## 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].