Optimal Consumption and Portfolio with Both Fixed and Proportional Transaction Costs

Bernt Øksendal · Agnès Sulem

185 citations0/0 files storedSIAM Journal on Control and Optimization

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Stochastic processes and financial applications Risk and Portfolio Optimization Probability and Risk Models

Abstract

We consider a market model with one risk-free and one risky asset, in which the dynamics of the risky asset are governed by a geometric Brownian motion. In this market we consider an investor who consumes from the bank account and who has the opportunity at any time to transfer funds between the two assets. We suppose that these transfers involve a fixed transaction cost k>0, independent of the size of the transaction, plus a cost proportional to the size of the transaction. The objective is to maximize the cumulative expected utility of consumption over a planning horizon. We formulate this problem as a combined stochastic control/impulse control problem, which in turn leads to a (nonlinear) quasi-variational Hamilton--Jacobi--Bellman inequality (QVHJBI). We prove that the value function is the unique viscosity solution of this QVHJBI. Finally, numerical results are presented.

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References

1971Optimum consumption and portfolio rules in a continuous-time model 1983Stochastic Differential Equations and Diffusion Processes 2003Stochastic Differential Equations 1990Portfolio Selection with Transaction Costs 1980Controlled Diffusion Processes 1994Optimal Investment and Consumption with Transaction Costs

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