Optimal reinsurance via BSDEs in a partially observable contagion model with jump clusters

We investigate the optimal reinsurance problem when the loss process exhibits jump clustering features and the insurance company has restricted information about the loss process. We maximize expected exponential utility of terminal wealth and show that an optimal solution exists. By exploiting both the Kushner-Stratonovich and Zakai approaches, we provide the equation governing the dynamics of the (infinite-dimensional) filter and characterize the solution of the stochastic optimization problem in terms of a BSDE, for which we prove existence and uniqueness of solution. After discussing the optimal strategy for a general reinsurance premium, we provide more explicit results for proportional reinsurance under the expected value premium principle

Max-Min optimization problem for Variable Annuities pricing

International audienceWe study the valuation of variable annuities for an insurer. We concentrate on two types of these contracts that are the guaranteed minimum death benefits and the guaranteed minimum living benefits ones and that allow the insured to withdraw money from the associated account. As for many insurance contracts, the price of variable annuities consists in a fee, fixed at the beginning of the contract, that is continuously taken from the associated account. We use a utility indifference approach to determine this fee and, in particular, we consider the indifference fee rate in the worst case for the insurer i.e. when the insured makes the withdrawals that minimize the expected utility of the insurer. To compute this indifference fee rate, we link the utility maximization in the worst case for the insurer to a sequence of maximization and minimization problems that can be computed recursively. This allows to provide an optimal investment strategy for the insurer when the insured follows the worst withdrawals strategy and to compute the indifference fee. We finally explain how to approximate these quantities via the previous results and give numerical illustrations of parameter sensibility

Quantitative Risk Management under the Interplay of Insurance and Financial Risks

En esta tesis, abordamos algunos problemas relacionados con la interacción de los riesgos de seguros y financieros. Primero, consideramos una compañía de seguros o financiera con la intención de asignar el capital de riesgo retenido para su cartera de inversión general entre sus constituyentes. Brevemente, suponemos que la compañía calcula el capital de riesgo a través de la medida de riesgo Haezendonck - Goovaerts, y establecemos la regla de asignación de capital única consistente con un enfoque RORAR (retorno sobre capital ajustado al riesgo). Además, presentamos algunas asintóticas y proponemos un estimador consistente para la regla de asignación de capital. Finalmente, realizamos algunos estudios numéricos. Luego, resolvemos el problema de valorar algunos derivados vinculados a la mortalidad empleando el enfoque de precios de indiferencia de utilidad. De manera sucinta, suponemos que el riesgo de mortalidad emana de una cartera de asegurados de vida, cuyas vidas restantes se modelan como tiempos aleatorios condicionalmente independientes. Al adaptar algunos resultados de la teoría del riesgo de crédito, calculamos una expresión explícita para el precio de indiferencia de la utilidad cuando el derivado es una combinación lineal de dotaciones puras. Al considerar una reclamación contingente más general, utilizamos técnicas de ecuaciones diferenciales estocásticas hacia atrás (BSDE) para caracterizar el precio de indiferencia en términos de una solución a un BSDE no lineal con un generador no Lipschitz. Finalmente, consideramos a un individuo con el objetivo de elegir de manera óptima sus estrategias de inversión, consumo y compra de seguros de vida en un mercado financiero completo. Al suponer que el criterio de optimización es la maximización de la utilidad esperada del individuo, la cual depende del estado de la economía, resolvemos el problema de elección óptima en una configuración general, que incluye varias funciones de utilidad empleadas en la literatura.In this thesis, we tackle some problems concerning the interplay of insurance and financial risks. First, we consider an insurance or financial company intending to allocate the risk capital withheld for its overall investment portfolio among its constituents. Shortly, we assume that the company computes the risk capital through the Haezendonck--Goovaerts risk measure, and we establish the unique capital allocation rule consistent with a RORAR (return on risk-adjusted capital) approach. Besides, we present some asymptotics and propose a consistent estimator for the capital allocation rule. Finally, we conduct some numerical studies. Then, we solve the problem of valuing some mortality-linked derivatives by employing the utility indifference pricing approach. Succinctly, we suppose that the mortality risk emanates from a portfolio of life insurance policyholders, whose remaining lifetimes are modeled as conditionally independent random times. By adapting some results from credit risk theory, we compute an explicit expression for the utility indifference price when the derivative is a linear combination of pure endowments. By considering a more general contingent claim, we use techniques of backward stochastic differential equations (BSDE) to characterize the indifference price in terms of a solution to a non-linear BSDE with a non-Lipschitz generator. Finally, we consider an individual aiming to optimally choose its investment, consumption, and life insurance purchase strategies in a complete financial market. By assuming that the optimality criterion is the maximization of the individual's expected state-dependent utility, we solve the optimal choice problem in a general setup, which includes several utility functions employed in the literature.Doctor en CienciaDoctorad

Dynamic Credit Investment in Partially Observed Markets

We consider the problem of maximizing expected utility for a power investor who can allocate his wealth in a stock, a defaultable security, and a money market account. The dynamics of these security prices are governed by geometric Brownian motions modulated by a hidden continuous time finite state Markov chain. We reduce the partially observed stochastic control problem to a complete observation risk sensitive control problem via the filtered regime switching probabilities. We separate the latter into pre-default and post-default dynamic optimization subproblems, and obtain two coupled Hamilton-Jacobi-Bellman (HJB) partial differential equations. We prove existence and uniqueness of a globally bounded classical solution to each HJB equation, and give the corresponding verification theorem. We provide a numerical analysis showing that the investor increases his holdings in stock as the filter probability of being in high growth regimes increases, and decreases his credit risk exposure when the filter probability of being in high default risk regimes gets larger