On The Ruin Problem With Investment When The Risky Asset Is A Semimartingale

TL;DR

This paper analyzes the ruin probability in investment scenarios where the business process is a Lévy process and the investment return is a semimartingale, providing bounds and conditions for ruin with various process assumptions.

Contribution

It introduces a general framework for the ruin problem with semimartingale returns and derives asymptotically optimal bounds, extending classical results to more complex stochastic models.

Findings

Power-law decrease of ruin probabilities with initial capital

Bounds are asymptotically optimal under certain conditions

Explicit ruin probability conditions for Brownian motion with negative drift

Abstract

In this paper, we study the ruin problem with investment in a general framework where the business part X is a L{\'e}vy process and the return on investment R is a semimartingale. We obtain upper bounds on the finite and infinite time ruin probabilities that decrease as a power function when the initial capital increases. When R is a L{\'e}vy process, we retrieve the well-known results. Then, we show that these bounds are asymptotically optimal in the finite time case, under some simple conditions on the characteristics of X. Finally, we obtain a condition for ruin with probability one when X is a Brownian motion with negative drift and express it explicitly using the characteristics of R.