On average losses in the ruin problem with fractional Brownian motion as input

Abstract

We consider the model $S_{t}=u+ct-B_{t}^{H}$, where u > 0, c > 0, $B_{t}^{H}$ is the fractional Brownian motion with Hurst parameter H, 0 < H < 1. We study the asymptotic behavior of average losses in the case of ruin, i.e. the asymptotic behavior of the conditional expected value $E\left( -\inf _{t\in\lbrack0,T]}S_{t}\left\vert \inf_{t\in\lbrack0,T]}S_{t}<0\right. \right)$ as u→ ∞ . Three cases are considered: the short time horizon, with T finite or growing much slower than u; the long time horizon, with T at or above the time of ruin, including infinity; and the intermediate time horizon, with T proportional to u but not growing as fast as in the long time horizon. As one of the examples, we derive an asymptotically optimal portfolio minimizing average losses in the case of two independent markets.