Special Session 134: Mean field stochastic control problems and related topics

Path-depending controlled mean-field coupled forward-backward SDEs. The associated stochastic maximum principle

Rainer RB Buckdahn Universite de Bretagne Occidentale France Co-Author(s):    Juan Li, Junsong Li, Chuanzhi Xing

Abstract: We present a new type of mean-field coupled forward-backward stochastic differential equations (MFFBSDEs). The novelty consists in the fact that the coefficients of both the forward and the backward SDEs depend not only on the controlled solution processes (X, Y, Z) at the current time t, but also on the law of the paths of (X, Y, u) of the solution process and the control process. The existence of the solution for such a MFFBSDE fully coupled through the law of the paths of (X, Y ) in the coefficients of both the forward and the backward equations is proved under rather general assumptions. Concerning the law, we just suppose the continuity under the 2-Wasserstein distance of the coefficients with respect to the law of (X, Y). The uniqueness is shown under Lipschitz assumptions and the non anticipativity of the law of X in the forward equation. The main part of the work is devoted to the study of the Pontryagin maximal principle for such a MFFBSDE. The dependence of the coefficients on the law of the paths of the solution processes and their control makes that a completely new and interesting criterion for the optimality of a stochastic control for the MFFBSDE is obtained. In particular, also the Hamiltonian is novel and quite different from that in the existing literature. Last but not least, under the assumption of convexity of the Hamiltonian we show that our optimality condition is not only necessary but also sufficient.

A stochastic porous media Schrodinger equation: Feynman-type motivation, well-posedness and control interpretation

Ioana Ciotir INSA Rouen Normanie France Co-Author(s):    Ioana Ciotir, Dan Goreac, Juan Li, Xinru Zhang

Abstract: This papers aim is threefold. First, using Feynmans path approach to the derivation of the classical Schrodingers equation in Feynman (Rev Mod Phys, 1948) and by introducing a slight path (or wave) dependency of the action, we derive a new class of equations of Schrodinger type where the driving operator is no longer the Laplace one but rather of complex porous media type. Second, using suitable concepts of monotonicity in the complex setting and on appropriate functional spaces, we show the existence and uniqueness of the solution to this type of equation. In the formulation of our equation, we adjoin possible measurement absolute errors translating in an additive Brownian perturbation and interactions between different waves translating in a mean-field (or McKean-Vlasov) dependency of drift coefficient. Finally, using Fitzpatricks characterization of maximal monotone operators (Schrodinger Phys Rev, 1926) we propose a Brezis-Ekeland-type characterization of the solution of the deterministic equation via a control problem. This is envisaged as a possible way to overcome strict monotonicity requirements in the complex setting.

Continuity problems for Backward Stochastic Differential Equations with singular terminal condition.

Laurent Denis Le Mans University France Co-Author(s):

Abstract: We study the continuity problem at the terminal time of the minimal supersolution of backward stochastic differential equation with singular terminal condition. In the continuous case, where the equation is driven by a Brownian motion we give a general criterion on the generator ensuring continuity up to terminal time. We also prove that the solution has a Malliavin derivative, which is the limit of the derivative of the approximating sequence and provide the asymptotic behavior of this derivative close to the terminal time. We apply this result to the regularity of the related partial differential equationby using the associated integro-partial differential equation. Finally, we prove that if there are jumps (i.e. the operator of the PDE is non local), we observe a propagation of the singularity, contrary to the continuous case (local operator). This talk is based on several joint works with D. Cacitti-Holland and A. Popier.

Cramer-Lundberg Models with Mean-Field Claims and Premia

Dan Goreac Laval University Canada Co-Author(s):    Hezhen Bao, Dan Goreac, Juan Li, and Radu Mitric

Abstract: This talk introduces McKean--Vlasov stochastic differential equations to model insurance reserves in a large, interacting pool of insurers. Our framework departs from the classical Cram\`er--Lundberg model by letting premiums and claim distributions adapt to the evolving cross-sectional distribution of capital, thereby capturing feedback and contagion effects. Using characteristic-function based integral and recursive formulas together with iterative moment calculations, we track distributional dynamics over time and design adaptive premium and systemic risk measures. These methods yield distribution-sensitive adjustment coefficients and supermartingale bounds for ruin probabilities, updating classical ruin theory for modern interconnected insurance markets.

A global stochastic maximum principle for mean-field forward-backward stochastic control systems with quadratic generators

Juan Li Shandong University Peoples Rep of China Co-Author(s):    Rainer Buckdahn, Juan Li, Yanwei Li, Yi Wang

Abstract: Our talk is devoted to the study of Peng`s stochastic maximum principle (SMP) for a stochastic control problem composed of a controlled forward stochastic differential equation (SDE) as dynamics and a controlled backward SDE which defines the cost functional. Our studies combine the difficulties which come, on one hand, from the fact that the coefficients of both the SDE and the backward SDE are of mean-field type (i.e., they do not only depend on the control process and the solution processes but also on their law), and on the other hand, from the fact that the coefficient of the BSDE is of quadratic growth in $Z$. Our SMP is novel, it extends in a by far non trivial way existing results on SMP. The talk is based on a joint work with Rainer Buckdahn (UBO, France), Yanwei Li (SDU, China), Yi Wang (SDU, China).

Particle Approximation for Conditional Mean Field SDEs

Yunzhang Li Fudan University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we construct a type of interacting particle systems to approximate a class of stochastic different equations whose coefficients depend on the conditional probability distributions of the processes given partial observations. After proving the well-posedness and regularity of the particle systems, we establish a quantitative convergence result for the empirical measures of the particle systems in the Wasserstein space, as the number of particles increases. Moreover, we discuss an Euler-Maruyama scheme of the particle system and validate its strong convergence. A numerical experiment is conducted to illustrate our results. This talk is based on the joint work with Kai Du and Yuyang Ye.

A Novel Approach to Pengs Maximum Principle for Mean Field SDE

Johan Benedikt Spille Technical University Berlin Germany Co-Author(s):    Johan Benedikt Spille, Wilhelm Stannat

Abstract: We present a novel approach to the proof of Peng`s maximum principle for McKean-Vlasov stochastic differential equations (SDE). The main step is the introduction of a third adjoint equation, a conditional McKean-Vlasov backward SDE, to accommodate the dualization of quadratic terms containing two independent copies of the first-order variational process. This is an intrinsic extension of the maximum principle from Peng for standard SDE and gives a conceptually consistent proof. Our approach will be useful in further extensions to the common noise setting and the infinite dimensional setting.

BSDEs and BSVIEs with anticipating generators

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

Abstract: For a backward stochastic differential equation (BSDE, for short), when the generator is not progressively measurable, it might not admit adapted solutions, shown by an example. However, for backward stochastic Volterra integral equations (BSVIEs, for short), the generators are allowed to be anticipating. This gives, among other things, an essential difference between BSDEs and BSVIEs. Under some proper conditions, the well-posedness of such BSVIEs is established. Further, the results are extended to path-dependent BSVIEs, in which the generators can depend on the future paths of unknown processes. An additional finding is that for path-dependent BSVIEs, in general, the situation of anticipating generators is not avoidable, and the adaptedness condition similar to that imposed for anticipated BSDEs by Peng and Yang [Anticipated backward stochastic differential equations, Ann. Probab., 37 (2009), pp. 877--902] is not necessary. Joint work with Jiongmin Yong and Chao Zhou.

OPTIMAL CONTROL OF SDES WITH MERELY MEASURABLE DRIFT: AN HJB APPROACH

Qingmeng Wei Northeast Normal University Peoples Rep of China Co-Author(s):

Abstract: The talk is about an optimal control problem for a diffusion whose drift and running cost are merely measurable in the state variable. Such low regularity rules out the use of Pontryagin's maximum principle and also invalidates the standard proof of the Bellman principle of optimality. We address these difficulties by analyzing the associated Hamilton--Jacobi--Bellman (HJB) equation. Using PDE techniques together with a policy iteration scheme, we prove that the HJB equation admits a unique strong solution, and this solution coincides with the value function of the control problem. Based on this identification, we establish a verification theorem and recover the Bellman optimality principle without imposing any additional smoothness assumptions. We further investigate a mollification scheme depending on a parameter $\\varepsilon > 0$. It turns out that the smoothed value functions $V_{\\varepsilon}$ may fail to converge to the original value function $V$ as $\\varepsilon \\to 0$, and we provide an explicit counterexample. To resolve this, we identify a structural condition on the control set. When the control set is countable, convergence $V_{\\varepsilon} \\to V$ holds locally uniformly.

Mean-field BSDEs with quadratic growth and related control problems

Jiaqiang Wen Southern University of Science and Technology Peoples Rep of China Co-Author(s):

Abstract: In this talk, I will present some of our recent works on general mean-field backward stochastic differential equations (BSDEs) with quadratic growth, including results on existence, uniqueness, comparison theorems, particle systems, and their relation with PDEs. Then, based on these theoretical results, I will discuss our recent work on stochastic optimal control problems and utility maximization problems.

Comparison theorems for mean-field BSDEs whose generators depend on the law of the solution (Y,Z)

Chuanzhi Xing Shandong University Peoples Rep of China Co-Author(s):    Juan Li, Zhanxin Li, Chuanzhi Xing

Abstract: For general mean-field backward stochastic differential equations (BSDEs) it is well-known that we usually do not have the comparison theorem if the coefficients depend on the law of $Z$-component of the solution process $(Y,Z)$. A natural question is whether general mean-field BSDEs whose coefficients depend on the law of $Z$ have the comparison theorem for some cases. In this talk we establish the comparison theorems for one-dimensional mean-field BSDEs whose coefficients also depend on the joint law of the solution process $(Y,Z)$. With the help of Malliavin calculus and a BMO martingale argument, we obtain two comparison theorems for different cases and a strong comparison result. In particular, in this framework, we compare not only the first component $Y$ of the solution $(Y,Z)$ for such mean-field BSDEs, but also the second component $Z$. After a discussion of mean-field BSDEs whose terminal condition and the driving coefficient are Malliavin differentiable, the results are extended in a second phase to the case without assumption of Malliavin differentiability. The talk is based on a joint work with Juan Li (SDU) and Zhanxin Li (SDU).

Maximum Principle for Partially Observed with Jump Observations and Controlled by $\alpha$-Stable L\`evy Processes

Jing Zhang Fudan University Peoples Rep of China Co-Author(s):

Abstract: We establish a stochastic maximum principle for partially observed optimal control problem, in which the state dynamic is driven by an $\alpha-$stable process ($1<\alpha<2$) and the observation process contains both Brownian and jump noises. By employing the separation principle, the original control problem is transformed into an infinite-dimensional setting, where the unnormalized conditional density of the state satisfies a fractional Zakai equation involving a fractional Laplacian and Poisson jumps. Under suitable assumptions, the well-posedness of the weak solution to the associated fractional forward-backward stochastic partial differential equation (FBSPDE) is proved within a Gelfand triple framework. The main result provides a necessary condition for optimality in the form of a Hamiltonian inequality, thereby extending earlier work to allow the control to enter the diffusion coefficient and the observation to include jumps.

The Ergodic Linear-Quadratic Optimal Control Problems with Random Periodic Coefficients

Qi Zhang Fudan University Peoples Rep of China Co-Author(s):    Jiacheng Wu, Qi Zhang

Abstract: In this talk, I introduce our recent work on the ergodic linear-quadratic closed-loop optimal control problems with random periodic coefficients. We put forward the random periodic mean-square exponentially stable condition, and prove the random periodicity of solutions to state equation based on it. Then we prove the existence and uniqueness of random periodic solutions to two types of backward stochastic differential equations which serve as stochastic Riccati equations in the procedure of completing the square. With the random periodicity of state equation and stochastic Riccati equations, the ergodic cost functional on infinite horizon is simplified to an equivalent cost functional over a single periodic interval without limit. Finally, the closed-loop optimal controls are explicitly given based on random periodic solutions to state equation and stochastic Riccati equations.