{\mathrm{\partial }}^2{w}_{\rho }}{\mathrm{\partial }t\mathrm{\partial }x}(t,x)$ is a spatially inhomogeneous white noise. Under some assumptions on the catalytic measure of the inhomogeneous Brownian sheet $w_{\rho }(t,x)$, we study the moment bounds for the solution. As a byproduct, we prove that the solution is weakly full intermittent based on the moment estimates of the solution. We also study the Hölder regularity of the solution with respect to the temporal and spatial variables, respectively.