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Section: New Results

Keywords : jump diffusions, stochastic control.

Stochastic control - Application in finance and assurance

Participants : B. Øksendal (Oslo University), D. Hernandez-Hernandez, M. Mnif, A. Ngo, P. Tankov, A. Sulem.

B. Øksendal (Oslo University) and A.Sulem have written a book on Stochastic control of Jump diffusions [12] . The main purpose was to give a rigorous, yet mostly nontechnical, introduction to the most important and useful solution methods of various types of stochastic control problems for jump diffusions and its applications. The types of control problems covered include classical stochastic control, optimal stopping, impulse control and singular control. Both the dynamic programming method and the maximum principle method are discussed, as well as the relation between them. Corresponding verification theorems involving the Hamilton-Jacobi Bellman equation and/or (quasi-)variational inequalities are formulated. There are also chapters on the viscosity solution formulation and numerical methods. The text emphazises applications, mostly to finance. All the main results are illustrated by examples and exercises appear at the end of each chapter with complete solutions.

In [39] , M.Mnif and A.Sulem study the optimal reinsurance policy and dividends distribution of an insurance company under excess of loss reinsurance. The objective of the insurer is to maximize the expected discounted dividends. They suppose that in the absence of dividend distribution, the reserve process of the insurance company follows a compound Poisson process. Existence and uniqueness results for the associated integro-differential variational inequality in the viscosity sense are given. The optimal strategy of reinsurance, the optimal strategy of dividends pay-out and the value function are then computed numerically.

A. Ngo, P. Tankov and A. Sulem are studing pricing and hedging in markets with jumps using utility maximization and indifference pricing. D. Hernandez-Hernandez has solved the problem of characterization of the indifference price of derivatives for stochastic volatility models.

D. Lamberton and Gilles Pagès have studied the rate of convergence of the classical two-armed bandit algorithm in [57] . They have also investigated another algorithm with a penalization procedure [56] .


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