Stochastic control under progressive enlargement of filtrations and applications to multiple defaults risk management

We formulate and investigate a general stochastic control problem under a progressive enlargement of filtration. The global information is enlarged from a reference filtration and the knowledge of multiple random times together with associated marks when they occur. By working under a density hypothesis on the conditional joint distribution of the random times and marks, we prove a decomposition of the original stochastic control problem under the global filtration into classical stochastic control problems under the reference filtration, which are determined in a finite backward induction. Our method revisits and extends in particular stochastic control of diffusion processes with finite number of jumps. This study is motivated by optimization problems arising in default risk management, and we provide applications of our decomposition result for the indifference pricing of defaultable claims, and the optimal investment under bilateral counterparty risk. The solutions are expressed in terms of BSDEs involving only Brownian filtration, and remarkably without jump terms coming from the default times and marks in the global filtration

Long time asymptotics for optimal investment

This survey reviews portfolio selection problem for long-term horizon. We consider two objectives: (i) maximize the probability for outperforming a target growth rate of wealth process (ii) minimize the probability of falling below a target growth rate. We study the asymptotic behavior of these criteria formulated as large deviations control pro\-blems, that we solve by duality method leading to ergodic risk-sensitive portfolio optimization problems. Special emphasis is placed on linear factor models where explicit solutions are obtained

Feynman-Kac representation of fully nonlinear PDEs and applications

The classical Feynman-Kac formula states the connection between linear parabolic partial differential equations (PDEs), like the heat equation, and expectation of stochastic processes driven by Brownian motion. It gives then a method for solving linear PDEs by Monte Carlo simulations of random processes. The extension to (fully)nonlinear PDEs led in the recent years to important developments in stochastic analysis and the emergence of the theory of backward stochastic differential equations (BSDEs), which can be viewed as nonlinear Feynman-Kac formulas. We review in this paper the main ideas and results in this area, and present implications of these probabilistic representations for the numerical resolution of nonlinear PDEs, together with some applications to stochastic control problems and model uncertainty in finance

Optimal investment with counterparty risk: a default-density modeling approach

We consider a financial market with a stock exposed to a counterparty risk inducing a drop in the price, and which can still be traded after this default time. We use a default-density modeling approach, and address in this incomplete market context the expected utility maximization from terminal wealth. We show how this problem can be suitably decomposed in two optimization problems in complete market framework: an after-default utility maximization and a global before-default optimization problem involving the former one. These two optimization problems are solved explicitly, respectively by duality and dynamic programming approaches, and provide a fine understanding of the optimal strategy. We give some numerical results illustrating the impact of counterparty risk and the loss given default on optimal trading strategies, in particular with respect to the Merton portfolio selection problem

Impulse control problem on finite horizon with execution delay

We consider impulse control problems in finite horizon for diffusions with decision lag and execution delay. The new feature is that our general framework deals with the important case when several consecutive orders may be decided before the effective execution of the first one. This is motivated by financial applications in the trading of illiquid assets such as hedge funds. We show that the value functions for such control problems satisfy a suitable version of dynamic programming principle in finite dimension, which takes into account the past dependence of state process through the pending orders. The corresponding Bellman partial differential equations (PDE) system is derived, and exhibit some peculiarities on the coupled equations, domains and boundary conditions. We prove a unique characterization of the value functions to this nonstandard PDE system by means of viscosity solutions. We then provide an algorithm to find the value functions and the optimal control. This easily implementable algorithm involves backward and forward iterations on the domains and the value functions, which appear in turn as original arguments in the proofs for the boundary conditions and uniqueness results

On the Use of Data Envelopment Analysis in Hedge Fund Performance Appraisal

This paper aims to show that Data Envelopment Analysis (DEA) is an efficient tool to assist investors in multiple criteria decision-making tasks like assessing hedge fund performance. DEA has the merit of offering investors the possibility to consider simultaneously multiple evaluation criteria with direct control over the priority level paid to each criterion. By addressing main methodological issues regarding the use of DEA in evaluating hedge fund performance, this paper attempts to provide investors sufficient guidelines for tailoring their own performance measure which reflect successfully their own preferences. Although these guidelines are formulated in the hedge fund context, they can also be applied to other kinds of investment funds.hedge fund, mutual fund, alternative investment, data envelopment analysis, performancemeasures, Sharpe ratio

Hedge fund behavior: An ex-post analysis

This paper aims to analyze hedge fund index behavior over the 9-year period ranging from January 1994 to December 2002 with help of various statistical measures. The results indicate that hedge fund returns are not normally distributed and exhibit first order autocorrelation, a phenomenon known as smoothing or stale price bias. Entire period correlations between 13 hedge fund indices and 85 market factors provide evidence that most of hedge fund styles show strong positive correlations with equity and real estate indices, and negative correlations with volatility index. Two exceptions are Dedicated Short Bias and Long Short Equity indices, which exhibit significant negative correlations with equity indices but positive correlations with volatility index. However, these correlations vary over time, depending on market conditions. The results also reveal that hedge funds generally underperform than the market in upward periods but do better than the market in downward ones. Dedicated Short Bias and Long Short Equity are the only ones that make loss in upward markets and make profits in downside market.hedge fund, alternative investment, performance measurement