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We consider a portfolio optimisation problem on infinite horizon when
the investment policy satisfies the drawdown constraint, which is the
wealth process of an investor is always above a threshold given as a
function of the past maximum of the wealth process. The preferences are
given by a utility function and investor aims to maximise an asymptotic
growth rate of her expected utility of wealth. This problem was firstly
considered by Grossman and Zhou [3] and solved for a Black-Scholes
market and linear drawdown constraint.
The main contribution of the paper is an equivalence result: the
constrained problem with utility U and drawdown function w has the same
value function as the unconstrained problem with utility UoF, where
function F is given explicitly in terms of w. This work was inspired by
ideas from [2], whose results are a special case of our work. We show
that the connection between constrained and unconstrained problems holds
for a much more general setup than their paper, i.e. a general
semimartingale market, larger class of utility functions and drawdown
function which is not necessarily linear. The paper greatly simplifies
previous approaches using the tools of Azema-Yor processes developed in
[1]. In fact we show that the optimal wealth process for constrained
problem can be found as an explicit Azema-Yor transformation of the
optimal wealth process for the unconstrained problem.
We further provide examples with explicit solution for complete and
incomplete markets.
[1] Carraro, L., Karoui, N. E., and Obloj, J. On Azema-Yor processes,
their optimal properties and the Bachelier-Drawdown equation, to appear in
Annals of Probability, 2011.
[2] Cvitanic, J., and Karatzas, I. On portfolio optimization under
drawdown constraints. IMA Volumes in Mathematics and Its Applications
65(3), 1994, 35-45
[3] Grossman, S. J., and Zhou, Z. Optimal investment strategies for
controlling drawdowns. Mathematical Finance 3(3), 1993, 241-276