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This paper investigates the nonparametric estimation of the functional coefficients of the FBSDEs with random terminal time, including the local constant and local linear estimators. We provide complete two-dimensional asymptotics in both the time span and the sampling interval, allowing for the precise characterization of their distribution. Moreover, the empirical likelihood (EL) method to construct the data-driven confidence intervals for these estimators is provided. Some numerical simulations investigate the finite-sample properties of the estimators and compare the performance of the EL method and the conventional method in constructing confidence intervals based on asymptotic normality.

This paper is devoted to proposing a new asymmetric risk-sensitive criterion involving different risk attitudes toward varying risk sources. The criterion can only be defined through the initial value of the minimal solutions of quadratic backward stochastic differential equations (BSDEs). Before uncovering the mean-variance representation for the introduced criterion by the variational approach, some axioms are given for the first time to characterize a variance decomposition of square integrable random variables. The stochastic control problems under this criterion are described as a kind of stochastic recursive control problems that includes controlled quadratic BSDEs. An asymmetric risk-sensitive global stochastic maximum principle is derived when the quadratic BSDEs are equipped with bounded data. A closed-form solution of a stochastic linear-quadratic risk-sensitive control problem is obtained by introducing a novel completion-of-squares technique for controlled quadratic BSDEs. In addition, a dynamic portfolio optimization problem featuring a stochastic return rate is provided as an application of the asymmetric risk-sensitive control., Comment: 29 pages; Mingshang Hu, Shaolin Ji, Rundong Xu, and Xiaole Xue contributed equally to this work

This paper focuses on finding approximate solutions to the stochastic optimal control problems where the state trajectory is subject to controlled stochastic differential equations permitting controls in the diffusion coefficients. An algorithm based on the method of successive approximations is described for finding a set of small measure, in which the control is varied finitely so as to reduce the value of the functional and, as the control domains are not necessarily convex, the second-order adjoint processes are introduced in each minimization step of the Hamiltonian. Under certain convexity conditions, we prove that the values of the cost functional descend to the global minimum as the number of iterations tends to infinity. In particular, a convergence rate for a class of linear-quadratic systems is available.

In this paper, we study a discrete-time stochastic optimal control problem under distribution uncertainty with convex control domain. By weak convergence method and Sion's minimax theorem, we obtain the variational inequality for cost functional under a reference probability $P^{\ast}$. Moreover, under the square integrability condition for noise and control, we establish the discrete-time stochastic maximum principle under $P^{\ast}$. Finally, we introduce a backward algorithm to calculate the reference probability $P^{\ast}$ and the optimal control $u^{\ast}$., Comment: 20 pages

In this paper, we obtain the existence and uniqueness theorem for backward stochastic differential equation driven by G-Brownian motion (G-BSDE) under degenerate case. Moreover, we propose a new probabilistic method based on the representation theorem of G-expectation and weak convergence to obtain the regularity of fully nonlinear PDE associated to G-BSDE., Comment: 34 pages

In this paper,we mainly focus on the numerical solution of high-dimensional stochastic optimal control problem driven by fully-coupled forward-backward stochastic differential equations (FBSDEs in short) through deep learning. We first transform the problem into a stochastic Stackelberg differential game problem (leader-follower problem), then a bi-level optimization method is developed where the leader's cost functional and the follower's cost functional are optimized alternatively via deep neural networks. As for the numerical results, we compute two examples of the investment-consumption problem solved through stochastic recursive utility models, and the results of both examples demonstrate the effectiveness of our proposed algorithm.

In this paper, we mainly focus on solving high-dimensional stochastic Hamiltonian systems with boundary condition, which is essentially a Forward Backward Stochastic Differential Equation (FBSDE in short), and propose a novel method from the view of the stochastic control. In order to obtain the approximated solution of the Hamiltonian system, we first introduce a corresponding stochastic optimal control problem such that the extended Hamiltonian system of the control problem is exactly what we need to solve, then we develop two different algorithms suitable for different cases of the control problem and approximate the stochastic control via deep neural networks. From the numerical results, comparing with the Deep FBSDE method developed previously from the view of solving FBSDEs, the novel algorithms converge faster, which means that they require fewer training steps, and demonstrate more stable convergences for different Hamiltonian systems.

In this paper, we study a stochastic recursive optimal control problem in which the value functional is defined by the solution of a backward stochastic differential equation (BSDE) under $\tilde{G}$-expectation. Under standard assumptions, we establish the comparison theorem for this kind of BSDE and give a novel and simple method to obtain the dynamic programming principle. Finally, we prove that the value function is the unique viscosity solution of a type of fully nonlinear HJB equation., Comment: 19 pages

Novel multi-step predictor-corrector numerical schemes have been derived for approximating decoupled forward-backward stochastic differential equations (FBSDEs). The stability and high order rate of convergence of the schemes are rigorously proved. We also present a sufficient and necessary condition for the stability of the schemes. Numerical experiments are given to illustrate the stability and convergence rates of the proposed methods.

Based on the stochastic maximum principle for the partially coupled forward-backward stochastic control system (FBSCS for short), a modified method of successive approximations (MSA for short) is established for stochastic recursive optimal control problems. The second-order adjoint processes are introduced in the augmented Hamiltonian minimization step since the control domain is not necessarily convex. Thanks to the theory of bounded mean oscillation martingales (BMO martingales for short), we give a delicate proof of the error estimate and then prove the convergence of the modified MSA algorithm. In a special case, we obtain a logarithmic convergence rate. When the control domain is convex and compact, a sufficient condition which makes the control returned from the MSA algorithm be a near-optimal control is given for a class of linear FBSCSs., Comment: 30 pages

This paper examines the stochastic maximum principle (SMP) for a forward-backward stochastic control system where the backward state equation is characterized by the backward stochastic differential equation (BSDE) with quadratic growth and the forward state at the terminal time is constrained in a convex set with probability one. With the help of the theory of BSDEs with quadratic growth and the bounded mean oscillation (BMO) martingales, we employ the terminal perturbation approach and Ekeland's variational principle to obtain a dynamic stochastic maximum principle. The main result has a wide range of applications in mathematical finance and we investigate a robust recursive utility maximization problem with bankruptcy prohibition as an example., Comment: 24 pages

We study a stochastic optimal control problem for forward-backward control systems with quadratic generators. In order to establish the first and second-order variational and adjoint equations, we obtain a new estimate for one-dimensional linear BSDEs with unbounded stochastic Lipschitz coefficients involving bounded mean oscillation martingales (BMO-martingales for short) and prove the solvability for a class of multi-dimensional BSDEs with this type. Finally, a new global stochastic maximum principle is deduced., Comment: 31 pages

In this paper, we aim to solve the high dimensional stochastic optimal control problem from the view of the stochastic maximum principle via deep learning. By introducing the extended Hamiltonian system which is essentially an FBSDE with a maximum condition, we reformulate the original control problem as a new one. Three algorithms are proposed to solve the new control problem. Numerical results for different examples demonstrate the effectiveness of our proposed algorithms, especially in high dimensional cases. And an important application of this method is to calculate the sub-linear expectations, which correspond to a kind of fully nonlinear PDEs.

In this paper, we study a generalized Kalman-Bucy filtering problem under uncertainty. The drift uncertainty for both signal process and observation process is considered and the attitude to uncertainty is characterized by a convex operator (convex risk measure). The optimal filter or the minimum mean square estimator (MMSE) is calculated by solving the minimum mean square estimation problem under a convex operator. In the first part of this paper, this estimation problem is studied under g-expectation which is a special convex operator. For this case, we prove that there exists a worst-case prior. Based on this worst-case prior we obtained the Kalman-Bucy filtering equation under g-expectation. In the second part of this paper, we study the minimum mean square estimation problem under general convex operators. The existence and uniqueness results of the MMSE are deduced., Comment: 24 pages

We study the Neyman-Pearson problem for convex expectations on L^{\infty}(\mu). The existence of the optimal test is given. Without assuming that the level sets of penalty functions are weakly compact, we prove that the optimal tests for convex expectations on L^{\infty}(\mu) are just the classical Neyman-Pearson tests between a fixed representative pair of simple hypotheses. Then we show that the Neyman-Pearson problem for convex expectations on L^{1}(\mu) can be solved similarly.

Recently, the deep learning method has been used for solving forward-backward stochastic differential equations (FBSDEs) and parabolic partial differential equations (PDEs). It has good accuracy and performance for high-dimensional problems. In this paper, we mainly solve fully coupled FBSDEs through deep learning and provide three algorithms. Several numerical results show remarkable performance especially for high-dimensional cases., Comment: 24 pages, 7 figures

In this paper, we consider the solvability problems for the fully coupled forward-backward stochastic difference equations (FBS{\Delta}Es) on spaces related to discrete time, finite state processes. On one hand, we provide the necessary and sufficient condition for the solvability of the linear FBS{\Delta}Es. On the other hand, under the assumption that the coefficients satisfy the monotone condition, we investigate the existence and uniqueness theorems for the general nonlinear FBS{\Delta}Es., Comment: 21 Pages

This paper is concerned with a generalized Kalman-Bucy filtering model and corresponding robust problem under model uncertainty. We find that this robust problem is equivalent to considering an estimate problem under some sublinear operator. Therefore, we turn to obtaining the minimum mean square estimator under a sublinear operator. By Girsanov theorem and minimax theorem, we obtain the optimal estimator $\hat{x}_{t}$ of the signal process $x_{t}$ for given time $t\in\lbrack0,T]$., Comment: 9 pages. arXiv admin note: substantial text overlap with arXiv:1905.01791

A generalized Kalman-Bucy model under model uncertainty and a corresponding robust problem are studied in this paper. We find that this robust problem is equivalent to an estimate problem under a sublinear operator. By Girsanov transformation and the minimax theorem, we prove that this problem can be reformulated as a classical Kalman-Bucy filtering problem under a new probability measure. The equation which governs the optimal estimator is obtained. Moreover, the optimal estimator can be decomposed into the classical optimal estimator and a term related to model uncertainty., Comment: 10 pages

In this paper, we study the minimum mean square estimator for non-bounded random variables under sublinear operators. The existence and uniqueness of the minimum mean square estimator are obtained. Several properties of the minimum mean square estimator for non-bounded random variables are proved under some mild assumptions., Comment: 13. arXiv admin note: text overlap with arXiv:1412.5736 by other authors

This paper is concerned with optimal control of stochastic fully coupled forward-backward linear quadratic (FBLQ) problems with indefinite control weight costs. In order to obtain the state feedback representation of the optimal control, we propose a new decoupling technique and obtain one kind of non-Riccati-type ordinary differential equations (ODEs). By applying the completion-of-squares method, we prove the existence of the solutions for the obtained ODEs under some assumptions and derive the state feedback form of the optimal control. For this FBLQ problem, the optimal control depends on the entire trajectory of the state process. Some sepcial cases are given to illustrate our results., Comment: 44

In this paper, we study the solvability problem for one kind of fully coupled forward-backward stochastic difference equations (FBS{\Delta}Es). With the help of the necessary and sufficient condition for the solvability of the linear FBS{\Delta}Es, under the monotone assumption, we obtain the existence and uniqueness theorem for the general nonlinear ones., Comment: 19 pages. arXiv admin note: text overlap with arXiv:1812.11283

In this paper, we study the maximum principle for stochastic optimal control problems of forward-backward stochastic difference systems (FBS{\Delta}Ss). Two types of FBS{\Delta}Ss are investigated. The first one is described by a partially coupled forward-backward stochastic difference equation (FBS{\Delta}E) and the second one is described by a fully coupled FBS{\Delta}E. By adopting an appropriate representation of the product rule and an appropriate formulation of the backward stochastic difference equation (BS{\Delta}E), we deduce the adjoint difference equation. Finally, the maximum principle for this optimal control problem with the control domain being convex is established., Comment: 24 pages

Hu et. al 2018 studied a stochastic optimal control problem for fully coupled forward-backward stochastic control systems with a nonempty control domain. By assuming a weakly coupled condition, they established an approach to obtain the first-order, second-order variational equations and the adjoint equations for the states X, Y and Z and deduced the global maximum principle. But it is well known that there are several different conditions such as monotonicity condition, weakly coupled condition and other conditions which can guarantee the existence and uniqueness of the solution to fully coupled FBSDEs. In this note, to overcome the limitations of assuming a specific condition, we propose two kinds of assumptions which can guarantee that the approach developed in Hu et. al 2018 is still applicable. Under these two kinds of assumptions, we obtain the global stochastic maximum principle., Comment: arXiv admin note: substantial text overlap with arXiv:1803.02109

In this paper, we study the existence and uniqueness of viscosity solutions to a kind of Hamilton-Jacobi-Bellman (HJB) equations combined with algebra equations. This HJB equation is related to a stochastic optimal control problem for which the state equation is described by a fully coupled forward-backward stochastic differential equation. By extending Peng's backward semigroup approach to this problem, we obtain the dynamic programming principle and show that the value function is a viscosity solution to this HJB equation. As for the proof of the uniqueness of viscosity solution, the analysis method in Barles, Buckdahn and Pardoux Baeles-BP usually does not work for this fully coupled case. With the help of the uniqueness of the solution to FBSDEs, we propose a novel probabilistic approach to study the uniqueness of the solution to this HJB equation. We obtain that the value function is the minimum viscosity solution to this HJB equation. Especially, when the coefficients are independent of the control variable or the solution is smooth, the value function is the unique viscosity solution.

Within the framework of viscosity solution, we study the relationship between the maximum principle (MP) in [9] and the dynamic programming principle (DPP) in [10] for a fully coupled forward-backward stochastic controlled system (FBSCS) with a nonconvex control domain. For a fully coupled FBSCS, both the corresponding MP and the corresponding Hamilton-Jacobi-Bellman (HJB) equation combine an algebra equation respectively. So this relationship becomes more complicated and almost no work involves this issue. With the help of a new decoupling technique, we obtain the desirable estimates for the fully coupled forward-backward variational equations and establish the relationship. Furthermore, for the smooth case, we discover the connection between the derivatives of the solution to the algebra equation and some terms in the first and second-order adjoint equations. Finally, we study the local case under the monotonicity conditions as in [14,27] and obtain the relationship between the MP in [27] and the DPP in [14].

We study a stochastic optimal control problem for fully coupled forward-backward stochastic control systems with a nonempty control domain. For our problem, the first-order and second-order variational equations are fully coupled linear FBSDEs. Inspired by Hu (Hu, Probability, Uncertainty and Quantitative Risk, 2(1) (2017):pp 1-20), we develop a new decoupling approach by introducing an adjoint equation which is a quadratic BSDE. By revealing the relations among the terms of the first-order Taylor's expansions, we estimate the orders of them and derive a global stochastic maximum principle which includes a completely new term. Applications to stochastic linear quadratic control problems are investigated.

This paper considers the stochastic linear quadratic optimal control problem in which the control domain is nonconvex. By the functional analysis and convex perturbation methods, we establish a novel maximum principle. The application of the proposed maximum principle is illustrated through a work-out example., Comment: 15 pages

We model learning in a continuous-time Brownian setting where there is prior ambiguity. The associated model of preference values robustness and is time-consistent. It is applied to study optimal learning when the choice between actions can be postponed, at a per-unit-time cost, in order to observe a signal that provides information about an unknown parameter. The corresponding optimal stopping problem is solved in closed-form, with a focus on two specific settings: Ellsberg's two-urn thought experiment expanded to allow learning before the choice of bets, and a robust version of the classical problem of sequential testing of two simple hypotheses about the unknown drift of a Wiener process. In both cases, the link between robustness and the demand for learning is studied., Comment: 35 pages

This paper studies the continuous time mean-variance portfolio selection problem with one kind of non-linear wealth dynamics. To deal the expectation constraint, an auxiliary stochastic control problem is firstly solved by two new generalized stochastic Riccati equations from which a candidate portfolio in feedback form is constructed, and the corresponding wealth process will never cross the vertex of the parabola. In order to verify the optimality of the candidate portfolio, the convex duality (requires the monotonicity of the cost function) is established to give another more direct expression of the terminal wealth level. The variance-optimal martingale measure and the link between the non-linear financial market and the classical linear market are also provided. Finally, we obtain the efficient frontier in closed form. From our results, people are more likely to invest their money in riskless asset compared with the classical linear market.

This paper concerns the recursive utility maximization problem. We assume that the coefficients of the wealth equation and the recursive utility are concave. Then some interesting and important cases with nonlinear and nonsmooth coefficients satisfy our assumption. After given an equivalent backward formulation of our problem, we employ the Fenchel-Legendre transform and derive the corresponding variational formulation. By the convex duality method, the primal "sup-inf" problem is translated to a dual minimization problem and the saddle point of our problem is derived. Finally, we obtain the optimal terminal wealth. To illustrate our results, three cases for investors with ambiguity aversion are explicitly worked out under some special assumptions.

This paper concerns the continuous time mean-variance portfolio selection problem with a special nonlinear wealth equation. This nonlinear wealth equation has a nonsmooth coefficient and the dual method developed in [6] does not work. We invoke the HJB equation of this problem and give an explicit viscosity solution of the HJB equation. Furthermore, via this explicit viscosity solution, we obtain explicitly the efficient portfolio strategy and efficient frontier for this problem. Finally, we show that our nonlinear wealth equation can cover three important cases.

This paper concerns the recursive utility maximization problem under partial information. We first transform our problem under partial information into the one under full information. When the generator of the recursive utility is concave, we adopt the variational formulation of the recursive utility which leads to a stochastic game problem and a characterization of the saddle point of the game is obtained. Then, we study the K-ignorance case and explicit saddle points of several examples are obtained. At last, when the generator of the recursive utility is smooth, we employ the terminal perturbation method to characterize the optimal terminal wealth., Comment: 20 pages

In this paper, the Neyman-Pearson lemma for general sublinear expectations is studied. We weaken the assumptions for sublinear expectations in [1] and give a completely new method to study this problem. Applying Mazur-Orlicz Theorem and the decomposition theorem of finitely additive set functions, we prove that the optimal test still has the reminiscent form as in the classical Neyman-Pearson lemma. Finally, for the special sublinear expectation which can be represented by a family of probability measures, we give a sufficient condition for the existence of the optimal test and show the form of the optimal test selected in L_{c}^1-space which is introduced by Peng [10] in his nonlinear-expectation framework., Comment: 15 pages

We study a stochastic recursive optimal control problem in which the cost functional is described by the solution of a backward stochastic differential equation driven by G-Brownian motion. Some of the economic and financial optimization problems with volatility ambiguity can be formulated as such problems. Different from the classical variational approach, we establish the maximum principle by the linearization and weak convergence methods., Comment: 29 pages

We define fully coupled forward-backward stochastic differential equations on spaces related to continuous time, finite state Markov Chains. Existence and uniqueness results of the fully coupled forward-backward stochastic differential equations on Markov Chains are obtained.

In this paper, we study a stochastic recursive optimal control problem in which the cost functional is described by the solution of a backward stochastic differential equation driven by G-Brownian motion. Under standard assumptions, we establish the dynamic programming principle and the related fully nonlinear HJB equation in the framework of G-expectation. Finally, we show that the value function is the viscosity solution of the obtained HJB equation., Comment: 29 pages. arXiv admin note: text overlap with arXiv:1306.1312

In this paper, we study a functional fully coupled forward-backward stochastic differential equations (FBSDEs). Under a new type of integral Lipschitz and monotonicity conditions, the existence and uniqueness of solutions for functional fully coupled FBSDEs is proved. We also investigate the relationship between the solution of functional fully coupled FBSDE and the classical solution of the path-dependent partial differential equation (P-PDE). When the solution of the P-PDE has some smooth and regular properties, we solve the related functional fully coupled FBSDE and prove the P-PDE has a unique solution.

In this paper, we study a stochastic recursive optimal control problem in which the objective functional is described by the solution of a backward stochastic differential equation driven by G-Brownian motion. Under standard assumptions, we establish the dynamic programming principle and the related Hamilton-Jacobi-Bellman (HJB) equation in the framework of G-expectation. Finally, we show that the value function is the viscosity solution of the obtained HJB equation.

In this paper, we study reflected backward stochastic difference equations (RBSDEs for short) with finitely many states in discrete time. The general existence and uniqueness result, as well as comparison theorems for the solutions, are established under mild assumptions. The connections between RBSDEs and optimal stopping problems are also given. Then we apply the obtained results to explore optimal stopping problems under $g$-expectation. Finally, we study the pricing of American contingent claims in our context., Comment: 29 pages

In this paper, we study the pricing of contingent claims under G-expectation. In order to accomodate volatility uncertainty, the price of the risky security is supposed to governed by a general linear stochastic differential equation (SDE) driven by G-Brownian motion. Utilizing the recently developed results of Backward SDE driven by G-Brownian motion, we obtain the superhedging and suberhedging prices of a given contingent claim. Explicit results in the Markovian case are also derived., Comment: 15 pages. arXiv admin note: substantial text overlap with arXiv:1212.5403, arXiv:1206.5889