Robust Impulse Control in Financial Markets with Drift Uncertainty and Transaction Costs

At the 2025 SIAM Conference on Analysis of Partial Differential Equations (SIAM PD25), which took place in Pittsburgh, Pa., in November, I presented a poster that detailed my ongoing work on robust impulse control problems motivated by quantitative finance. The project explores how optimal trading strategies are shaped by the combined effects of transaction costs and model uncertainty.

In many financial applications, continuous trading is impractical due to transaction costs [1]. This motivates the study of impulse control problems, where stakeholders make decisions at discrete times and incur fixed costs [4]. While classical impulse control models assume that the drift of an asset is known, in practice this parameter is not known with certainty and is subject to misspecification. Continuous control problems under model uncertainty have also been studied [2]. Taken together, transaction costs and drift uncertainty motivate a robust impulse control framework.

The Classical Impulse Control Problem

We begin with the classical impulse control formulation, in which the drift is assumed to be known. The state process \(X_t\) represents the value of the investor’s holding in a risky asset and follows a one-dimensional diffusion process, though the most common model is geometric Brownian motion. The investor is allowed to sell a portion of the asset at discrete intervention times. Mathematically, we model this with an impulse policy that consists of a sequence of intervention times and corresponding liquidation amounts. When an intervention occurs, the value of the holding is reduced by the chosen amount, and each intervention incurs a fixed transaction cost \(K > 0.\) The objective is to maximize the expected discounted reward over all admissible impulse policies. Thus, the value function is defined as \(V(x) = \sup_{\pi} J(x;\pi),\) where \(J(x;\pi)\) denotes the expected discounted reward with initial state \(x\) under policy \(\pi\) and is given by

\[J(x;\pi) = \mathbb E_x\!\left[ \int_0^\infty e^{-\rho t} f_0(X_t)\,dt + \sum_{i\ge1} e^{-\rho \tau_i}({\xi}_{i}- K) \right].\tag1\]

Here \(f_0(X_t)\) represents a running reward, such as a continuous dividend yield that is proportional to the value of the holding. The parameter \(\rho > 0\) denotes the discounted rate, while \({\xi}_i\) represents the amount of the asset liquidated at the intervention time \(\tau_i\). Each intervention generates an immediate payoff equal to the amount liquidated, minus the fixed transaction cost. The problem is therefore determining the optimal policy that maximizes the expected discounted reward.

Because each intervention incurs a fixed cost, the investor must act strategically. Naturally, this leads to the \((w,y)\)-strategy, which is characterized by two levels: \(0 < w < y < \infty.\) Threshold-type policies of this form arise naturally in many impulse control problems [3]. When the value of the holding reaches the upper threshold \(y,\) the investor sells a portion of the asset so that the value is reduced to a lower level \(w;\) as long as the holding value remains below \(y,\) no intervention occurs.

Under the appropriate assumptions on the diffusion and reward structure, a resolvent representation leads to the explicit reward of a \((w,y)\)-strategy:

\[J(x;w,y)={\mathcal{g}_0}(x)+\psi(x)\,F(w,y).\tag2\]

Here \(\mathcal{g}_0\) is the resolvent that corresponds to the running reward for the uncontrolled process and \(\psi\) is a fundamental solution of the homogeneous equation associated with the diffusion generator. The non-linear function \(F(w,y)\) depends on the intervention thresholds and captures the net gains from resetting the holding from \(y\) to \(w,\) accounting for liquidation proceeds and transaction costs.

Since \(\mathcal{g}_0\) and \(\psi\) depend only on \(x,\) maximizing the expected discounted reward over all \((w,y)\) strategies reduces to maximizing \(F(w,y)\) over \(0 < w < y < \infty.\) Under appropriate conditions, we can show that there exists a pair \((w^*,y^*)\) such that

\[F(w^*,y^*)={\quad}\underset{0 < w < y < \infty}{\textrm{sup}}{\quad}F(w,y).\tag3\]

Given the maximizer \((w^*,y^*)\), the corresponding threshold strategy is an optimal \((w,y)\)-policy. More importantly, it allows us to explicitly construct a solution to the Hamilton-Jacobi-Bellman Quasi-Variational Inequality (HJB-QVI), which together with a verification theorem leads to the explicit forms of the value function and the \((w^*,y^*)\)-policy is an optimal impulse control among all admissible controls.

Robust Impulse Control Under Drift Uncertainty

In contrast to the classical formulation discussed above, where the drift of the asset is assumed to be known, in practice this parameter must be estimated from historical data and is therefore subject to model uncertainty. If the true drift differs from the estimated value, strategies designed under the classical model may perform poorly.

This work introduces a robust impulse control framework that accounts for possible misspecification of the asset drift. In this setting, the investor evaluates strategies across a range of plausible models rather than under a single assumed drift. The objective is to identify policies that perform well even under unfavorable specifications of the drift.

Intuitively, the presence of uncertainty may lead the investor to adopt more conservative intervention strategies, adjusting the threshold levels that determine when liquidation occurs. The robust impulse control problem introduces substantially greater challenges than the classical formulation because the investor must account for possible misspecification of the drift while determining optimal intervention policies. Incorporating model uncertainty has important implications for practical decision-making in financial markets, where parameters must be estimated and may be unreliable. At the same time, the robust framework may lead to further theoretical developments in impulse control. Understanding how optimal intervention policies change when the investor guards against adverse drift specifications is the focus of the ongoing phase of this project.

Temitope Comfort Iroko delivered a poster presentation on this research at the 2025 SIAM Conference on Analysis of Partial Differential Equations (PD25), which took place last November in Pittsburgh, Pa. She received funding to attend PD25 through a SIAM Student Travel Award. To learn more about Student Travel Awards or to submit an application, visit the online page. 

SIAM Student Travel Awards are made possible in part by the generous support of our community. To make a gift to the Student Travel Fund, visit the SIAM website. 

Acknowledgments: I am grateful to my advisor, Chao Zhu, for his guidance throughout this project. This work was supported in part by the Northwestern Mutual Data Science Institute. 

References
[1] Almgren, R. & Chriss, N. (2001). Optimal execution of portfolio transactions. Journal of Risk, 3, 5–40.
[2] Chakraborty, P., Cohen, A., & Young, V.R. (2023). Optimal dividends under model uncertainty. SIAM J. Financ. Math., 14(2), 497–524.
[3] Helmes, K. L., Stockbridge, R. H., & Zhu, C., (2015). A measure approach for continuous inventory models: discounted cost criterion.  SIAM J. Control Optim., 53(4), 2100–2140.
[4] Korn, R. (1999). Some applications of impulse control in mathematical finance. Math. Methods Oper. Res., 50(3), 493–518.