On the ruin problem with investment when the risky asset is a semimartingale

In this paper, we study the ruin problem with investment in a general framework where the business part $X$ is a Lévy process and the return on investment $R$ is a semimartingale. Under some conditions, we obtain upper and lower bounds on the finite and infinite time ruin probabilities as well as the logarithmic asymptotic for them. When $R$ is a Lévy process, we retrieve some well-known results. Finally, we obtain conditions on the exponential functionals of $R$ for ruin with probability $1$, and we express these conditions using the semimartingale characteristics of $R$ in the case of Lévy processes.