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

Abstract

We consider the model S t = u + ct − B H t , where u > 0, c > 0, B H t 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 valueThree 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.

On the γ-reflected processes with fBm input*

“…In this contribution, three scenarios of T u will be distinguished. In view of (4) the asymptotic mean of the ruin time (equal to t 0 u) and the asymptotic standard deviation A(u) (see (5)) should be used as a scaling parameter for T u leading to the definition of the following three scenarios:…”

“…c) As discussed in [5] also of interest is the investigation of the maximum losses given that ruin occurs, which, in our setup, is defined as…”

“…From a practical point of view, a more interesting quantity is the finite-time ruin probability. The case of the surplus dependent time interval lies thus in between and is of both practical and theoretical interests; results for the ruin probability in this case (with γ = 0) are initially derived in [5], see Theorem 2.1 below.…”

“…With motivation from [5] and [9], as a continuation of the investigation of the aforementioned papers we shall analyze the ruin probability and the conditional ruin time of W γ over the surplus dependent time interval [0, T u ] letting u → ∞. In the literature, commonly the case T = ∞ is considered since for many models explicit calculations of the ruin probability is possible.…”

On the distribution of storage processes from the class $V(𝜙,𝜓)$

“…A considerable number of papers are devoted nowadays to studies of the behavior of the data traffic in computer networks (see, for example, [1,2,4,8,10]). A special place in this topic is occupied by the problem of estimating the reliability when transferring the data.…”