Stochastic impulse control with discounted and ergodic optimization criteria: A comparative study for the control of risky holdings

We consider a single-asset investment fund that in the absence of transactions costs would hold a constant amount of wealth in the risky asset. In the presence of market frictions wealth is allowed to fluctuate within a control band: Its upper (lower) boundary is chosen so that gains (losses) from adjustments to the target minus (plus) fixed plus proportional transaction costs maximize (minimize) a power utility function. We compare stochastic impulse control policies derived via ergodic and discounted optimization criteria. For the solution of the ergodic problem we use basic tools from the theory of diffusions whereas the discounted problem is solved after being characterized as a system of quasi-variational inequalities. For both versions of the problem, derivation of the control bands pertains to the numerical solution of a system of nonlinear equations. We solve numerous such systems and present an extensive comparative sensitivity analysis with respect to the parameters that characterize investor’s preferences and market behavior.Transaction costs; stochastic impulse control; ergodic criteria

Controlling the risky fraction process with an ergodic criterion

This article examines a tracking problem, similar to the one presented in Pliska and Suzuki (Quantitative Finance, 2004): an investor would keep constant proportions of her wealth in different assets if markets were frictionless; however, in the presence of fixed and proportional transaction costs her implementation problem is to keep asset proportions close to the target levels whilst avoiding too much intervention costs. Instead of minimizing discounted tracking error plus transaction costs over an infinite horizon, the optimization objective here is minimization of long run tracking error plus intervention costs per unit time. This ergodic problem is treated via combining basic tools from diffusion theory and nonlinear optimization techniques. A comparative sensitivity analysis of the ergodic and discounted problems is undertaken.

Investment strategies and compensation of a mean-variance optimizing fund manager

This paper introduces a general continuous-time mathematical framework for solution of dynamic mean–variance control problems. We obtain theoretical results for two classes of functionals: the first one depends on the whole trajectory of the controlled process and the second one is based on its terminal-time value. These results enable the development of numerical methods for mean–variance problems for a pre-determined risk-aversion coefficient. We apply them to study optimal trading strategies pursued by fund managers in response to various types of compensation schemes. In particular, we examine the effects of continuous monitoring and scheme’s symmetry on trading behavior and fund performance

Optimal portfolio choice with benchmarks

We construct an algorithm that makes it possible to numerically obtain an investor’s optimal portfolio under general preferences. In particular, the objective function and risks constraints may be driven by benchmarks (reflecting state-dependent preferences). We apply the algorithm to various classic optimal portfolio problems for which explicit solutions are available and show that our numerical solutions are compatible with them. This observation allows us to conclude that the algorithm can be trusted as a viable way to deal with portfolio optimisation problems for which explicit solutions are not in reach

Robust Multiple Stopping -- A Pathwise Duality Approach

In this paper we develop a solution method for general optimal stopping problems. Our general setting allows for multiple exercise rights, i.e., optimal multiple stopping, for a robust evaluation that accounts for model uncertainty, and for general reward processes driven by multi-dimensional jump-diffusions. Our approach relies on first establishing robust martingale dual representation results for the multiple stopping problem which satisfy appealing pathwise optimality (almost sure) properties. Next, we exploit these theoretical results to develop upper and lower bounds which, as we formally show, not only converge to the true solution asymptotically, but also constitute genuine upper and lower bounds. We illustrate the applicability of our general approach in a few examples and analyze the impact of model uncertainty on optimal multiple stopping strategies