We study a problem of optimal consumption and portfolio selection in a market where the logreturns of the uncertain assets are not necessarily normally distributed. The natural models then involve pure-jump Lévy processes as driving noise instead of Brownian motion like in the Black and Scholes model. The state constrained optimization problem involves the notion of local substitution and is of singular type. The associated Hamilton-Jacobi-Bellman equation is a nonlinear first order integro-differential equation subject to gradient and state constraints. We characterize the value function of the singular stochastic control problem as the unique constrained viscosity solution of the associated Hamilton-JacobiBellman equation. This characterization is obtained in two main steps. First, we prove that the value function is a constrained viscosity solution of an integrodifferential variational inequality. Second, to ensure that the characterization of the value function is unique, we prove a new comparison (uniqueness) result for the state constraint problem for a class of integro-differential variational inequalities. In the case of HARA utility, it is possible to determine an explicit solution of our portfolio-consumption problem when the Lévy process posseses only negative jumps. This is, however, the topic of a companion paper [7].

We study Merton's classical portfolio optimization problem for an investor who can trade in a risk-free bond and a stock. The goal of the investor is to allocate money so that her expected utility from terminal wealth is maximized. The special feature of the problem studied in this paper is the inclusion of stochastic volatility in the dynamics of the risky asset. The model we use is driven by a superposition of nonGaussian Ornstein-Uhlenbeck processes and it was recently proposed and intensively investigated for real market data by Barndorff-Nielsen and Shephard (2001). Using the dynamic programming method, explicit trading strategies and expressions for the value function via Feynman-Kac formulas are derived and verified for power utilities.Some numerical examples are also presented.

We determine the optimal portfolio management rules for a portfolio selection problem with consumption which incorporates the notions of durability and intertemporal substitution. The logreturns of the uncertain assets are not necessarily normally distributed. The natural models then involve Lévy processes as the driving noise instead of the more frequently used Brownian motion. The optimization problem is a state constrained singular stochastic control problem and the associated Hamilton-Jacobi-Bellman equation is a non linear second order degenerate elliptic integro-differential equation subject to gradient and state constraints. For utility functions of HARA type, we calculate the optimal investment and consumption policies together with an explicit expression for the value function. Also for the classical Merton problem, which is a special case of our optimization problem, we provide explicit policies. Instead of relying on a classical verification theorem, we verify our results within a viscosity solution framework. This framework is an adaption of the one used in our companion paper [4], which is devoted to a characterization of the value function as the unique constrained viscosity solution of the associated Hamilton-Jacobi-Bellman equation in the case of general Utilities and pure-jump Lévy processes.

We investigate an infinite horizon investment-consumption model in which a single agent consumes and distributes her wealth between a risk-free asset (bank account) and several risky assets (stocks) whose prices are governed by Lévy (jump-diffusion) processes. We suppose that transactions between the assets incur a transaction cost proportional to the size of the transaction. The problem is to maximize the total utility of consumption under Hindy-Huang-Kreps intertemporal preferences. This portfolio optimization problem is formulated as a singular stochastic control problem and is solved using dynamic programming and the theory of viscosity Solutions. The associated dynamic programming equation is a second order degenerate elliptic integro-differential variational inequality subject to a State constraint boundary condition. The main result is a characterization of the value function as the unique constrained viscosity solution of the dynamic programming equation. Emphasis is put on providing a framework that allows for a general dass of Lévy processes, Owing to the complexity of our investment-consumption model, it is not possible to derive closed form Solutions for the value function. Hence the optimal policies cannot be obtained in closed form from the first order conditions for the dynamic programming equation. Therefore we have to resort to numerical methods for computing the value function as well as the associated optimal policies. In view of the viscosity solution theory, the analysis found in this paper will ensure the convergence of a large dass of numerical methods for the investment-consumption model in question.

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We prove existence of optimal investment-consumption strategies for an infinite horizon portfolio optimization problem in a Lévy market with intertemporal substitution and transaction costs. This paper complements our previous work [4], which established that the valne function can be uniquely characterized as a constrained viscosity solution of the associated Hamilton-Jacobi-Bellman equation (but [4] left open the question of existence of optimal strategies). In this paper, we also give an alternative proof of the viscosity solution property of the value function. This proof exploits the existence of optimal strategies and is consequently simpler than the one proposed in [4].

In [7], we proved that the American (call/put) option valuation problem can be stated in terms of one single semilinear Black and Scholes partial differential equation set in a fixed domain. The semilinear Black and Scholes equation constitutes a starting point for designing and analyzing a variety of "easy to implement" numerical schemes for computing the value of an American option. To demonstrate this feature, we propose and analyze an upwind finite difference scheme of "predictor-corrector type" for the semilinear Black and Scholes equation. We prove that the approximate solutions generated by the predictor-corrector scheme respect the early exercise constraint and that they converge uniformly to the American option value. A numerical example is also presented. Besides the predictor-corrector schemes, other methods for constructing approximate solution sequences are discussed and analyzed.