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.