State-Dependent Singular Control and Its HJB Quasi-Variational Inequality
Summary
This document frames a stochastic control problem in which a regular control acts through the drift and diffusion, while a nondecreasing singular control changes the state through a coefficient that depends on both time and the current state. Its central question is how that state dependence affects the dynamic programming derivation and verification of the associated Hamilton–Jacobi–Bellman quasi-variational inequality. The author compares a general dynamic programming treatment with a textbook discussion and a financial application involving tax timing, seeking a rigorous source for the formulation.
The document does not provide a derivation, proof, or empirical result; it is a request for references prompted by difficulties adapting existing arguments. Its value is in identifying a modeling and mathematical issue that researchers may need to investigate when applying singular control methods. The discussion is specialized and does not establish which cited approach is correct or resolve the concerns about verification. It is most relevant to quantitative researchers studying stochastic control and derivative or financial decision models, rather than to traders seeking a ready-to-use strategy.
Key ideas
- The state dynamics combine regular controls with a nondecreasing singular control.
- The singular control coefficient depends on time and the current state.
- The question concerns deriving and verifying the associated HJB quasi-variational inequality.
- The document requests rigorous references and does not supply a solution.
Tags
Full text
# Reference request for singular stochastic control problem with singular control coefficient depending on the state
# Reference request for singular stochastic control problem with singular control coefficient depending on the state
I am interested in papers/books discussing singular stochastic control problems where the dynamics are of the form $$ dX_t = b(t, X_t, u_t)dt + \sigma(t, X_t, u_t)dW_t + G(t, X_t)d\xi_t, \tag{1} $$ where $(u_t)_{0\leq t \leq T}$ is the regular control and $(\xi_t)_{0\leq t \leq T}$ is the singular control (a process which is nondecreasing and right continuous with left limits). My emphasis is on the fact that $G$ depends on both $t$ and $X_t$. I am interested in papers like [1], which provide a general treatment of the problem, focusing on the associated HJB equation (which is really a quasi variational inequality). The problem with [1] is that their $G$ depends only on $t$. I have tried to adapt their deduction of the HJB to my case, but encountered some problems.
I have read [2, Section 3.5.5], in which Carmona addresses dynamics like in equation 1. However, I think there are some issues with his proof of the verification theorem for the HJB equation and he does not present any references for it.
This search of mine started because I was reading [3] and, in this paper, the authors deal with a singular control problem with dynamics as in equation 1 and solve the associated HJB equation, but do not provide any references for why the HJB equation is what they claim it is.
References
1 - Haussmann, U. G., & Suo, W. (1995). Singular Optimal Stochastic Controls II: Dynamic programming. SIAM Journal on Control and Optimization, 33(3), 937–959. https://doi.org/10.1137/S0363012993250529
2 - Carmona, R. (2016). Lectures on BSDEs, Stochastic Control, and Stochastic Differential Games with Financial Applications. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.9781611974249
3 - Dai, M., Liu, H., Yang, C., & Zhong, Y. (2015). Optimal Tax Timing with Asymmetric Long-Term/Short-Term Capital Gains Tax. Review of Financial Studies, 28(9), 2687–2721. https://doi.org/10.1093/rfs/hhv024Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.