Søren Asmussen

Controlled diffusion models for optimal dividend pay-out

Abstract The reserve r ( t ) of an insurance company at time t is assumed to be governed by the stochastic differential equation dr ( t ) = ( μ − a ( t )) d t + σ d w ( t ), where w is standard Brownian motion, μ , σ > 0 constants and a ( t ) the rate of dividend payment at time t (0 acts as absorbing barrier for r ( t )). The function a ( t ) is subject to dynamic allocation and the objective is to find the one which maximizes EJ x ( a (·)), where J x = ∫ 0 ∞ e − ct a ( t ) d t is the total (discounted) pay-out of dividend and x refers to r (0) = x . Two situations are considered: 1. (a) The dividend rate is restricted so that the function a ( t ) varies in [0, a 0 ] for some a 0 a 0 is smaller than some critical value, the optimal strategy is to always pay the maximal dividend rate a 0 . Otherwise, the optimal policy prescribes to pay nothing when the reserve is below some critical level m , and to pay maximal dividend rate a 0 when the reserve is above m . 2. (b) The dividend rate is unrestricted so that a ( t ) is allowed to vary in all of (0, ∞). Then the optimal strategy is of singular control type in the sense that it prescribes to pay out whatever amount exceeds some critical level m , but not pay out dividend when the reserve is below m .

Risk theory in a Markovian environment

Abstract We consider risk processes t t⩾0 with the property that the rate β of the Poisson arrival process and the distribution of B of the claim sizes are not fixed in time but depend on the state of an underlying Markov jump process {Zt } t⩾0 such that β=β i and B=Bi when Zt=i . A variety of methods, including approximations, simulation and numerical methods, for assessing the values of the ruin probabilities are studied and in particular we look at the Cramer-Lundberg approximation and diffusion approximations with correction terms. The mathematical framework is Markov-modulated random walks in discrete and continuous time, and in particular Wiener-Hopf factorisation problems and conjugate distributions (Esscher transforms) are involved.

Optimal risk control and dividend distribution policies. Example of excess-of loss reinsurance for an insurance corporation

Abstract. We consider a model of a financial corporation which has to find an optimal policy balancing its risk and expected profits. The example treated in this paper is related to an insurance company with the risk control method known in the industry as excess-of-loss reinsurance. Under this scheme the insurance company divert part of its premium stream to another company in exchange of an obligation to pick up that amount of each claim which exceeds a certain level a. This reduces the risk but it also reduces the potential profit. The objective is to make a dynamic choice of a and find the dividend distribution policy, which maximizes the cumulative expected discounted dividend pay-outs. We use diffusion approximation for this optimal control problem, where two situations are considered:(a) The rate of dividend pay-out are unrestricted and in this case mathematically the problem becomes a mixed singular-regular control problem for diffusion processes. Its analytical part is related to a free boundary (Stephan) problem for a linear second order differential equation. The optimal policy prescribes to reinsure using a certain retention level (depending on the reserve) and pay no dividends when the reserve is below some critical level

Stationary distributions for fluid flow models with or without brownian noise

x_1

Regenerative Stochastic Simulation.

and to pay out everything that exceeds

Rare events simulation for heavy-tailed distributions

x_1

Simulation of Ruin Probabilities for Subexponential Claims

. Reinsurance will stop at a level

Improved algorithms for rare event simulation with heavy tails

x_0\leq x_1

Asymptotics for sums of random variables with local subexponential behaviour

depending on the claim size distribution.(b) The rate of dividend pay-out is bounded by some positive constant

Large deviations results for subexponential tails, with applications to insurance risk

M<\infty