Multidimensional Backward Stochastic Differential Equations Research Articles

Multidimensional backward stochastic differential equations with rough drifts

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 p -rough stochastic integral for p ∈ [ 2 , 3 ) p \in \left [2, 3\right ) , 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.

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.

Lp solutions of general time interval BSDEs with generators satisfying a p-order weak stochastic-monotonicity condition

This article is interested in solutions of general time interval backward stochastic differential equations (BSDEs) with generator g satisfying a p-order weak stochastic-monotonicity condition and a general growth condition in y. An existence and uniqueness result of the Lp solution is proved for the multidimensional BSDEs where the generator g is stochastic-Lipschitz in z. And, a general comparison theorem and an existence and uniqueness theorem of the Lp solution is established for the one-dimensional BSDEs under a stochastic uniform-continuity condition of the generator g in z. These results strengthen some known works. The stochastic Gronwall-type inequality and the stochastic Bihari-type inequality play an important role in the whole article.

{{varvec{L}}}^{{varvec{p}}}-Solutions and Comparison Results for L\xe9vy-Driven Backward Stochastic Differential Equations in a Monotonic, General Growth Setting

We present a unified approach to L^p-solutions (p > 1) of multidimensional backward stochastic differential equations (BSDEs) driven by Lévy processes and more general filtrations. New existence, uniqueness and comparison results are obtained. The generator functions obey a time-dependent extended monotonicity (Osgood) condition in the y-variable and have general growth in y. Within this setting, the results generalize those of Royer, Yin and Mao, Yao, Kruse and Popier, and Geiss and Steinicke.

W-symmetries of backward stochastic differential equations, preservation of simple symmetries and Kozlov’s theory

In this paper, we study the determining equations of W-symmetries for backward stochastic differential equations (BSDEs). We have proved that each simple random symmetry of multi-dimensional BSDE can be preserved under simple deterministic change of coordinates. Finally, we study the use of symmetries in studying BSDEs, and find that symmetries of BSDE can not form Lie algebra.

Multidimensional BSDE with Poisson jumps of Osgood type

Abstract This paper is devoted to solve a multidimensional backward stochastic differential equation with jumps in finite time horizon. Under linear growth generator, we prove existence and uniqueness of solution.

Lp Solutions of BSDEs with a New Kind of Non-Lipschitz Coefficients

In this paper, we are interested in solving multidimensional backward stochastic differential equations (BSDEs) with a new kind of non-Lipschitz coefficients. We establish an existence and uniqueness result of the Lp (p > 1) solutions, which includes some known results as its particular cases.

Multi-dimensional BSDEs driven by G-Brownian motion and related system of fully nonlinear PDEs

In this paper, we study the well-posedness of multi-dimensional backward stochastic differential equations driven by G-Brownian motion (G-BSDEs). The existence and uniqueness of solutions are obtained via a contraction argument for Y component and a backward iteration of local solutions. Furthermore, we show that, the solution of multi-dimensional G-BSDE in a Markovian framework provides a probabilistic formula for the viscosity solution of a system of nonlinear parabolic partial differential equations.

Multi-dimensional optimal trade execution under stochastic resilience

We provide a general framework for analysing multi-dimensional portfolio liquidation problems with instantaneous and persistent price impact and stochastic resilience. We show that the value function can be described by a system of multi-dimensional backward stochastic Riccati differential equations (BSRDEs) with a singular terminal condition. We prove the existence of a solution to the BSRDE system and characterise both the value function and the optimal strategy in terms of that solution. We prove that the solution to the liquidation problem can be approximated by the solutions to a sequence of unconstrained problems with increasing penalisation of open positions at the terminal time. Our proof is based on a novel a priori estimate for the approximating BSRDE systems, from which we infer the convergence of the optimal trading strategies for the unconstrained models to an admissible liquidation strategy for the original problem.

Existence, Uniqueness and Stability of $$L^1$$ L 1 Solutions for Multidimensional Backward Stochastic Differential Equations with Generators of One-Sided Osgood Type

We establish a general existence and uniqueness result of $$L^1$$ solution for a multidimensional backward stochastic differential equation (BSDE for short) with generator g satisfying a one-sided Osgood condition as well as a general growth condition in y, and a Lipschitz condition together with a sublinear growth condition in z, which improves some existing results. In particular, we put forward and prove a stability theorem of the $$L^1$$ solutions for the first time. A new type of $$L^1$$ solution is also investigated. Some delicate techniques involved in the relationship between convergence in $$L^1$$ and in probability and dividing appropriately the time interval play crucial roles in our proofs.

Formula omitted] solutions of backward stochastic differential equations with jumps

Given p∈(1,2), we study Lp solutions of a multi-dimensional backward stochastic differential equation with jumps (BSDEJ) whose generator may not be Lipschitz continuous in (y,z)-variables. We show that such a BSDEJ with p-integrable terminal data admits a unique Lp solution by approximating the monotonic generator by a sequence of Lipschitz generators via convolution with mollifiers and using a stability result.

Lp Solutions of Backward Stochastic Differential Equations with Jumps

Given $p \in (1, 2)$, we study $L^p$-solutions of a multi-dimensional backward stochastic differential equation with jumps (BSDEJ) whose generator may not be Lipschitz continuous in $(y,z)-$variables. We show that such a BSDEJ with a p-integrable terminal data admits a unique $L^p$ solution by approximating the monotonic generator by a sequence of Lipschitz generators via convolution with mollifiers and using a stability result.

General time interval BSDEs under the weak monotonicity condition and nonlinear decomposition for general g-supermartingales

The main purpose of this paper is to prove an existence and uniqueness result for solutions of a multidimensional backward stochastic differential equation (BSDE) with a general time interval (including the deterministic and stochastic cases), where the generator g of the BSDE is weakly monotonic and of general growth in y, and Lipschitz continuous in z, both non-uniformly with respect to t. And, the corresponding comparison theorem for the solutions of one-dimensional BSDEs is provided. As applications, we establish a nonlinear Doob-Meyer’s decomposition theorem for general continuous g-supermartingales under an additional assumption of the generator g. Some new problems in our setting arise naturally and are well overcome. These results generalize and improve some known works.

An existence theorem for multidimensional BSDEs with mixed reflections

In this note, we consider the pricing problem for a type of real option, which gives the right to switch investment modes and abandon the investment project before its maturity. The value of this option can be characterized by solutions to multidimensional backward stochastic differential equations (BSDEs) with both normal and oblique reflections, whose coefficients are of linear growth and are left-Lipschitz with respect to (w.r.t) y and Lipschitz w.r.t. z. We provide an existence theorem of minimal solutions for BSDEs in this framework.

On the stability theorem of $L^{p}$ solutions for multidimensional BSDEs with uniform continuity generators in $z$

In this paper, we first establish an existence and uniqueness result of $L^{p}$ ($p>1$) solutions for multidimensional backward stochastic differential equations (BSDEs) whose generator $g$ satisfies a certain one-sided Osgood condition with a general growth in $y$ as well as a uniform continuity condition in $z$, and the $i$th component ${}^{i}g$ of $g$ depends only on the $i$th row ${}^{i}z$ of matrix $z$ besides $(\omega,t,y)$. Then we put forward and prove a stability theorem for $L^{p}$ solutions of this kind of multidimensional BSDEs. This generalizes some known results.

Multi-dimensional backward stochastic differential equations of diagonally quadratic generators

In this paper, we study a multi-dimensional BSDE with a “diagonally” quadratic generator, the quadratic part of whose ith component depends only on the ith row of the second unknown variable. Local and global solutions are given, which seem to be the first systematic (positive) results on the general solvability of multi-dimensional quadratic BSDEs. In our proofs, it is natural and crucial to apply both John–Nirenberg and reverse Hölder inequalities for BMO martingales. Our results are finally illustrated to solve the system of “diagonally” quadratic BSDEs arising from a nonzero-sum risk-sensitive stochastic differential game, which answers the open problem posed in El Karoui and Hamadène [Stochastic Process. Appl. 107 (2003), page 164].

Formula omitted] solutions of multidimensional BSDEs with weak monotonicity and general growth generators

In this paper, we first establish the existence and uniqueness of Lp(p>1) solutions for multidimensional backward stochastic differential equations (BSDEs) under a weak monotonicity condition together with a general growth condition in y for the generator g. Then, we overview several conditions related closely to the weak monotonicity condition and compare them in an effective way. Finally, we put forward and prove a stability theorem and a comparison theorem of Lp(p>1) solutions for this kind of BSDEs.

A general existence and uniqueness result on multidimensional BSDEs

A general existence and uniqueness result on multidimensional BSDEs XU Shao-ya, FAN Sheng-jun (1. College of Sciences, China University of Mining and Technology, Xuzhou Jiangsu 221116, China; 2. School of Mathematical Sciences, Fudan University, Shanghai 200433, China) Abstract: This paper established a general existence and uniqueness result of solutions for multidimensional backward stochastic differential equations (BSDEs) whose generators satisfy the weakly monotonic condition in y, which generalizes some existing results.

Existence and Uniqueness of Multidimensional BSDEs and of Systems of Degenerate PDEs with Superlinear Growth Generator

In the first part of this paper, we deal with the unique solvability of multidimensional backward stochastic differential equations (BSDEs) with a $p$-integrable terminal condition $(p>1)$ and a superlinear growth generator. We introduce a new local condition on the generator (assumption (H.4)) and then show that it ensures the existence and uniqueness, as well as the $L^p$-stability, of solutions. The assumptions that we impose on the generator are local in the three variables $y, z, \omega $, and therefore we also cover the BSDEs with stochastic Lipschitz coefficient. Our conditions on the generator go beyond all existing ones in the literature. For instance, the generator is not assumed uniformly continuous and therefore cannot satisfy the classical Osgood condition. Furthermore, it could be neither locally monotone in the $y$-variable nor locally Lipschitz in the $z$-variable. Although we are focused on multidimensional BSDEs, our results on uniqueness and stability are new even for one-dimensional BS...

A new comparison theorem of multidimensional BSDEs

In this paper, we first study a property about the generator g of Backward Stochastic Differential Equation (BSDE) when the price of contingent claims can be represented by a multidimensional BSDE in the no-arbitrage financial market. Furthermore, motivated by the behavior of agents in finance market, we introduce a new total order on ℝ n and obtain a necessary and sufficient condition for comparison theorem of multidimensional BSDEs under this order. We also give some further results for .