Special Session 59: Backward Stochastic Volterra Integral Equations and Time Inconsistent Optimal Control Problems

Maximum principle for optimal control problems of stochastic Volterra equations with singular kernels

Yushi Hamaguchi Kyoto University Japan Co-Author(s):

Abstract: In this talk, we consider optimal control problems of stochastic Volterra equations (SVEs) with singular kernels and general (non-convex) control domain, and demonstrate a general maximum principle by means of the spike variation technique. We first show a Taylor type expansion of the controlled SVE with respect to the spike variation, where the convergence rates of the expansions are characterized by the singularity of the kernel. Next, assuming that the kernel is completely monotone, we convert the variational SVEs appearing in the expansion to their infinite-dimensional lifts. Then, we derive new kinds of first and second order adjoint equations in the infinite-dimensional framework and obtain a necessary condition for optimal controls. Furthermore, we discuss some relationships between the infinite dimensional adjoint equations and backward stochastic Volterra integral equations.

Asset Pricing with $\alpha$-maxmim Expected Utility Model

Xuedong He The Chinese University of Hong Kong Hong Kong Co-Author(s):    Jiacheng Fan and Xuedong He and Ruocheng Wu

Abstract: We study an asset pricing problem in which a representative agent trades a risky stock, a risk-free asset, and human capital to maximize her preference value of consumption represented by the $\alpha$-maxmin expected utility model. This preference model is known to lead to time inconsistency, so we consider intra-personal equilibrium for the representative agent and define the market equilibrium to the set of asset prices under which the intra-personal equilibrium strategy clears the market. We prove that there exists a unique market equilibrium and the asset prices are determined by the solution to a second-order ordinary differential equation. Finally, we conduct comparative statics to study the effect of the agent`s ambiguity attitude on the asset prices. This is a joint work with Jiacheng Fan and Ruocheng Wu.

Dynamic Portfolio Choice with Illiquid Securities: An Infinite-Horizon Stochastic LQ Framework

Ali Lazrak UBC Canada Co-Author(s):    Ali Lazrak, Hanxiao Wand and Jiongmin Yong

Abstract: In this paper, we provide a stochastic linear-quadratic (LQ, for short) control approach to the portfolio choice model introduced by Garleanu and Pedersen (2016). We first solve the original model in Garleanu and Pedersen ( 2016) by the classical stochastic LQ control theory in infinite horizon.To capture the present bias, we then generalize the model to the case with non-constant discounting, which is an infinite-horizon time-inconsistent stochastic LQ optimal control problem with nonhomogeneous terms. With the dynamic game point of view, we rigorously develop an approach to finding the so-called equilibrium trading intensity, which is time-consistent and satisfies the local optimality. The solvability of the associated equilibrium algebra quasi-Riccati equations and infinite-horizon extended backward stochastic Volterra integral equations are established.

Deep learning algorithms with iteration policy for fully nonlinear BSPDEs

Jingtang Ma Southwestern University of Finance and Economics Peoples Rep of China Co-Author(s):    Haofei Wu and Harry Zheng

Abstract: In this talk, I will present the recent work on the deep learning algorithms for solving the nonlinear backward stochastic partial differential equations (BSPDEs). In particular we focus on continuous-time optimal investment (utility maximization) under the rough volatility models (stochastic Volterra integral equations) which are non-Markovian. The optimal value is expressed by a nonlinear BSPDE. The deep learning algorithms with iteration policy are proposed to solve the nonlinear BSPDE and analyzed in regards to the convergence.

Solving Coupled Nonlinear Forward-backward Stochastic Differential Equations: An Optimization Perspective with Backward Measurability Loss

Yuanhua Ni Nankai University Peoples Rep of China Co-Author(s):    Yutian Wang, Xun Li

Abstract: This paper aims to extend the BML method proposed in [Probabilistic Framework of Howard`s Policy Iteration: BML Evaluation and Robust Convergence Analysis, IEEE TAC, 2024, vo.69, no.8, pp.5200-5215] to make it applicable to more general coupled nonlinear FBSDEs. We interpret BML from the fixed-point iteration perspective and show that optimizing BML is equivalent to minimizing the distance between two consecutive trial solutions in a fixed-point iteration. Thus, this paper provides a theoretical foundation for an optimization-based approach to solving FBSDEs. We also empirically evaluate the method through four numerical experiments.

Classical Differentiability of BSVIEs and Dynamic Capital Allocations

Ludger Overbeck Justus-Liebig-University/Institute of Mathematics Germany Co-Author(s):    Eduard Kromer

Abstract: Backward stochastic Volterra integral equations are used in Mathematical Finance and Risk Theory as a tool to define dynamic risk measures. We will adress the corresponding topic of Capital allocation, which requires some differentiablility of BSVIE. Capital allocations have been studied in conjunction with static risk measures in various papers. The dynamic case has been studied only in a discrete-time setting. We address the problem of allocating risk capital to subportfolios in a continuous-time dynamic context. For this purpose we introduce a classical differentiability result for backward stochastic Volterra integral equations and apply this result to derive continuous-time dynamic capital allocations. Moreover, we study a dynamic capital allocation principle that is based on backward stochastic differential equations and derive the dynamic gradient allocation for the dynamic entropic risk measure. As a consequence we finally provide a representation result for dynamic risk measures that is based on the full allocation property of the Aumann-Shapley allocation, which is also new in the static case.

On the Solvability of Second-order Backward Stochastic Volterra Integral Equations and Equilibrium HJB Equations

Chi Seng Pun Nanyang Technological University Singapore Co-Author(s):    Qian Lei, Chi Seng Pun

Abstract: This paper addresses the solvability of a broad class of nonlocal second-order backward stochastic differential equations featuring two temporal parameters or backward stochastic Volterra integral equations (2BSVIEs). These equations arise in the characterization of equilibrium strategies and corresponding value functions for time-inconsistent (TIC) stochastic control problems, where agents` present- or state-biased preferences violate Bellman`s principle of optimality. In such contexts, our formulation extends the scope of existing work by allowing both the drift and volatility of the underlying state process to be controllable, and considering objective functionals that depend on both the initial time and state. The comprehensive nature of our 2BSVIE framework requires moving away from a purely probabilistic approach for demonstrating solvability, directing us instead towards an analytical method grounded in partial differential equations (PDEs). Specifically, we employ a continuity method and Banach`s fixed-point arguments within custom-designed Banach spaces to establish the well-posedness and regularity of solutions for a class of PDEs with nonlocality in both time and space (nPDEs). Subsequently, we derive a Feynman--Kac-type formula using It\^{o}`s lemma to establish a relationship between the solutions of the 2BSVIEs and the nPDEs, thereby proving the solvability of the general 2BSVIEs. These solvability results significantly advance the understanding of long-standing open problems in equilibrium Hamilton-Jacobi-Bellman (HJB) equations and TIC controls. Finally, we present two globally solvable financial examples.

Optimal Controls for FBSDEs: Time-Inconsistency and Time-Consistent Solutions

Hanxiao Wang Shenzhen University Peoples Rep of China Co-Author(s):    Jiongmin Yong, Chao Zhou

Abstract: This talk is concerned with an optimal control problem for a forward-backward stochastic differential equation (FBSDE, for short) with a recursive cost functional determined by a backward stochastic Volterra integral equation (BSVIE, for short). It is found that such an optimal control problem is time-inconsistent in general, even if the cost functional is reduced to a classical Bolza type one as in Peng (AMO 1993), Lim-Zhou (SICON 2001), and Yong (SICON 2010). Therefore, instead of finding a global optimal control (which is time-inconsistent), we will look for a time-consistent and locally optimal equilibrium strategy, which can be constructed via the solution of an associated equilibrium Hamilton-Jacobi-Bellman (HJB, for short) equation. A verification theorem for the local optimality of the equilibrium strategy is proved by means of the generalized Feynman-Kac formula for BSVIEs and some stability estimates of the representation parabolic partial differential equations (PDEs, for short). Under certain conditions, it is proved that the equilibrium HJB equation, which is a nonlocal PDE, admits a unique classical solution. As applications, the linear-quadratic problems, a mean-variance model, a social planner problem with heterogeneous Epstein-Zin utilities, and a Stackelberg game are briefly mentioned. In particular, we will show an interesting phenomenon in the social planner problem. We remark that our framework can cover not only the optimal control problems for FBSDEs studied in Peng (AMO 1993), Lim-Zhou (SICON 2001), Yong (SICON 2010), and so on, but also the problems of the general discounting and some nonlinear appearance of conditional expectations for the terminal state, studied in Yong (MCRF 2012, ICM 2014) and Bjork-Khapko-Murgoci (FS 2017).

A general maximum principle for optimal control of stochastic differential delay systems

tianxiao wang Sichuan University Peoples Rep of China Co-Author(s):    Weijun Meng, Jingtao Shi, Jifeng Zhang

Abstract: In this talk, we discuss the general maximum principle for a stochastic optimal control problem where the control domain is an arbitrary non-empty set and all the coefficients (especially the diffusion term and the terminal cost) contain the control and state delay. In order to overcome the difficulty of dealing with the cross term of state and its delay in the variational inequality, we propose a new method: transform a delayed variational equation into a Volterra integral equation without delay inspired by [Y. Hamaguchi, Appl. Math. Optim., 87 (2023), 42], and introduce novel first-order, second-order adjoint equations via the backward stochastic Volterra integral equation theory established in [T. Wang and J. Yong, SIAM J. Control Optim., 61 (2023), 3608-3634]. Finally we express these two kinds of adjoint equations in more compact anticipated backward stochastic differential equation types for several special yet typical control systems.

Extended mean-field control problems with Poissonian common noise: Stochastic maximum principle and Hamiltonian-Jacobi-Bellman equation

Xiaoli Wei Harbin Insitute of Technology Peoples Rep of China Co-Author(s):    Lijun Bo, Jingfei Wang, Xiang Yu

Abstract: This paper studies mean-field control problems with state-control joint law dependence and Poissonian common noise. We develop the stochastic maximum principle (SMP) and establish its connection to the Hamiltonian-Jacobi-Bellman (HJB) equation on the Wasserstein space. The presence of the conditional joint law in the McKean-Vlasov dynamics and its discontinuity caused by the Poissonian common noise bring us new technical challenges. To develop the SMP when the control domain is not necessarily convex, we first consider a strong relaxed control formulation that allows us to perform the first-order variation. We also propose the technique of extension transformation to overcome the compatibility issues arising from the joint law in the relaxed control formulation. By further establishing the equivalence between the relaxed control formulation and the strict control formulation, we obtain the SMP for the original problem with strict controls. In the part to investigate the HJB equation, we formulate an auxiliary control problem subjecting to a controlled measure-valued dynamics with Poisson jumps, which allows us to derive the HJB equation of the original problem by a newly established equivalence result. We also show the connection between the SMP and the HJB equation and present an illustrative example of linear quadratic extended mean-field control with Poissonian common noise.

Almost strong equilibria for time-inconsistent stopping problems under finite horizon in continuous time

Zhou Zhou The University of Sydney Australia Co-Author(s):    Zhou Zhou

Abstract: We consider time-inconsistent stopping problems for a continuous-time Markov chain under finite time horizon with non-exponential discounting. We provide an example indicating that strong equilibria may not exist in general. As a result, we propose a notion of equilibrium called almost strong equilibrium (ASE), which is a weak equilibrium and satisfies the condition of strong equilibria except at the boundary points of the associated stopping region. We provide an iteration procedure and show that this procedure leads to an ASE. Moreover, we prove that this ASE is the unique ASE among all regular stopping policies under finite horizon. In contrast, we show that strong equilibria (and thus ASE) exist and may not be unique for the infinite horizon case. Furthermore, we show that the limit of the finite-horizon ASE, as the time horizon goes to infinity, is a weak equilibrium for the infinite-horizon problem and may not be a strong equilibrium or ASE.