Special Session 160: Recent progress on stochastic analysis and stochastic control with applications

An Asymptotic Expansion of the Profit Function of a Class of Impulse Control Problems

Kurt L Helmes Humboldt Universitaet zu Berlin Germany Co-Author(s):

Abstract: We derive an explicit asymptotic expansion for the profit function in a class of long-term average stochastic impulse control problems for one-dimensional diffusions with two revenue streams: intervention rewards and state-dependent subsidies. Optimality results characterize the long-term average optimal value $F^*$ as the maximum of a nonlinear function on $\mathbb{R}^2$ and the optimal threshold policy as its argmax. When the diffusion`s left boundary is an entrance point, the profit function $J^*(T,x)$, representing the expected accumulated optimal profit up to time $T$, satisfies $J^*(T,x) \approx F^* T + G(x) - \langle G, \nu^* \rangle$, where $\nu^*$ is the stationary distribution of the optimally controlled system and $G$ the relative value function. The correction term $G(x) - \langle G, \nu^* \rangle$ can also be expressed using the first two moments of the profit contributions and the average cycle time of the optimal process. This interpretable representation aids applications and reveals connections to cumulative renewal processes. The results are joint work with R. Stockbridge and C. Zhu and are based on Long-Term Average Impulse and Singular Control of a Growth Model with Two Revenue Sources (2026a, https://arxiv.org/pdf/2601,09646.pdf) and Long-Term Average Impulse Control with Mean Field Interactions (2026b, https://arxiv.org/pdf/2505.11345.pdf).

An Asymptotic Expansion of the Profit Function of a Class of Impulse Control Problems\end

Kurt L Helmes Humboldt Universitaet zu Berlin Germany Co-Author(s):

Abstract: We derive an explicit asymptotic expansion for the profit function in a class of long-term average stochastic impulse control problems for one-dimensional diffusions with two revenue streams: intervention rewards and state-dependent subsidies. Optimality results characterize the long-term average optimal value $F^*$ as the maximum of a nonlinear function on $\mathbb{R}^2$ and the optimal threshold policy as its argmax. When the diffusion`s left boundary is an entrance point, the profit function $J^*(T,x)$, representing the expected accumulated optimal profit up to time $T$, satisfies $J^*(T,x) \approx F^* T + G(x) - \langle G, \nu^* \rangle$, where $\nu^*$ is the stationary distribution of the optimally controlled system and $G$ the relative value function. The correction term $G(x) - \langle G, \nu^* \rangle$ can also be expressed using the first two moments of the profit contributions and the average cycle time of the optimal process. This interpretable representation aids applications and reveals connections to cumulative renewal processes. The results are joint work with R. Stockbridge and C. Zhu and are based on Long-Term Average Impulse and Singular Control of a Growth Model with Two Revenue Sources (2026a, https://arxiv.org/pdf/2601,09646.pdf) and Long-Term Average Impulse Control with Mean Field Interactions (2026b, https://arxiv.org/pdf/2505.11345.pdf).

Large deviation principles for fully coupled multiscale multivalued stochastic systems

Huijie Qiao Southeast University Peoples Rep of China Co-Author(s):

Abstract: This study focuses on large deviation principles for fully coupled multiscale multivalued stochastic systems, in which the slow component is governed by a multivalued stochastic differential equation and the fast component is described by a general stochastic differential equation. First, we establish the large deviation principle for the slow component at any fixed time by leveraging viscosity solutions of second-order Hamilton-Jacobi-Bellman equations involving multivalued operators. Subsequently, we illustrate the theoretical results through a concrete finance example.

Averaging Principle for Fully Coupled Two-Time-Scale Stochastic Systems with Infinite Integral-Type Delays and Application to Near-Optimal Control

Meilin Tang Department of Mathematics Huazhong University of Science and Technology Peoples Rep of China Co-Author(s):    Fuke Wu

Abstract: This paper establishes the averaging principle for a class of fully coupled two-time-scale stochastic functional differential equations involving infinite integral-type delays with exponential kernels. The presence of delays leads to the lack of the Markov property, which introduces essential difficulties. Noting the special structure of the infinite delay, by using the variable substitution technique, this paper transforms the original system into a higher-dimensional system without delay. For this higher-dimensional system, we establish the existence and uniqueness of the invariant probability measure for the fixed-$X$ equation and continuous dependence on the parameter $X$. Moreover, the uniform moment boundedness of the fast-varying and slow-varying variables is also proved. Based on these results, by using the martingale problem formulation, we prove the weak convergence of the slow-varying process. As an important application, we investigate a two-time-scale stochastic optimal control problem. By using the established averaging principle and the relaxed control framework, we prove the convergence of the value function and construct near-optimal controls for the original system by using the optimal control of the limit problem. Finally, the example of an optimal advertising problem is presented to demonstrate the applicability of the obtained results.

Minimizing the Ruin Probability with Irreversible Reinsurance and Investments

Gu Wang Worcester Polytechnic Institute USA Co-Author(s):

Abstract: The insurer chooses the investment and reinsurance strategy to minimize the probability of ruin, in a diffusion model for the uncontrolled surplus. The reinsurance contract is irreversible and costly with transaction cost at purchase and premium proportional to the size of ceded risk. The optimal strategy entails constant dollar amount invested in the financial asset correlated with the actuarial risk and reinsurance purchase only when the surplus is greater than the endogenous free boundary. For sufficiently large level of ceded risk, reinsurance becomes ineffective for reducing ruin probability due to the long term premium commitment

Recent progress on optimal control of partially observable forward-backward stochastic system

Guangchen Wang Shandong University Peoples Rep of China Co-Author(s):

Abstract: This talk is concerned with recent progress on optimal control of forward-backward stochastic system, where the information available to controller is provided by noisy observation. A backward separation method is originally proposed and is used to establish a fundamental research framework for such a class of problems. The framework is also applicable to other important scenarios.

Fast-Slow Coupled Forward-Backward Stochastic Differential Equations

Fuke Wu Huazhong University of Science and Technology Peoples Rep of China Co-Author(s):    Yihao Sheng, George Yin

Abstract: This work focuses on fast-slow coupled forward-backward stochastic differential equations (FBSDEs). Firstly, the well-posedness of such systems is established and some necessary estimates are provided. Subsequently, the averaging principle with strong convergence for both forward and backward components is established by using Khasminskii`s time discretization technique. Finally, this averaging principle is applied to examine the singularly perturbed linear-quadratic stochastic control problem.

Averaging principles for nonautonomous multiscale McKean-Vlasov stochastic systems

Jie Xiang School of Mathematics, Southeast University Peoples Rep of China Co-Author(s):    Huijie Qiao

Abstract: In this paper, we study a class of nonautonomous multiscale McKean-Vlasov stochastic systems. By using the nonautonomous Poisson equation, we establish the strong and weak averaging principles with explicit convergence rates. In general, the coefficients of the averaged equations depend on the scaling parameter $\varepsilon$. Additionally, when the fast coefficients are asymptotically convergent or time-periodic, we show that the slow component converges, in the strong and weak senses, to the solution of an averaged equation with coefficients independent of $\varepsilon$.

Pairs Trading under Geomertic Brownian Motions

Qing Zhang University of Georgia USA Co-Author(s):

Abstract: This talk studies optimal strategies for pairs trading, in which a long position is taken in a weaker stock and a short position in a stronger one following price divergence. Stock prices are modeled by general geometric Brownian motions, and transaction costs are incorporated. The optimal trading policy is characterized by threshold curves derived from Hamilton Jacobi Bellman equations. Several recent developments are also reported, including pairs trading with stop-loss constraints and trading under additional market constraints.

Convergence of Proximal Policy Gradient Method for Problems with Control Dependent Diffusion Coefficients

Harry Zheng Imperial College England Co-Author(s):    Ashley Davey

Abstract: We prove convergence of the proximal policy gradient method for a class of constrained stochastic control problems with control in both the drift and diffusion of the state process. The problem requires either the running or terminal cost to be strongly convex, but other terms may be non-convex. The inclusion of control-dependent diffusion introduces additional complexity in regularity analysis of the associated backward stochastic differential equation. We provide sufficient conditions under which the control iterates converge linearly to the optimal control, by deriving representations and estimates of the solution to adjoint BSDEs. We introduce numerical algorithms that implement this method using deep learning and ODE-based techniques. These approaches enable high accuracy and scalability for stochastic control problems in higher dimensions. We provide numerical examples to demonstrate the accuracy and validate the theoretical convergence guarantees of the algorithms.

Ergodic McKean-Vlasov Games: Verification Theorems and Linear-Quadratic Applications

Chao Zhu University of Wisconsin-Milwaukee USA Co-Author(s):    Qingshuo Song, Gu Wang, and Zuo Quan Xu

Abstract: This work develops a general framework for two-player ergodic nonzero-sum stochastic differential games with McKean-Vlasov dynamics. A verification theorem is established, linking solutions of a system of coupled Hamilton-Jacobi-Bellman (HJB) Master equations to Nash equilibria, which are characterized via an auxiliary bias optimal control problem formulated on the space of probability measures. The paper demonstrates that the value functions of this auxiliary control problem are uniquely determined, up to an additive constant, by the uniqueness of the invariant measure associated with the optimally controlled state process. The framework is further illustrated in a Linear-Quadratic-Gaussian (LQG) setting, where explicit solutions to the Master equations are obtained by exploiting their polynomial structure in the measure variables.