Narrow Results

We consider a stochastic-factor financial model wherein the asset price and the stochastic-factor processes depend on an observable Markov chain and exhibit an affine structure. We are faced with a finite investment horizon and derive optimal dynamic investment strategies that maximize the investor's expected utility from terminal wealth. To this end we apply Merton's approach, because we are dealing with an incomplete market. Based on the semimartingale characterization of Markov chains, we first derive the Hamilton–Jacobi–Bellman (HJB) equations that, in our case, correspond to a system of coupled nonlinear partial differential equations (PDE). Exploiting the affine structure of the model, we derive simple expressions for the solution in the case with no leverage, i.e. no correlation between the Brownian motions driving the asset price and the stochastic factor. In the presence of leverage, we propose a separable ansatz that leads to explicit solutions. General verification results are also proved. The results are illustrated for the special case of a Markov-modulated Heston model.

In this work, we propose a model for the extraction of a nonrenewable resource in an economy where, initially, only one agent is enabled to perform extraction tasks. However, at certain nonpredictable (random) times, more companies receive the government’s approval for extracting the country’s resources. We provide a setup suitable for the use of standard dynamic programming results for both, the competitive and cooperative schemes; we develop the corresponding HJB equations, prove a verification theorem, and give an example. Our framework is inspired by the trends that oil industries are experiencing in countries like Mexico and Russia.

In this paper we study the pricing problem for a class of Universal Variable Life (UVL) insurance products, using the idea of “Principle of Equivalent Utility”. The main features of the UVL products include the varying (death) benefit based on both tradable and non-tradable investment incomes and “multiple decrement” cases. We formulate the pricing problem as stochastic control problems, and derive the corresponding HJB equations for the value functions. In the case of exponential utilities, we obtain the explicit pricing formulae in terms of the global, bounded solutions of a class of semilinear parabolic PDEs with exponential growth. The general insurance models with multiple decrements and random time benefit payments are discussed as well.

The paper surveys recent results on the finite element approximation of Hamilton-Jacobi-Bellman equations. Various methods are analyzed and error estimates in the maximum norm are derived. Also, a finite element monotone iterative scheme for the computation of the approximate solution is given and its geometrical convergence proved.