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Combining Stochastic Control and Optimal Stopping in Wealth Management

Article Quant Q&A · Author: Richard
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Summary

The document formulates a wealth-management problem in which an investor chooses a consumption stream and risky-asset holdings while also deciding when to stop. Wealth earns the risk-free return, receives a Brownian-driven risky investment component, and is reduced by consumption. The objective combines discounted utility from consumption before stopping with a terminal payoff based on wealth at the stopping time.

The response classifies this as a combined optimal stopping and stochastic control problem and points to a general treatment in a textbook on stochastic control with jump diffusions. It suggests adapting that framework by omitting jumps for the stated model. No derivation, solution, or verification conditions are provided, so the document serves as a pointer to the relevant theory rather than a worked method. Applying the approach would still require specifying admissible controls, boundary conditions, and assumptions that ensure the value function and stopping rule are well defined.

Key ideas

  • The investor jointly chooses consumption, risky holdings, and a stopping time.
  • The objective adds discounted consumption utility to a payoff determined by wealth at stopping.
  • This is a combined optimal stopping and stochastic control problem.
  • A jump-diffusion treatment may be adapted by removing the jump terms for a continuous wealth model.
  • The response refers readers to a general reference but does not derive a solution.

Tags

StatisticsRisk managementPortfolio construction
Full text
# Stochastic control (HJB) for wealth process involving stopping times


# Stochastic control (HJB) for wealth process involving stopping times












Given a wealth process that evolves as $$d w_t = r w_t dt + \theta_t ( \sigma dW_t + (\mu-r) dt) - c_t dt.$$ where $\theta_t$ is the worth of holding at time $t$ and $c_t$ is the consumption stream.

Also, we define smooth functions $u,F: [0, +\infty) \rightarrow \mathbb{R}$.

How can we optimise the following:

$$V(w) = \sup_{c \geq 0, \, \theta, \, \tau} \mathbb{E} \bigg[ \int_0^{\tau} e^{- \rho t} u (c_t) dt + e^{-\rho \tau} F(w_{\tau}) \bigg| w_0 =w \bigg],$$ where $\tau$ is a stopping time.

The traditional method of using the HJB and martingale principle of optimal control does not seem to work in this case, when stopping time is involved.

Any suggestions on how to optimise this?

## Answer by Leon (score 3)

https://quant.stackexchange.com/a/25228

This is a standard combined optimal stopping and optimal stochastic control problem.

You are looking for a control $u=(\theta,c)$ and stopping time $\tau$ which maximize the pefrormance functional of the form:

$$J^{(u,\tau)}(w):=\mathbb{E}^{w} \bigg[ \int_0^{\tau} e^{- \rho t} u (c_t) dt + e^{-\rho \tau} F(w_{\tau})\chi_{\{\tau<+\infty\}}\bigg]$$

I will not write down the whole theory (HJB verification theorem is rather long) but everything what you need you can find in this book (in even more general form, I mean when dynamics of $w_{t}$ includes jumps. For your problem just ignore the jumps in the below book.):

Authors: Bernt Oksendal, Agnes Sulem

Title: Applied Stochastic Control of Jump Diffusions

Chapter: 4. Combined Optimal Stopping and Stochastic Control of Jump Diffusions

Page: 65

Second Edition

Shown 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.