Exponential Utility Maximization in an Incomplete Market with Defaults

Abstract

In this paper, we study the indifference pricing of a contingent claim via the maximization of exponential utility over a set of admissible strategies. We consider a financial market with a default time inducing a discontinuity in the price of stocks. We first consider the case of strategies valued in a compact set. Using a verification theorem, we show that in the case of bounded coefficients the value function of the exponential utility maximization problem can be characterized as the solution of a Lipschitz BSDE (backward stochastic differential equation). Then, we consider the case of non constrained strategies. By using dynamic programming technics, we state that the value function is the maximal subsolution of a BSDE. Moreover, the value function is the limit of a sequence of processes, which are the value functions associated with some subsets of bounded admissible strategies. In the case of bounded coefficients, these approximating processes are the solutions of Lipschitz BSDEs, which leads to possible numerical computations. These properties can be applied to the indifference pricing problem and they can be generalized to the case of several default times or a Poisson process.

Information and Optimal Investment in Defaultable Assets

“…Moreover, our approach is different in framework and we do not use HJB type solutions. In [25] we find a study of a problem similar to ours. The approach is however entirely different as in this case backward stochastic differential equations are involved.…”

Information uncertainty related to marked random times and optimal investment

“…The optimization for the ordinary agent is standard (see for example Lim and Quenez (2011)). For the insider, we follow Amendinger et al (1998Amendinger et al ( , 2003 to solve the problem.…”

“…In the optimization problem with random default times, it is often supposed that the random time satisfies the intensity hypothesis (e.g., Lim and Quenez (2011) and Kharroubi et al (2013)) or the density hypothesis (e.g., Blanchet-Scalliet et al (2008), Jeanblanc et al (2015), and Jiao et al (2013)), so that it is a totally inaccessible stopping time in the market filtration. In particular, in Jiao et al (2013), we consider marked random times where the random mark represents the loss at default and we suppose that the vector of default time and mark admits a conditional density.…”

Portfolio optimization under model uncertainty and BSDE games

“…The method described in this section is basically well known albeit maybe not in the general context of Itô Lévy processes (see in particular [11]). For completeness we give a detailed exposition below.…”

“…Although the relation between stochastic control and BSDEs is well known (see e.g. Chapter 7 of [20] and the recent paper [11]), the application to stochastic differential games is new. Using comparison theorems for BSDEs withh jumps we arrive at tractable criteria for the solution of such games, in the form of a kind of non-Markovian analogue of the HJBI equation (Theorem 3.1).…”