Approximation Schemes for Mixed Optimal Stopping and Control Problems with Nonlinear Expectations and Jumps

Abstract

We propose a class of numerical schemes for mixed optimal stopping and control of processes with infinite activity jumps and where the objective is evaluated by a nonlinear expectation. Exploiting an approximation by switching systems, piecewise constant policy timestepping reduces the problem to nonlocal semi-linear equations with different control parameters, uncoupled over individual time steps, which we solve by fully implicit monotone approximations to the controlled diffusion and the nonlocal term, and specifically the Lax-Friedrichs scheme for the nonlinearity in the gradient. We establish a comparison principle for the switching system and demonstrate the convergence of the schemes, which subsequently gives a constructive proof for the existence of a solution to the switching system. Numerical experiments are presented for a recursive utility maximization problem to demonstrate the effectiveness of the new schemes.2010 AMS subject classifications: 65M06, 65M12, 62L15, 93E20, 91G80where t ∈ [0, T ] is the initial time of the control problem, α is an admissible control process and τ is a stopping time, and X α,t,x is a controlled stochastic differential equation (SDE) of the form:The positive constant r denotes the discount rate, and the functions ξ and f represent the terminal payoff and the instantaneous reward function, respectively. Under certain regularity assumptions on the coefficients, one can demonstrate that the value function u satisfies a nonlocal Hamilton-Jacobi-Bellman variational inequality (HJBVI) in the viscosity sense.These results are extended in [15] to a setting where the linear expectation E is replaced by a nonlinear expectation E α,t,x generated by a BSDE with jumps, u(t, x) = sup τ sup α E α,t,x t,τ [ξ(τ, X α,t,x τ )].

Improved order 1/4 convergence for piecewise constant policy approximation of stochastic control problems

“…Piecewise constant policy time stepping has been used in a numerical method for solving Hamilton-Jacobi-Bellman equations in [13], where the computational advantage comes from the fact that over the time intervals in which the policy is constant, only linear PDEs have to be solved. This has been extended to mixed optimal stopping and control problems with nonlinear expectations and jumps in [6]. A further benefit lies in the inherent parallelism so that the linear problems with different controls can be solved on parallel processors.…”

A penalty scheme and policy iteration for nonlocal HJB variational inequalities with monotone nonlinearities

“…Works covering specific extensions to the aforementioned references include [7,40] for an application of policy iteration together with penalization to solve HJB obstacle problems with linear drivers, to [15] for schemes to HJB obstacle problems with Lipschitz drivers based on piecewise constant policy time stepping, and to [41] for policy iteration for (finite-dimensional) static HJB equations with Fréchet-differentiable concave drivers and finite control sets.…”

A Numerical Scheme for the Quantile Hedging Problem

“…To approximate v n , we consider an adaptation of the PCPT scheme in [31,3,38], and especially [20], to our setting, as described below.…”

“…An important question, from a numerical perspective, is to understand how to fix the parameters n and \pi in relation to each other. The theoretical difficulty here is to obtain a precise rate of convergence for the approximations given in Proposition 2.3 and Theorem 2.7, along the lines of the continuous dependence estimates with respect to the control discretization in [28,20], and estimates of the approximation by piecewise constant controls as in [30,29]. To answer this question in our general setting is a challenging task, extending also to error estimates for the full discretization in the next section, which is left for further research.…”

“…(ii) For the nonlinearity f , we used the Lax--Friedrichs scheme [15,20], adding the \theta (v + + v --2v) term in the definition of \widehat F to enforce monotonicity.…”