Optimal investments for robust utility functionals in complete market models

We introduce a systematic approach to the problem of maximizing the robust utility of the terminal wealth of an admissible strategy in a general complete market model, where the robust utility functional is defined by a set \cQ of probability measures. Our main result shows that this problem can be reduced to determining a "least favorable" measure Q_0\in\cQ, which is universal in the sense that it does not depend on the particular utility function. The robust problem is thus equivalent to a standard utility maximization problem with respect to the "subjective" probability measure $Q_0$. By using the Huber-Strassen theorem from robust statistics, it is shown that $Q_0$ always exists if \cQ is the core of a 2-alternating upper probability. We also discuss the problem of robust utility maximization with uncertain drift in a Black-Scholes market and the case of "weak information" as studied by Baudoin (2002)

Robust Maximization of Consumption with Logarithmic Utility

We analyze the stochastic control approach to the dynamic maximization of the robust utility of consumption and investment. The robust utility functionals are defined in terms of logarithmic utility and a dynamically consistent convex risk measure. The underlying market is modeled by a diffusion process whose coefficients are driven by an external stochastic factor process. Our main results give conditions on the minimal penalty function of the robust utility functional under which the value function of our problem can be identified with the unique classical solution of a quasilinear PDE within a class of functions satisfying certain growth conditions.Robust utility maximization, optimal consumption, stochastic factor model, stochastic control, convex risk measure, dynamic consistency, Hamilton-Jacobi-Bellman equation

Optimal Investments for Risk- and Ambiguity-Averse Preferences: A Duality Approach

Ambiguity, also called Knightian or model uncertainty, is a key feature in financial modeling. A recent paper by Maccheroni et al. (2004) characterizes investor preferences under aversion against both risk and ambiguity. Their result shows that these preferences can be numerically represented in terms of convex risk measures. In this paper we study the corresponding problem of optimal investment over a given time horizon, using a duality approach and building upon the results by Kramkov and Schachermayer (1999, 2001).Model uncertainty, ambiguity, convex risk measures, optimal investments, duality theory

A Control Approach to Robust Utility Maximization with Logarithmic Utility and Time-Consistent Penalties

We propose a stochastic control approach to the dynamic maximization of robust utility functionals that are defined in terms of logarithmic utility and a dynamically consistent convex risk measure. The underlying market is modeled by a diffusion process whose coefficients are driven by an external stochastic factor process. In particular, the market model is incomplete. Our main results give conditions on the minimal penalty function of the robust utility functional under which the value function of our problem can be identified with the unique classical solution of a quasilinear PDE within a class of functions satisfying certain growth conditions. The fact that we obtain classical solutions rather than viscosity solutions is important for the use of numerical algorithms, whose applicability is demonstrated in examples.Optimal investment, model uncertainty, incomplete markets, stochastic volatility, coherent risk measure, convex risk measure, optimal control, convex duality

Robust Optimal Control for a Consumption-investment Problem

We give an explicit PDE characterization for the solution of the problem of maximizing the utility of both terminal wealth and intertemporal consumption under model uncertainty. The underlying market model consists of a risky asset, whose volatility and long-term trend are driven by an external stochastic factor process. The robust utility functional is defined in terms of a HARA utility function with risk aversion parameter 0Optimal Consumption, Robust Control, Model Uncertainty, Incomplete Markets, Stochastic Volatility, Coherent Risk Measures, Convex Duality

Time--consistent investment under model uncertainty: the robust forward criteria

We combine forward investment performance processes and ambiguity averse portfolio selection. We introduce the notion of robust forward criteria which addresses the issues of ambiguity in model specification and in preferences and investment horizon specification. It describes the evolution of time-consistent ambiguity averse preferences. We first focus on establishing dual characterizations of the robust forward criteria. This offers various advantages as the dual problem amounts to a search for an infimum whereas the primal problem features a saddle-point. Our approach is based on ideas developed in Schied (2007) and Zitkovic (2009). We then study in detail non-volatile criteria. In particular, we solve explicitly the example of an investor who starts with a logarithmic utility and applies a quadratic penalty function. The investor builds a dynamical estimate of the market price of risk $\hat \lambda$ and updates her stochastic utility in accordance with the so-perceived elapsed market opportunities. We show that this leads to a time-consistent optimal investment policy given by a fractional Kelly strategy associated with $\hat \lambda$. The leverage is proportional to the investor's confidence in her estimate $\hat \lambda$