Topics on forward investment theory
Keywords: Forward Utility
Portfolio Theory

In this thesis, we study three topics in optimal portfolio selection that are relevant to the theory of forward investment performance processes.
In Chapter 1, we develop a connection between the classical mean-variance optimisation and time-monotone forward performance processes for infinitesimal trading times. Namely, we consider consecutive mean-variance problems and we show that, for an appropriate choice of the corresponding mean-variance trade-off coefficients, the wealth process that is generated converges (as the trading interval goes to zero) to the optimal wealth process generated by a time-monotone forward performance process. The choice of the trade-off coefficients is made in accordance to the evolution of the risk tolerance process of the forward performance process. This result allows us to provide a fresh view on the issue of time-consistency of mean-variance analysis, for we propose a method to update mean-variance risk preferences forward in time. As a by-product, our convergence theorem generalises a result by Gyöngy (1998) on the convergence of the Euler scheme for SDEs. We also provide novel results on the Lipschitz regularity of the local risk tolerance function of forward investment performance processes. The material in this chapter is joint work with Marek Musiela and Thaleia Zariphopoulou.
Chapter 2 combines forward investment theory and partial information. Specifically, we construct forward investment performance processes in models where the drift is a random variable distributed according to a known distribution. The forward performance processes we consider are of the type U(t,x) = u(t,x, R_t), where R. denotes the process of cumulative excess returns, and u(t,x,z):[0,∞) × &Ropf; imes &Ropf;N &xrarr; &Ropf; is such that u(t,.,z) is a utility function satisfying Inada's conditions. We derive the Hamilton-Jacobi-Bellman (HJB) equation for u(.). The HJB equation is linearised into the ill-posed heat equation; then, using the multidimensional version of Widder's theorem, we fully characterise the solutions to this equation in terms of a collection of positive measures; the result is an integral representation of the convex conjugate function of u(t,.,z). We construct several examples, and we show how these can be combined, in the dual domain, to generate mixtures of forward investment performance processes. We also show that the volatility of these processes is intrinsic, in that it is not generated by changes of numéraire/measure.
In Chapter 3, we provide an extension of the Black-Litterman model to the continuous time setting. Our extension is different from, and complements that of, Frey, Gabih, and Wunderlich (2012) and Davis and Lleo (2013). Specifically, we develop a novel robust estimator of instantaneous expected returns which is continuously shrunk towards the predictions of an asset pricing theory, such as the CAPM. We derive this estimator fairly explicitly and study some of its properties. As in the Black-Litterman model, such an estimator can be used to make optimal asset allocation problems in continuous time more robust with respect to estimation errors. We provide explicit solutions to the problem of maximising expected power utility of terminal wealth, when our estimator is used to estimate the drift. As an example, we illustrate our results explicitly in the case of a multifactor model, where Arbitrage Pricing Theory predicts that alphas should be approximately zero.