Comonotonic approximation to periodic investment problems under stochastic drift

We investigate periodic investment problems under a Black-Scholes market with stochastic drift. The decision maker invests a series of positive amounts at finitely predetermined time spots, to maximize the expected terminal wealth while controlling its downside risk as measured by the Condition Value at Risk (CVaR). It turns out that the increment for unit wealth on the whole path can be divided into two parts: the increment corresponding to the stochastic drift and that corresponding to the Brownian Motion. A comonotonic approximation is proposed for the second part, and an upper bound is provided for the CVaR of the first part, which construct together a closed-form approximation of the terminal wealth under the risk measure of CVaR. We further decompose the problem into a sequence of sub-problems whose optimal solutions are explicit and follow fractional Kelly Strategy. Numerical and empirical results illustrate the performance of our methodology.