Abstract
This paper is concerned with modeling, analysis, and numerical methods for stochastic optimal control of an illiquid stock position build-up. The stock price model is based on a geometric Brownian motion formulation, in which the drift is allowed to be purchase-rate dependent to capture the “price impact” of heavy share accumulation over time. The expected fund (or capital) availability has an upper bound. A Lagrange multiplier method is used to treat the constrained control problem. The stochastic control problem is analyzed and a verification theorem is developed. Although optimality is proved, a closed-form solution is virtually impossible to obtain. As a viable alternative, approximation schemes are developed, which consist of inner and outer approximations. The inner approximation is a numerical procedure for obtaining optimal strategies based on a fixed parameter of the Lagrange multiplier. The outer approximation is a stochastic approximation algorithm for obtaining the optimal Lagrange multiplier. Convergence analysis is provided for both the inner and outer approximations. Finally, numerical examples are provided to illustrate our results.
Published Version
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