Optimal Strategies for Prudent Investors

We find the optimal investment strategy for an individual who seeks to minimize one of four objectives: (1) the probability that his wealth reaches a specified ruin level {\it before} death, (2) the probability that his wealth reaches that level {\it at} death, (3) the expectation of how low his wealth drops below a specified level {\it before} death, and (4) the expectation of how low his wealth drops below a specified level {\it at} death. Young (2004) showed that under cri...

We assume that an individual invests in a financial market with one riskless and one risky asset, with the latter's price following geometric Brownian motion as in the Black-Scholes model. Under a constant rate of consumption, we find the optimal investment strategy for the individual who wishes to minimize the probability that her wealth drops below some fixed proportion of her maximum wealth to date, the so-called probability of {\it lifetime drawdown}. If maximum wealth is...

This paper investigates optimal portfolio strategies in a financial market where the drift of the stock returns is driven by an unobserved Gaussian mean reverting process. Information on this process is obtained from observing stock returns and expert opinions. The latter provide at discrete time points an unbiased estimate of the current state of the drift. Nevertheless, the drift can only be observed partially and the best estimate is given by the conditional expectation gi...

This paper derives a portfolio decomposition formula when the agent maximizes utility of her wealth at some finite planning horizon. The financial market is complete and consists of multiple risky assets (stocks) plus a risk free asset. The stocks are modelled as exponential Brownian motions with drift and volatility being Ito processes. The optimal portfolio has two components: a myopic component and a hedging one. We show that the myopic component is robust with respect to ...

Optimal execution of portfolio transactions is the essential part of algorithmic trading. In this paper we present in simple analytical form the optimal trajectory for risk-averse trader with the assumption of exponential market recovery and short-time investment horizon.

We determine the optimal robust investment strategy of an individual who targets at a given rate of consumption and seeks to minimize the probability of lifetime ruin when she does not have perfect confidence in the drift of the risky asset. Using stochastic control, we characterize the value function as the unique classical solution of an associated Hamilton-Jacobi-Bellman (HJB) equation, obtain feedback forms for the optimal investment and drift distortion, and discuss thei...

This paper investigates well posedness of utility maximization problems for financial markets where stock returns depend on a hidden Gaussian mean reverting drift process. Since that process is potentially unbounded well posedness cannot be guaranteed for utility functions which are not bounded from above. For power utility with relative risk aversion smaller than those of log-utility this leads to restrictions on the choice of model parameters such as the investment horizon ...

This survey is an introduction to asymptotic methods for portfolio-choice problems with small transaction costs. We outline how to derive the corresponding dynamic programming equations and simplify them in the small-cost limit. This allows to obtain explicit solutions in a wide range of settings, which we illustrate for a model with mean-reverting expected returns and proportional transaction costs. For even more complex models, we present a policy iteration scheme that allo...

We find the optimal investment strategy to minimize the expected time that an individual's wealth stays below zero, the so-called {\it occupation time}. The individual consumes at a constant rate and invests in a Black-Scholes financial market consisting of one riskless and one risky asset, with the risky asset's price process following a geometric Brownian motion. We also consider an extension of this problem by penalizing the occupation time for the degree to which wealth i...

In this article we consider the infinite-horizon Merton investment-consumption problem in a constant-parameter Black - Scholes - Merton market for an agent with constant relative risk aversion R. The classical primal approach is to write down a candidate value function and to use a verification argument to prove that this is the solution to the problem. However, features of the problem take it outside the standard settings of stochastic control, and the existing primal verifi...