Forward–backward SDEs with distributional coefficients

Forward–backward stochastic differential equations (FBSDEs) have attracted significant attention since they were introduced, due to their wide range of applications, from solving non-linear PDEs to pricing American-type options. Here, we consider two new classes of multidimensional FBSDEs with distributional coefficients (elements of a Sobolev space with negative order). We introduce a suitable notion of solution and show its existence and in certain cases its uniqueness. Moreover we establish a link with PDE theory via a non-linear Feynman–Kac formula. The associated semi-linear parabolic PDE is the same for both FBSDEs, also involves distributional coefficients and has not previously been investigated.     

Forward–Backward–Half Forward Dynamical Systems for Monotone Inclusion Problems with Application to v-GNE

Journal of Optimization Theory and Applications - In this paper, the first-order forward–backward–half forward dynamical systems associated with the inclusion problem consisting of...     

Forward–backward systems for expected utility maximization

In this paper we deal with the utility maximization problem with general utility functions including power utility with liability. We derive a new approach in which we reduce the resulting control problem to the study of a system of fully-coupled Forward–Backward Stochastic Differential Equations (FBSDEs) that promise to be accessible to numerical treatment.     

Lagrangian Penalization Scheme with Parallel Forward–Backward Splitting

We propose a new iterative algorithm for the numerical approximation of the solutions to convex optimization problems and constrained variational inequalities, especially when the functions and operators involved have a separable structure on a product space, and exhibit some dissymmetry in terms of their component-wise regularity. Our method combines Lagrangian techniques and a penalization scheme with bounded parameters, with parallel forward–backward iterations. Conveniently combined, these techniques allow us to take advantage of the particular structure of the problem. We prove the weak convergence of the sequence generated by this scheme, along with worst-case convergence rates in the convex optimization setting, and for the strongly non-degenerate monotone operator case. Implementation issues related to the penalization of the constraint set are discussed, as well as applications in image recovery and non-Newtonian fluids modeling. A numerical illustration is also given, in order to prove the performance of the algorithm.     

Forward–backward quasi-Newton methods for nonsmooth optimization problems

The forward–backward splitting method (FBS) for minimizing a nonsmooth composite function can be interpreted as a (variable-metric) gradient method over a continuously differentiable function which we call forward–backward envelope (FBE). This allows to extend algorithms for smooth unconstrained optimization and apply them to nonsmooth (possibly constrained) problems. Since the FBE can be computed by simply evaluating forward–backward steps, the resulting methods rely on a similar black-box oracle as FBS. We propose an algorithmic scheme that enjoys the same global convergence properties of FBS when the problem is convex, or when the objective function possesses the Kurdyka–?ojasiewicz property at its critical points. Moreover, when using quasi-Newton directions the proposed method achieves superlinear convergence provided that usual second-order sufficiency conditions on the FBE hold at the limit point of the generated sequence. Such conditions translate into milder requirements on the original function involving generalized second-order differentiability. We show that BFGS fits our framework and that the limited-memory variant L-BFGS is well suited for large-scale problems, greatly outperforming FBS or its accelerated version in practice, as well as ADMM and other problem-specific solvers. The analysis of superlinear convergence is based on an extension of the Dennis and Moré theorem for the proposed algorithmic scheme.     

The Forward–Backward Algorithm and the Normal Problem

The forward–backward splitting technique is a popular method for solving monotone inclusions that have applications in optimization. In this paper, we explore the behaviour of the algorithm when the inclusion problem has no solution. We present a new formula to define the normal solutions using the forward–backward operator. We also provide a formula for the range of the displacement map of the forward–backward operator. Several examples illustrate our theory.     

Discrete-time approximation of decoupled Forward–Backward SDE with jumps

We study a discrete-time approximation for solutions of systems of decoupled Forward–Backward Stochastic Differential Equations (FBSDEs) with jumps. Assuming that the coefficients are Lipschitz-continuous, we prove the convergence of the scheme when the number of time steps n$n$ goes to infinity. The rate of convergence is at least n^−1/2+ε$n^{-1/2+\epsilon }$, for any ε>0$\epsilon >0$. When the jump coefficient of the first variation process of the forward component satisfies a non-degeneracy condition which ensures its inversibility, we achieve the optimal convergence rate n^−1/2$n^{-1/2}$. The proof is based on a generalization of a remarkable result on the path-regularity of the solution of the backward equation derived by Zhang [J. Zhang, A numerical scheme for BSDEs, Annals of Applied Probability 14 (1) (2004) 459–488] in the no-jump case.     

Four-Operator Splitting via a Forward–Backward–Half-Forward Algorithm with Line Search

In this article, we provide a splitting method for solving monotone inclusions in a real Hilbert space involving four operators: a maximally monotone, a monotone-Lipschitzian, a cocoercive, and a monotone-continuous operator. The proposed method takes advantage of the intrinsic properties of each operator, generalizing the forward–backward–half-forward splitting and the Tseng’s algorithm with line search. At each iteration, our algorithm defines the step size by using a line search in which the monotone-Lipschitzian and the cocoercive operators need only one activation. We also derive a method for solving nonlinearly constrained composite convex optimization problems in real Hilbert spaces. Finally, we implement our algorithm in a nonlinearly constrained least-square problem and we compare its performance with available methods in the literature.

     

On the Stability of a Forward–Backward Heat Equation

In this paper we examine spectral properties of a family of periodic singular Sturm?CLiouville problems which are highly non-self-adjoint but have purely real spectrum. The problem originated from the study of the lubrication approximation of a viscous fluid film in the inner surface of a rotating cylinder and has received a substantial amount of attention in recent years. Our main focus will be the determination of Schatten class inclusions for the resolvent operator and regularity properties of the associated evolution equation.     

Forward–Backward Penalty Scheme for Constrained Convex Minimization Without Inf-Compactness

In order to solve constrained minimization problems, Attouch et al. propose a forward–backward algorithm that involves an exterior penalization scheme in the forward step. They prove that every sequence generated by the algorithm converges weakly to a solution of the minimization problem if either the objective function or the penalization function corresponding to the feasible set is inf-compact. Unfortunately, this assumption leaves out problems that are not coercive, as well as several interesting applications in infinite-dimensional spaces. The purpose of this short article is to show this convergence result without the inf-compactness assumption.     

Forward–Backward Stochastic Differential Games and Stochastic Control under Model Uncertainty

We study optimal stochastic control problems with jumps under model uncertainty. We rewrite such problems as stochastic differential games of forward–backward stochastic differential equations. We prove general stochastic maximum principles for such games, both in the zero-sum case (finding conditions for saddle points) and for the nonzero sum games (finding conditions for Nash equilibria). We then apply these results to study robust optimal portfolio-consumption problems with penalty. We establish a connection between market viability under model uncertainty and equivalent martingale measures. In the case with entropic penalty, we prove a general reduction theorem, stating that a optimal portfolio-consumption problem under model uncertainty can be reduced to a classical portfolio-consumption problem under model certainty, with a change in the utility function, and we relate this to risk sensitive control. In particular, this result shows that model uncertainty increases the Arrow–Pratt risk aversion index.     

Fast Kalman Filtering and Forward–Backward Smoothing via a Low-Rank Perturbative Approach

Kalman filtering-smoothing is a fundamental tool in statistical time-series analysis. However, standard implementations of the Kalman filter-smoother require O(d³) time and O(d²) space per time step, where d is the dimension of the state variable, and are therefore impractical in high-dimensional problems. In this article we note that if a relatively small number of observations are available per time step, the Kalman equations may be approximated in terms of a low-rank perturbation of the prior state covariance matrix in the absence of any observations. In many cases this approximation may be computed and updated very efficiently (often in just O(k²d) or O(k²d + kdlog?d) time and space per time step, where k is the rank of the perturbation and in general k ? d), using fast methods from numerical linear algebra. We justify our approach and give bounds on the rank of the perturbation as a function of the desired accuracy. For the case of smoothing, we also quantify the error of our algorithm because of the low-rank approximation and show that it can be made arbitrarily low at the expense of a moderate computational cost. We describe applications involving smoothing of spatiotemporal neuroscience data. This article has online supplementary material.     

Forward–backward SDEs with jumps and classical solutions to nonlocal quasilinear parabolic PDEs

We obtain an existence and uniqueness theorem for fully coupled forward–backward SDEs (FBSDEs) with jumps via the classical solution to the associated quasilinear parabolic partial integro-differential equation (PIDE), and provide the explicit form of the FBSDE solution. Moreover, we embed the associated PIDE into a suitable class of non-local quasilinear parabolic PDEs which allows us to extend the methodology of Ladyzhenskaya et al. (1968) to non-local PDEs of this class. Namely, we obtain the existence and uniqueness of a classical solution to both the Cauchy problem and the initial–boundary value problem for non-local quasilinear parabolic second-order PDEs.     

Forward–backward stochastic differential equations with monotone functionals and mean field games with common noise

We consider a system of forward–backward stochastic differential equations (FBSDEs) with monotone functionals. We show that such a system is well-posed by the method of continuation similarly to Peng and Wu (1999) for classical FBSDEs. As applications, we prove the well-posedness result for a mean field FBSDE with conditional law and show the existence of a decoupling function. Lastly, we show that mean field games with common noise are uniquely solvable under a linear-convex setting and weak-monotone cost functions and prove that the optimal control is in a feedback form depending only on the current state and conditional law.     

Maximum Principles of Markov Regime-Switching Forward–Backward Stochastic Differential Equations with Jumps and Partial Information

This paper presents three versions of maximum principle for a stochastic optimal control problem of Markov regime-switching forward–backward stochastic differential equations with jumps. First, a general sufficient maximum principle for optimal control for a system, driven by a Markov regime-switching forward–backward jump–diffusion model, is developed. In the regime-switching case, it might happen that the associated Hamiltonian is not concave and hence the classical maximum principle cannot be applied. Hence, an equivalent type maximum principle is introduced and proved. In view of solving an optimal control problem when the Hamiltonian is not concave, we use a third approach based on Malliavin calculus to derive a general stochastic maximum principle. This approach also enables us to derive an explicit solution of a control problem when the concavity assumption is not satisfied. In addition, the framework we propose allows us to apply our results to solve a recursive utility maximization problem.     

Linear Forward—Backward Stochastic Differential Equations

The problem of finding adapted solutions to systems of coupled linear forward—backward stochastic differential equations (FBSDEs, for short) is investigated. A necessary condition of solvability leads to a reduction of general linear FBSDEs to a special one. By some ideas from controllability in control theory, using some functional analysis, we obtain a necessary and sufficient condition for the solvability of a class of linear FBSDEs. Then a Riccati-type equation for matrix-valued (not necessarily square) functions is derived using the idea of the Four-Step Scheme (introduced in [11] for general FBSDEs). The solvability of such a Riccati-type equation is studied which leads to a representation of adapted solutions to linear FBSDEs. Accepted 29 April 1997     

Maximum Principle for Markov Regime-Switching Forward–Backward Stochastic Control System with Jumps and Relation to Dynamic Programming

This paper presents a sufficient stochastic maximum principle for a stochastic optimal control problem of Markov regime-switching forward–backward stochastic differential equations with jumps. The relationship between the stochastic maximum principle and the dynamic programming principle in a Markovian case is also established. Finally, applications of the main results to a recursive utility portfolio optimization problem in a financial market are discussed.     

Variable Metric Forward–Backward Algorithm for Minimizing the Sum of a Differentiable Function and a Convex Function

We consider the minimization of a function G defined on \({ \mathbb{R} } ^{N}\) , which is the sum of a (not necessarily convex) differentiable function and a (not necessarily differentiable) convex function. Moreover, we assume that G satisfies the Kurdyka–?ojasiewicz property. Such a problem can be solved with the Forward–Backward algorithm. However, the latter algorithm may suffer from slow convergence. We propose an acceleration strategy based on the use of variable metrics and of the Majorize–Minimize principle. We give conditions under which the sequence generated by the resulting Variable Metric Forward–Backward algorithm converges to a critical point of G. Numerical results illustrate the performance of the proposed algorithm in an image reconstruction application.