SAMS Subject Classification Number: 25

We consider an optimal portfolio and consumption choice problem which incorporates the average past consumption. The investor consumes and allocates her wealth between a risk-free asset and a risky asset. The objective is to find an allocation process and a consumption pattern which optimises the expected utility of the average past consumption. As in [1], [2], we formulate the portfolio optimisation problem as a singular stochastic control problem and is solved using dynamic programming and the theory of viscosity solutions. The value function is characterised as the unique constrained viscosity solution of the corresponding integro-differential variational inequality. For investors having a hyperbolic absolute risk aversion utility functions, we provide explicit consumption and allocation choices.

[1] F.E., Karlsen, K.H., and Reikvam, K. Optimal portfolio management rules in a non-Gaussian market with durability and intertemporal substitution. Finance and Stochastics 5 (2001) 447–467.

[2] Bo, L., Wang, Y., and Yang, X. Optimal portfolio and consumption selection with default risk. Frontiers of Mathematics in China, 7(6) (2012) 1019–1042.