Quadratic Backward Stochastic Differential Equations Research Articles

Malliavin Calculus and Its Application to Robust Optimal Investment for an Insider

In the theory of portfolio selection, there are few methods that effectively address the combined challenge of insider information and model uncertainty, despite numerous methods proposed for each individually. This paper studies the problem of the robust optimal investment for an insider under model uncertainty. To address this, we extend the Itô formula for forward integrals by Malliavin calculus, and use it to establish an implicit anticipating stochastic differential game model for the robust optimal investment. Since traditional stochastic control theory proves inadequate for solving anticipating control problems, we introduce a new approach. First, we employ the variational method to convert the original problem into a nonanticipative stochastic differential game problem. Then we use the stochastic maximum principle to derive the Hamiltonian system governing the robust optimal investment. In cases where the insider information filtration is of the initial enlargement type, we derive the closed-form expression for the investment by using the white noise theory when the insider is ’small’. When the insider is ’large’, we articulate a quadratic backward stochastic differential equation characterization of the investment. We present the numerical result and conduct an economic analysis of the optimal strategy across various scenarios.

Multi-dimensional backward stochastic differential equations of diagonally quadratic generators: The general result

This paper is devoted to a general solvability of a multi-dimensional backward stochastic differential equation (BSDE) of a diagonally quadratic generator g(t,y,z), by relaxing the assumptions of Hu and Tang [15] on the generator and terminal value. More precisely, the generator g(t,y,z) can have more general growth and continuity in the first unknown variable y in the local solution; while in the global solution, the generator g(t,y,z) can have a skew sub-quadratic but in addition “strictly and diagonally” quadratic growth in the second unknown variable z, or the terminal value can be unbounded but the generator g(t,y,z) is “diagonally dependent” on the variable z (i.e., the i-th component gi of the generator g only depends on the i-th row zi of z for each i=1,⋯,n). Three new results are established on the local and global solutions when the terminal value is bounded and the generator g is subject to some general assumptions. When the terminal value is unbounded but is of exponential moments of arbitrary order, an existence and uniqueness result is given under the assumptions that the generator g(t,y,z) is Lipschitz continuous in y, and varies with z in a “diagonal”, “component-wisely convex or concave”, and “quadratically growing” way, which seems to be the first general solvability of system of quadratic BSDEs with unbounded terminal values. This generalizes and strengthens some existing results via some new ideas.

Malliavin Regularity of Non-Markovian Quadratic BSDEs and Their Numerical Schemes

We study both Malliavin regularity and numerical approximation schemes for a class of quadratic backward stochastic differential equations (QBSDEs for short) in cases where the terminal data need not be a function of a forward diffusion. By using the connection between the QBSDE under study and some backward stochastic differential equations (BSDEs) with global Lipschitz coefficients, we firstly prove Lq, (q≥2) existence and uniqueness results for QBSDE. Secondly, the Lp-Hölder continuity of the solutions is established for (q>4 and 2≤p<q2). Then, we analyze some numerical schemes for our systems and establish their rates of convergence. Moreover, our results are illustrated with three examples.

Global existence for quadratic FBSDE systems and application to stochastic differential games

In this note, we extend some recent results on systems of backward stochastic differential equations (BSDEs) with quadratic growth to the case of coupled forward-backward stochastic differential equations (FBSDEs). We work in a Markovian setting, and use results from the quadratic BSDE literature together with PDE techniques to obtain a-priori estimates which lead to an existence result. We also identify a general class of stochastic differential games whose corresponding FBSDE systems are covered by our main existence result. This leads to the existence of Markovian Nash equilibria for such games.

Quadratic [formula omitted]-BSDEs with convex generators and unbounded terminal conditions

In this paper, we first study one-dimensional quadratic backward stochastic differential equations driven by G-Brownian motions (G-BSDEs) with unbounded terminal values. With the help of a θ-method of Briand and Hu (2008) and nonlinear stochastic analysis techniques, we propose an approximation procedure to prove existence and uniqueness result when the generator is convex (or concave) and terminal value is of exponential moments of arbitrary order. Finally, we also establish the well-posedness of multi-dimensional G-BSDEs with diagonally quadratic generators.

Local existence and uniqueness of solutions to quadratic BSDEs with weak monotonicity and general growth generators

In this paper, we study quadratic backward stochastic differential equations (QBSDEs for short) with generators satisfying a weak monotonicity and a general growth condition with respect to the first variable. We provide the local existence and uniqueness of solutions to multi-dimensional QBSDEs. We use the convolution approach, the Picard iteration, the truncation and the Bihari inequality.

Quantitative stability and numerical analysis of Markovian quadratic BSDEs with reflection

<p style='text-indent:20px;'>We study the quantitative stability of solutions to Markovian quadratic reflected backward stochastic differential equations (BSDEs) with bounded terminal data. By virtue of bounded mean oscillation martingale and change of measure techniques, we obtain stability estimates for the variation of the solutions with different underlying forward processes. In addition, we propose a truncated discrete-time numerical scheme for quadratic reflected BSDEs and obtain the explicit rate of convergence by applying the quantitative stability result.</p>

Deep Neural Networks Algorithms for Stochastic Control Problems on Finite Horizon: Numerical Applications

This paper presents several numerical applications of deep learning-based algorithms for discrete-time stochastic control problems in finite time horizon that have been introduced in Huré et al. (2018). Numerical and comparative tests using TensorFlow illustrate the performance of our different algorithms, namely control learning by performance iteration (algorithms NNcontPI and ClassifPI), control learning by hybrid iteration (algorithms Hybrid-Now and Hybrid-LaterQ), on the 100-dimensional nonlinear PDEs examples from Weinan et al. (2017) and on quadratic backward stochastic differential equations as in Chassagneux and Richou (2016). We also performed tests on low-dimension control problems such as an option hedging problem in finance, as well as energy storage problems arising in the valuation of gas storage and in microgrid management. Numerical results and comparisons to quantization-type algorithms Qknn, as an efficient algorithm to numerically solve low-dimensional control problems, are also provided.

Systems of quasilinear parabolic equations in [formula omitted] and systems of quadratic backward stochastic differential equations

The objective of this paper is two-fold. The first objective is to complete the former work of Bensoussan and Frehse [2]. One big limitation of this paper was the fact that they are systems of PDE. on a bounded domain. One can expect solutions to be bounded, since one looks for smooth solutions. This is a very important property for the development of the method. It is true also that solutions which exist in a bounded domain may fail to exist on Rn, because of the lack of bounds. We give conditions so that the results of [2] can be extended to Rn. The second objective is to consider the BSDE (Backward stochastic differential equations) version of the system of PDE. This is the objective of a more recent work of Xing and Z̆itković [8]. They consider systems of BSDE with quadratic growth, which is a well-known open problem in the BSDE literature. Since the BSDE are Markovian, the problem is equivalent to the analytic one. However, because of this motivation the analytic problem is in Rn and not on a bounded domain. Xing and Z̆itković developed a probabilistic approach. The connection between the analytic problem and the BSDE is not apparent. Our objective is to show that the analytic approach can be completely translated into a probabilistic one. Nevertheless probabilistic concepts are also useful, after their conversion into the analytic framework. This is in particular true for the uniqueness result.

On the singular risk-sensitive stochastic maximum principle

We consider a risk-sensitive stochastic control problem of a nonlinear system in which the variable control has two components, the first being absolutely continuous and the second is singular. The author generalizes the result obtained in risk neutral into risk-sensitive performance cost. The auxiliary process has been introduced to solve the problem of risk sensitive. The relationship between the expected exponential utility and the quadratic backward stochastic differential equation has been given and proved. Actually, I have proved in details two results, the first is necessary optimality condition, while the second is sufficient optimality condition for risk sensitive performance cost. Two examples to carried out to illustrate our main results of risk-sensitive control problem under nonlinear stochastic dynamics with exponential quadratic cost function with numerical investigation, while the second will be in the exponential utility optimization problem.

An incomplete equilibrium with a stochastic annuity

We prove the global existence of an incomplete, continuous-time finite-agent Radner equilibrium in which exponential agents optimise their expected utility over both running consumption and terminal wealth. The market consists of a traded annuity, and along with unspanned income, the market is incomplete. Set in a Brownian framework, the income is driven by a multidimensional diffusion and in particular includes mean-reverting dynamics. The equilibrium is characterised by a system of fully coupled quadratic backward stochastic differential equations, a solution to which is proved to exist under Markovian assumptions. We also show that the equilibrium allocations lead to Pareto-optimal allocations only in exceptional situations.

Systems of Ergodic BSDEs Arising in Regime Switching Forward Performance Processes

We introduce and solve a new type of quadratic backward stochastic differential equation (BSDE) systems defined in an infinite time horizon, called ergodic BSDE systems. Such systems arise naturall...

On the uniqueness of solutions to quadratic BSDEs with non-convex generators and unbounded terminal conditions

We prove a uniqueness result of the unbounded solution for a quadratic backward stochastic differential equation whose terminal condition is unbounded and whose generator $g$ may be non-Lipschitz continuous in the state variable $y$, non-convex (non-concave) in the state variable $z$, and instead satisfies a strictly quadratic condition and an additional assumption. The key observation is that if the generator is strictly quadratic, then the quadratic variation of the first component of the solution admits an exponential moment. Typically, a Lipschitz perturbation of some convex (concave) function satisfies the additional assumption mentioned above. This generalizes some results obtained in \cite{BriandHu2006PTRF} and \cite{BriandHu2008PTRF}.

Expected Utility Maximization Problem Under State Constraints and Model Uncertainty

We study a general robust utility maximization problem from terminal wealth and consumption under state constraints. Our framework includes financial models with constrained portfolios, labor income and large investor models. We state the existence and the uniqueness of the consumption–investment strategy by studying the associated quadratic backward stochastic differential equation. We characterize the optimal control by using the duality method and deriving a dynamic maximum principle.

Representation Theorem for Generators of Quadratic BSDEs

In this paper, we establish a general representation theorem for generator of backward stochastic differential equation (BSDE), whose generator has a quadratic growth in z. As some applications, we obtain a general converse comparison theorem of such quadratic BSDEs and uniqueness theorem, translation invariance for quadratic g-expectation.

A stability approach for solving multidimensional quadratic BSDEs

We establish an existence and uniqueness result for a class of multidimensional quadratic backward stochastic differential equations (BSDEs). This class is characterized by constraints on some uniform a priori estimates on solutions of a sequence of approximated BSDEs. We also present effective examples of applications. Our approach relies on the strategy developed by Briand and Elie in [Stochastic Process. Appl. 123 2921–2939] concerning scalar quadratic BSDEs.

Quadratic backward stochastic differential equations driven by [formula omitted]-Brownian motion: Discrete solutions and approximation

In this paper, we consider backward stochastic differential equations driven by G-Brownian motion (GBSDEs) under quadratic assumptions on coefficients. We prove the existence and uniqueness of solution for such equations. On the one hand, a priori estimates are obtained by applying the Girsanov type theorem in the G-framework, from which we deduce the uniqueness. On the other hand, to prove the existence of solutions, we first construct solutions for discrete GBSDEs by solving corresponding fully nonlinear PDEs, and then approximate solutions for general quadratic GBSDEs in Banach spaces.

A Global Stochastic Maximum Principle for Fully Coupled Forward-Backward Stochastic Systems

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 forward-backward stochastic differential equations. Inspired by Hu [Probab. Uncertain. Quant. Risk, 2 (2017), pp. 1--20], we develop a new decoupling approach by introducing an adjoint equation which is a quadratic backward stochastic differential equation. By revealing the relations among the terms of the first-order Taylor 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.

Time-Consistent Mean-Variance Reinsurance-Investment Problems Under Unbounded Random Parameters: BSDE and Uniqueness

To strike the best balance between insurance risk and profit, insurers transfer insurable risk through reinsurance and enhance yield by participating into the financial market. The long-term commitment of insurance contracts makes insurers necessary to consider time-consistent (TC) reinsurance-investment policies. Using the open-loop TC mean-variance (MV) reinsurance-investment framework, we investigate the equilibrium reinsurance-investment problems for the financial market with unbounded random coefficients or, specifically, an unbounded risk premium. We characterize the problem via a backward stochastic differential equation (BSDE) framework. An explicit solution to the equilibrium strategies is derived for a constant risk aversion under a general class of stochastic models, embracing the constant elasticity of variance (CEV) and Ornstein-Uhlenbeck (OU) processes as special cases. For state-dependent risk aversions, the problem is related to the existence of a solution to a quadratic BSDE with unbounded parameters. A semi-closed form solution is derived, up to the solution to a nonlinear partial differential equation. By examining properties of the equilibrium strategies numerically, we find that the reinsurance decision is greatly affected by the market situation under the state-dependent risk aversion case. We prove the uniqueness of equilibrium strategies for both cases.

A class of globally solvable Markovian quadratic BSDE systems and applications

We establish existence and uniqueness for a wide class of Markovian systems of backward stochastic differential equations (BSDE) with quadratic nonlinearities. This class is characterized by an abstract structural assumption on the generator, an a priori local-boundedness property, and a locally-Hölder-continuous terminal condition. We present easily verifiable sufficient conditions for these assumptions and treat several applications, including stochastic equilibria in incomplete financial markets, stochastic differential games and martingales on Riemannian manifolds.