Optimal control and the associated second-order Hamilton-Jacobi-Bellman (HJB) equation are studied for unbounded stochastic evolution systems in Hilbert spaces. A new notion of viscosity solution, featured by absence of B-continuity, is introduced for the second-order HJB equation in the sense of Crandall and Lions, and is shown to coincide with the classical solutions and to satisfy a stability property. The value functional is proved to be the unique continuous viscosity solution to the second-order HJB equation, with the coefficients being not necessarily B-continuous. Our result provides a new theory of viscosity solutions to the HJB equation for optimal control of stochastic evolutionary equations-driven by a linear unbounded operator-in a Hilbert space, and removes the B-continuity assumption on the coefficients which is used in the existing literature.

Abstract. With Ekeland’s variational principle, we prove a general stochastic maximum principle (SMP) for square-integrable optimal control of linearly growing stochastic differential systems subject to a quadratically growing cost functional. The diffusion coefficient is allowed to depend on the control variable, and the admissible control range is allowed to be nonconvex. We relax the existing assumptions of finite moments of arbitrary order on optimal control (see [S. G. Peng, SIAM J. Control Optim., 28 (1990), pp. 966–979]), and of control-bounded coefficients (see [J. M. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Appl. Math. (N. Y.) 43, Springer, 1999, pp. 114 and 118]), so that the typical linear quadratic optimal stochastic control problem is included to satisfy all the assumptions of our SMP.

ABSTRACT We present a comprehensive theory on the existence and uniqueness of adapted solutions to a one‐dimensional nonlinear backward stochastic differential equation (1D BSDE for short), and assume that the generator has a unilateral linear or super‐linear growth in the first unknown variable , and has an at most quadratic growth in the second unknown variable . We develop a unified methodology, featured by the test function method and the a priori estimate technique, to establish several existence theorems and comparison theorems, which immediately yield corresponding existence and uniqueness results. We also overview relevant known results and give some practical applications of our theoretical results. Finally, we list some open problems on the well‐posedness of 1D BSDEs.

A Backward Stochastic Differential Equation (BSDE) with a Peano-type generator, is known to have infinitely many solutions when the terminal value is vanishing, and is shown to have possibly multiple solutions even when the terminal value is not vanishing but nonnegative. In this paper, we study the uniqueness of adapted solutions of such a BSDE when the terminal value is almost surely positive. Two methods are developed. The first one is to connect the BSDE to an optimal stochastic control problem: under suitable integrability of the terminal values, with a verification argument, we prove that the first component of the adapted solution pair is the value process for the optimal stochastic control problem. The second one appeals to a change of variables, and is more inclined to analysis: by a change of variables, the original BSDE is reduced to a convex quadratic BSDE, and then using the θ-difference method, we give a sharp result in some special case, which includes the BSDE governing the well-known Kreps-Porteus utility.

n this article, from the viewpoint of the control theory, we discuss the relationshipsamong the commonly used monotonicity conditions that ensure the well-posedness of the solutionsarising from problems of mean field games (MFGs) and mean field type control (MFTC). Wefirst introduce the well-posedness of general forward-backward stochastic differential equations(FBSDEs) defined on some suitably chosen Hilbert spaces under the beta-monotonicity. We thenpropose a monotonicity condition for the MFG, namely, partitioning the running cost functionalinto two parts, so that both parts still depend on the control and the state distribution, yet onesatisfies a strong convexity and a small mean field effect condition, while the other has a newlyintroduced displacement quasi-monotonicity. To the best of our knowledge, the latter quasi typecondition has not yet been discussed in the contemporary literature, and it can be considered as a bitmore general monotonicity condition than those commonly used. Besides, for the MFG, we showthat convexity and small mean field effect condition for the first part of running cost function a land the quasi-monotonicity condition for the second part together imply the beta-monotonicity andthus the well-posedness for the associated FBSDEs. For the MFTC problem, we show that the beta-monotonicity for the corresponding FBSDEs is simply the convexity assumption on the costfunctional. Finally, we consider a more general setting where the drift functional is allowed to benon-linear for both MFG and MFTC problems.

In this paper, we consider a mean field game (MFG) with a major and N minor agents. We first consider the limiting problem and allow the coefficients to vary with the conditional distribution in a nonlinear way. We use the stochastic maximum principle to transform the limiting control problem into a system of two coupled conditional distribution dependent forward-backward stochastic differential equations (FBSDEs), and prove the existence and uniqueness result of the FBSDEs when the dependence between major agent and minor agents is sufficiently weak. We then use the solution of the limiting problem to construct an $ \mathcal {O}(N<^>{-\frac {1}{2}}) $ O(N-12)-Nash equilibrium for the MFG with a major and N minor agents.

This paper is devoted to a general solvability of multi-dimensional non-Markovian backward stochastic differential equations (BSDEs) with interactively quadratic generators. General structures of the generator g are posed for both local and global existence and uniqueness results on a BSDE, which admit a general growth of the generator g(omega, t, y, z) in the state variable y, and a quadratic growth of the ith components gi(omega, t, y, z) both in the jth row zj of the state variable z for j =6 i (which we call the "interactively quadratic" growth) and in the ith row zi of z. We first establish an existence and uniqueness result on local bounded solutions and then several existence and uniqueness results on global bounded and unbounded solutions. They improve several existing works in the non-Markovian setting, and also incorporate some interesting examples, which include global bounded solution of backward stochastic Burger differential systems and a partial answer to the problem posed in Jackson [25] (see Example 2.16(ii) and Corollary 2.25, respectively). A comprehensive study on the bounded solution of scalar-valued quadratic BSDEs with unbounded stochastic parameters is carried out for deriving our main results.

We study the well-posedness of a system of forward-backward stochastic differential equations (FBSDEs) corresponding to a degenerate mean field type control problem, when the diffusion coefficient depends on the state together with its measure and also the control. Degenerate mean field type control problems are rarely studied in the literature. Our method is based on a lifting approach which embeds the control problem and the associated FBSDEs in Wasserstein spaces into certain Hilbert spaces. We use a continuation method to establish the solvability of the FBSDEs and that of the G\^ateaux derivatives of this FBSDEs. We then explore the regularity of the value function in time and in measure argument, and we also show that it is the unique classical solution of the associated Bellman equation. We also study the higher regularity of the linear functional derivative of the value function, by then, we obtain the classical solution of the mean field type master equation.

In this paper, we study the Cauchy problem for backward stochastic partial differential equations (BSPDEs) involving the fractional Laplacian operator. First, by employing the martingale representation theorem and the fractional heat kernel, we construct an explicit form of the solution for fractional BSPDEs with space-invariant coefficients, thereby demonstrating the existence and uniqueness of the strong solution. Then utilizing the freezing coefficients method as well as the continuation method, we establish Ho"\lder estimates and well-posedness for general fractional BSPDEs with the coefficients depending on space-time variables. As an application, we use the fractional adjoint BSPDEs to investigate stochastic optimal control of the partially observed systems driven by \alpha -stable Le'\vy processes.

This paper aims to study the relationship between the maximum principle and the dynamic programming principle for recursive optimal control problem of stochastic evolution equations, where the control domain is not necessarily convex and the value function may be nonsmooth. By making use of the notion of conditionally expected operator-valued backward stochastic integral equations, we establish a connection between the first and second-order adjoint processes in MP and the general derivatives of the value function. Under certain additional assumptions, the value function is shown to be $C^{1,1}$-regular. Furthermore, we discuss the smooth case and present several applications of our results.

In this paper, we study general mean-field backward stochastic differential equations (BSDEs, for short) with quadratic growth. First, using some new ideas, we prove the existence and uniqueness of local and global solutions for a one-dimensional mean-field BSDE when the generator g(t,Y, Z,PY, PZ) has quadratic growth in Z and the terminal value is bounded. Second, we derive a comparison theorem for general mean-field BSDEs by applying the Girsanov transform. Third, within this framework, we use the mean-field BSDE to provide a probabilistic representation of the viscosity solution for a nonlocal partial differential equation (PDE, for short) as an extended nonlinear Feynman-Kac formula, which yields the existence and uniqueness of the solution to the PDE. Finally, we prove the convergence of the particle systems to general mean-field BSDEs with quadratic growth and give the corresponding convergence rate.

This paper is devoted to a new construction of the two-dimensional sine-Gordon model on bounded domains by a novel normalization technique in the finite ultraviolet regime. Our methodology involves a family of backward stochastic differential equations (BSDEs for short) driven by a cylindrical Wiener process, whose generators are purely quadratic functions of the second unknown variable. The terminal conditions of the quadratic BSDEs are uniformly bounded and converge in probability to the real part of imaginary multiplicative chaos tested against an arbitrarily given test function, which helps us describe our sine-Gordon measure through some delicate estimates concerning bounded mean oscillation martingales. As the ultraviolet cutoffs are vanishing, the quadratic BSDEs converge to a quadratic BSDE that completely characterizes the absolute continuity of our sine-Gordon measure with respect to the law of Gaussian free fields. Our approach can also be used effectively to establish the connection between our sine-Gordon measure and the scaling limit of correlation functions of the critical planar XOR-Ising model and to prove the weak convergence of the normalized charge distributions of two-dimensional log-gases.

In this article, we study the global-in-time well-posedness of second order mean field games (MFGs) with both nonlinear drift functions simultaneously depending on the state, distribution and control variables, and the diffusion term depending on both state and distribution. Besides, the diffusion term is allowed to be degenerate, unbounded and even nonlinear in the distribution, but it does not depend on the control. First, we establish the global well-posedness of the corresponding forward-backward stochastic differential equations (FBSDEs), which arise from the maximum principle under a so-called β-monotonicity commonly used in the optimal control theory. The β-monotonicity admits more interesting cases, as representative examples including but not limited to the displacement monotonicity, the small mean field effect condition or the Lasry-Lions monotonicity; and ensures the well-posedness result in diverse non-convex examples. In our settings, we pose assumptions directly on the drift and diffusion coefficients and the cost functionals, rather than indirectly on the Hamiltonian, to make the conditions more visible. Our probabilistic method tackles the nonlinear dynamics with a linear but infinite dimensional version, and together with our recently proposed cone property for the adjoint processes, following in an almost straightforward way the conventional approach to the classical stochastic control problem, we derive a sufficiently good regularity of the value functional, and finally show that it is the unique classical solution to the MFG master equation. Our results require fairly few conditions on the functional coefficients for solution of the MFG, and a bit more conditions – which are least stringent in the contemporary literature – for classical solution of the MFG master equation.

In this article, we apply a probabilistic approach to study general mean field type control (MFTC) problems with jump-diffusions, and give the first global-in-time solution. We allow the drift coefficient b and the diffusion coefficient σ to nonlinearly depend on the state, distribution and control variables, and both can be unbounded and possibly degenerate; besides, the jump coefficient γ is allowed to be non-constant. To tackle the non-linear and control-dependent diffusion σ, we further formulate a joint cone property and estimates for both processes P and Q of the corresponding adjoint process (where (P,Q,R) is the solution triple of the associated adjoint process as a backward stochastic differential equation with jump), in contrast to our previous single cone property of the only process P. We study first the system of forward-backward stochastic differential equations (FBSDEs) with jumps arising from the maximum principle, and then the related Jacobian flows, which altogether yield the classical regularity of the value function and thus allow us to show that the value function is the unique classical solution of the HJB integro-partial differential equation. Most importantly, our proposed probabilistic approach can apparently handle the MFTC problem driven by a fairly general process far beyond Brownian motion, in a relatively easier manner than the existing analytic approach.

In this paper, we study a multidimensional backward stochastic differential equation (BSDE) with an additional rough drift (rough BSDE), and give the existence and uniqueness of the adapted solution, either when the terminal value and the geometric rough path are small, or when each component of the rough drift only depends on the corresponding component of the first unknown variable (but we drop the one-dimensional assumption of Diehl and Friz [Ann. Probab. 40 (2012), pp. 1715-1758]). We also introduce a new notion of the p-rough stochastic integral for p is an element of [2 , 3), and then succeed in giving-through a fixed-point argument-a general existence and uniqueness result on a multidimensional rough BSDE with a general square-integrable terminal value, allowing the rough drift to be random and time-varying but having to be linear; furthermore, we connect it to a system of rough partial differential equations.

In this article, a notion of viscosity solutions is introduced for fully nonlinear second order path-dependent partial differential equations in the spirit of [Zhou, Ann. Appl. Probab., 33 (2023), 5564-5612]. We prove the existence, comparison principle, consistency and stability for the viscosity solutions. Application to path-dependent stochastic differential games is given.

We prove the existence and uniqueness of non-negative entropy solutions of the obstacle problem for stochastic porous media equations. The core of the method is to combine the entropy formulation with the penalization method.

This paper is devoted to the solvability of Markovian quadratic backward stochastic differential equations（BSDEs for short） with bounded terminal conditions. The generator is allowed to have an unbounded sub-quadratic growth in the second unknown variable z. The existence and uniqueness results are given to these BSDEs. As an application, an existence result is given to a system of coupled forward-backward stochastic differential equations with measurable coefficients.

We consider a class of stochastic optimal control problems with partial observation, and study their approximation by discrete-time control problems. We establish a convergence result by using weak convergence technique of Kushner and Dupuis [Numerical Methods for Stochastic Control Problems in Continuous Time (2001), Springer-Verlag, New York], together with the notion of relaxed control rule introduced by El Karoui, Huu Nguyen and Jeanblanc-Picqu\'e [SIAM J. Control Optim., 26 (1988) 1025-1061]. In particular, with a well chosen discrete-time control system, we obtain a first implementable numerical algorithm (with convergence) for the partially observed control problem. Moreover, our discrete-time approximation result would open the door to study convergence of more general numerical approximation methods, such as machine learning based methods. Finally, we illustrate our convergence result by the numerical experiments on a partially observed control problem in a linear quadratic setting.