Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: A viscosity solution approach

Abstract

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].

Viscosity Solutions for a System of Integro-PDEs and Connections to Optimal Switching and Control of Jump-Diffusion Processes

“…In some recent developments [9,10], Barles & Jakobsen used solutions of certain switching systems to generate suitable approximations of the viscosity solution of the Bellman equation associated with the optimal control of diffusion processes. In a future work we will adapt this approach to the nonlocal Bellman equation of controlled jump-diffusion processes, which is drawing a lot of interests these days due to its applications in mathematical finance (see for example [3], [2], [14], [15], [19] and the references therein). To derive error estimates like those in [9,10] for the nonlocal Bellman equation we need to have at our disposal a viscosity solution theory for switching systems of the type (1.1).…”

On optimal control of capital injections by reinsurance and investments

“…For introduction to viscosity solutions see for example Bardi and Capuzzo-Dolcetta [6] or Crandall et al [17]. The proof techniques used in the case δ ≥ 0, one finds in Benth et al [8], Azcue and Muler [5] and Albrecher and Thonhauser [1]. In the special case δ = 0 one can show the existence and uniqueness of the classical solution to the HJB equation using the proof techniques from Schmidli [70].…”

“…In the proof of the theorem below we used the technique proposed by Benth et al [8], which was also successfully applied by Albrecher and Thonhauser [1] and Azcue and Muler [5].…”

Non-linear degenerate integro-partial differential evolution equations related to geometric Lévy processes and applications to backward stochastic differential equations

“…Empirical work shows that the normal distribution poorly fits the logreturn data for, e.g., stock prices. Among other things the data show heavier tails than predicted by the normal distribution, and it has in recent years been suggested to model logreturns by generalized hyperbolic distributions (see the references in [6,7,8,10,26,9] for relevant works).…”

“…An existence result and a comparison principle among uniformly continuous and at most linearly growing viscosity sub-and supersolutions of fully nonlinear parabolic integro-partial differential equations of the Bellman type are proved in [25], see also [21] for some other existence results. The Bellman equations (variational inequalities) associated with some singular stochastic control problems arising in finance are studied in [7,8]. In [16], the authors prove a "non-local" maximum principle for semicontinuous viscosity sub-and supersolutions of integro-partial differential equations, which should be compared with the "local" maximum principle for semicontinuous functions [11].…”