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Optimal Trading with Stochastic Liquidity and Volatility

We consider the problem of mean-variance optimal agency execution strategies, when the market li- quidity and volatility vary randomly in time. Under specific assumptions for the stochastic processes satisfied by these parameters, we construct a Hamilton-Jacobi-Bellman equation for the optimal cost and strategy. We solve this equation numerically and illustrate optimal strategies for varying risk aversion. These strategies adapt optimally to the instantaneous variations of market quality.