A Class of Stochastic Control Problems with State Constraints

Battle Count: 55

The paper has indirect relevance to quantitative trading. The linear-quadratic structure with state constraints connects to: (1) risk-sensitive portfolio optimization (Remark 2.13 explicitly recovers risk-sensitive functionals); (2) constrained trading strategies where the portfolio state must remain in a feasible region; (3) the Schrödinger bridge connection to optimal transport in finance; (4) Doob's h-transform for conditioning processes. However, the paper is primarily a theoretical contribution to stochastic control and does not directly address trading strategies, market microstructure, or empirical financial applications. The state constraint framework could model no-short-selling or position limits but this is not explicitly developed.

High implementation complexity due to: (1) the optimal control α* = -σ^T ∇u/(2u) blows up at the boundary of C, requiring careful numerical handling; (2) computing u requires evaluating expectations of exponential functionals of killed diffusions; (3) verifying Assumptions 2.5, 2.6, and 2.7 requires deep knowledge of potential theory and PDE regularity; (4) the strong formulation requires the control to be adapted to the Brownian filtration; (5) Monte Carlo simulation of killed processes with exponential payoffs can suffer from variance issues; (6) the HJB equation has singular boundary conditions. For the explicit examples (2.9-2.11), implementation is straightforward using standard normal CDF and PDF.

The paper provides fully explicit closed-form formulae for the value function and optimal control in several examples (Examples 2.9, 2.10, 2.11). The main theorem (Theorem 2.8) gives a general probabilistic representation v = -2 ln u where u is defined via an expectation of the killed process. Monte Carlo simulation is suggested for numerical computation. However, the theoretical framework requires checking several technical assumptions (Assumptions 2.5, 2.6, 2.7) which may be non-trivial in practice. No code repository is provided.