Present value distributions with applications to ruin theory and stochastic equations

Conditioned Stochastic Differential Equations: Theory and Applications

We generalize the notion of brownian bridge. More precisely, we study a standard brownian motion for which a certain functional is conditioned to follow a given law. Such processes appear as weak solutions of stochastic differential equations which we call conditioned stochastic differential equations. The link with the theory of initial enlargement of filtration is made and after a general pre...

Stochastic bounds on distributions of optimal value functions with applications to pert, network flows and reliability

Meilijson and Nadas [1979] have obtained stochastic bounds in the convex majorisation sense to the critical path length of a project network with random activity durations. In this paper we present those results in a more general framework and, using similar techniques, obtain bounds for shortest route, maximal flow and reliability system lifetime. Subject classification: #488 Bounds for stocha...

Large Deviations and Ruin Probabilities for Solutions to Stochastic Recurrence Equations with Heavy-tailed Innovations

In this paper we consider the stochastic recurrence equation Yt = AtYt−1 + Bt for an i.i.d. sequence of pairs (At ,Bt ) of nonnegative random variables, where we assume that Bt is regularly varying with index κ > 0 and EAt < 1. We show that the stationary solution (Yt ) to this equation has regularly varying finite-dimensional distributions with index κ . This implies that the partial sums Sn =...

Stochastic Differential Equations with Applications

Random Recurrence Equations and Ruin in a Markov-dependent Stochastic

We develop sharp large deviation asymptotics for the probability of ruin in a Markov-dependent stochastic economic environment and study the extremes for some related Markovian processes which arise in financial and insurance mathematics, related to perpetuities and the ARCH(1) and GARCH(1,1) time series models. Our results build upon work of Goldie [Ann. Appl. Probab. 1 (1991) 126–166], who ha...