Inria / Raweb 2004
Team: Mathfi

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Team : mathfi

Section: Application Domains

Keywords: Portfolio optimisation, transaction costs.

Portfolio optimisation

Participants: T. Bielecki (Northeastern Illinois University, Chicago), J.-Ph. Chancelier, B. Øksendal (University of Oslo), S. Pliska (University of Illinois at Chicago), A. Sulem, M. Taksar (Stony Brook University New York).

We consider a model of n risky assets (called Stocks) whose prices are governed by logarithmic Brownian motions, which can eventually depend on economic factors and one riskfree asset (called Bank). Consider an investor who has an initial wealth invested in Stocks and Bank and who has ability to transfer funds between the assets. When these transfers involve transaction costs, this problem can be formulated as a singular or impulse stochastic control problem.

In one type of models, the objective is to maximize the cumulative expected utility of consumption over a planning horizon [43] . Another type of problem is to consider a model without consumption and to maximize a utility function of the growth of wealth over a finite time horizon [41]. Finally, a third class of problem consists in maximizing a long-run average growth of wealth [42].

Dynamic programming lead to variational and quasivariational inequalities which are studied theoretically by using the theory of viscosity solutions and numerically by finite difference approximations and policy iteration type algorithms.

The case of fixed costs is studied in [46] and [71]. In [22] we develop methods of risk sensitive impulsive control theory in order to solve an optimal asset allocation problem with transaction costs and a stochastic interest rate.

In the case of jump diffusion markets, dynamic programming lead to integrodifferential equations. In the absence of transaction costs, the problem can be solved explicitly [62]: the optimal portfolio is to keep the fraction invested in the risky assets constantly equal to some optimal value. In the case of proportional transaction costs, there exists a no transaction region D with the shape of a cone with vertex at the origin, such that it is optimal to make no transactions as long as the position is in D and to sell stocks at the rate of local time (of the reflected process) at the upper/left boundary of D and purchase stocks at the rate of local time at the lower/right boundary [85]. These results generalize the results obtained in the no jump case. The case of portfolio optimisation with partial observation is studied in [90][45][58], [11].