First Passage Risk Probability Minimization for Piecewise Deterministic Markov Decision Processes

This paper is an attempt to study the minimization problem of the risk probability of piecewise deterministic Markov decision processes(PDMDPs)with unbounded transition rates and Borel spaces.Different from the expected discounted and average criteria in the existing literature,we consider the risk probability that the total rewards produced by a system do not exceed a prescribed goal during a first passage time to some target set,and aim to find a policy that minimizes the risk probability over the class of all history-dependent policies.Under suitable conditions,we derive the optimality equation(OE)for the probability criterion,prove that the value function of the minimization problem is the unique solution to the OE,and establish the existence ofε(≥0)-optimal policies.Finally,we provide two examples to illustrate our results.

CLASSICAL RISK MODEL WITH THRESHOLD DIVIDEND STRATEGY

In this article, a threshold dividend strategy is used for classical risk model. Under this dividend strategy, certain probability of ruin, which occurs in case of constant barrier strategy, is avoided. Using the strong Markov property of the surplus process and the distribution of the deficit in classical risk model, the survival probability for this model is derived, which is more direct than that in Asmussen（2000, P195, Proposition 1.10）. The occupation time of non-dividend of this model is also discussed by means of Martingale method.

Ruin Probabilities of a Surplus Process Described by PDMPs

In this paper we mainly study the ruin probability of a surplus process described by a piecewise deterministic Markov process （PDMP）. An integro-differential equation for the ruin probability is derived. Under a certain assumption, it can be transformed into the ruin probability of a risk process whose premiums depend on the current reserves. Using the same argument as that in Asmussen and Nielsen, the ruin probability and its upper bounds are obtained. Finally, we give an analytic expression for ruin probability and its upper bounds when the claim-size is exponentially distributed.

On the Gerber-Shiu Discounted Penalty Function for a Surplus Process Described by PDMPs

In this paper, we investigate the Gerber-Shiu discounted penalty function for the surplus process described by a piecewise deterministic Markov process （PDMP）. We derive an integral equation for the Gerber-Shiu discounted penalty function, and obtain the exact solution when the initial surplus is zero. Dickson formulae are also generalized to the present surplus process.

Smoothness of Certain Functions in Two Kinds of Risk Models with a Barrier Dividend Strategy

In this paper,we study the smoothness of certain functions in two kinds of risk models with a barrier dividend strategy.Mainly using technique from the piecewise deterministic Markov processes theory,we prove that the function is continuously differentiable in the first risk model.Using the weak infinitesimal generator method of Markov processes,we prove that the function is twice continuously differentiable in the second risk model.Intego-differential equations satisfied by them are derived.