Michael I. Taksar

Regenerative Analysis and Steady State Distributions for Markov Chains

We apply regenerative theory to derive certain relations between steady state probabilities of a Markov chain. These relations are then used to develop a numerical algorithm to find these probabilities. The algorithm is a modification of the Gauss-Jordan method, in which all elements used in numerical computations are nonnegative; as a consequence, the algorithm is numerically stable.

A Diffusion Model for Optimal Portfolio Selection in the Presence of Brokerage Fees

We consider a financial market model with two assets. One has deterministic rate of growth, while the rate of growth of the second asset is governed by a Brownian motion with drift. We can shift money from one asset to another; however, there are losses of money brokerage fees involved in shifting money from the risky to the nonrisky asset. We want to maximize the expected rate of growth of funds. n nIt is proved that an optimal policy keeps the ratio of funds in risky and nonrisky assets within a certain interval with minimal effort.

Deterministic equivalent for a continuous-time linear-convex stochastic control problem

We consider a finite-horizon control model with additive input. There are two convex functions which describe the running cost and the terminal cost within the system. The cost of input is proportional to the input and can take both positive and negative values. It is shown that there exists a deterministic control problem whose optimal cost is the same as the one in the stochastic control problem. The optimal policy for the stochastic problem consists of keeping the process as close to the optimal deterministic trajectory as possible.

Excess-of-loss reinsurance under taxes and fixed costs

We consider the problem of dividend optimization for an insurance company which can use the excess-of-loss reinsur- ance to control its risk. The decrease of risk results in a loss of potential profits in view of the necessity to diverge a part of the premiums to the reinsurance company. In addition to reinsurance the decision is made about the time and the amount of dividends to be paid out to shareholders. Each time when the dividends are paid a set-up cost of K is incurred independent of the amount distributed. In addition the dividends are taxed at the rate of 1 − k ,0 <k< 1. The resulting problem becomes a mixed regular-impulse stochastic control problem for a controlled diffusion process. We solve this problem and find the optimal policy. We give an economic interpretation to the solution obtained. The solution reveals an interesting dependence of the optimal policy on the parameters of the model. We also discuss an extension of this problem to the case when there are restrictions on the level of reinsurance available and show how one can construct the value function and the optimal policy in this case.

Optimal excess-of-loss reinsurance under borrowing constraints

In this paper, we study an optimal dynamic control problem of an insurance company with excess-of-loss reinsurance and investment. Three practical borrowing constraints are studied individually: (B1) borrowed dollar amount is no more than a borrowing limit K; (B2) borrowed proportion to surplus level is no more than k; and (B3) borrowing rate is higher than risk free rate of return (saving rate). The optimal criterion is to minimize probability of ruin. Classical stochastic control theory is applied to solve the problem. Under each of the constraints, minimal ruin probability functions are obtained in closed form by solving Hamilton–Jacobi–Bellman (HJB) equations. Their associated optimal reinsurance–investment control policies are found as well.

INFINITE EXCESSIVE AND INVARIANT MEASURES

In the paper [10] the following problem was considered. Given a contraction semigroup Tt on a Borel space d and an excessive measure ν, when is it possible find another contraction semigroup Tt such that n n

Optimal consumption and investment policies with bankruptcy modelled by a diffusion with delayed reflection

widetilde{{{T_t}}} > {T_t}

Optimal risk control and dividend distribution for a financial corporation with policy constraints

n n(1.1.1) n nand n n