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Weak Solutions of Forward–Backward SDE's
Fabio Antonelli,Jin Ma +1 more
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TLDR
In this article, the authors studied a class of forward-backward stochastic differential equations with functional-type terminal conditions and proved the existence and uniqueness of the strong adapted solution in the usual sense.Abstract:
In this note we study a class of forward–backward stochastic differential equations (FBSDE for short) with functional-type terminal conditions. In the case when the time duration and the coefficients are “compatible” (e.g., the time duration is small), we prove the existence and uniqueness of the strong adapted solution in the usual sense. In the general case we introduce a notion of weak solution for such FBSDEs, as well as two notions of uniqueness. We prove the existence of the weak solution under mild conditions, and we prove that the Yamada–Watanabe Theorem, that is, pathwise uniqueness implies uniqueness in law, as well as the Principle of Causality also hold in this context.read more
Citations
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Journal ArticleDOI
The Yamada-Watanabe-Engelbert theorem for general stochastic equations and inequalities
TL;DR: A general version of the Yamada-Watanabe and Engelbert results relating existence and uniqueness of strong and weak solutions for stochastic equations is given in this paper, where the results apply to a wide variety of non-convex and convex problems.
Journal ArticleDOI
Weak existence and uniqueness for forward–backward SDEs
TL;DR: In this article, the existence and uniqueness of weak solutions to a suitable class of non-degenerate deterministic FBSDEs with a one-dimensional backward component is established.
Journal ArticleDOI
On Weak Solutions of Backward Stochastic Differential Equations
TL;DR: In this article, the concept of weak solutions of a certain type of backward stochastic differential equations was discussed and a general existence result was given using weak convergence in the Meyer-Zheng topology.
Journal ArticleDOI
The wellposedness of fbsdes
TL;DR: In this paper, the authors investigated the well-posedness of a class of Forward-Backward SDEs and showed that they have the following properties: (i) arbitrary time duration; (ii) random coefficients; (iii) degenerate forward diffusion; and (iv) no monotonicity condition.
Journal ArticleDOI
Weak solutions for forward--backward SDEs--a martingale problem approach
Jin Ma,Jianfeng Zhang,Ziyu Zheng +2 more
TL;DR: In this paper, a new notion of Forward-Backward Martingale Problem (FBMP) was proposed and its relationship with the weak solution to the forward-backward stochastic differential equations was studied.
References
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Book
Brownian Motion and Stochastic Calculus
TL;DR: In this paper, the authors present a characterization of continuous local martingales with respect to Brownian motion in terms of Markov properties, including the strong Markov property, and a generalized version of the Ito rule.
Journal ArticleDOI
User’s guide to viscosity solutions of second order partial differential equations
TL;DR: The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorem, and continuous dependence may now be proved by very efficient and striking arguments as discussed by the authors.
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TL;DR: In this article, Stochastic Differential Equations and Diffusion Processes are used to model the diffusion process in stochastic differential equations. But they do not consider the nonlinearity of diffusion processes.
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Adapted solution of a backward stochastic differential equation
TL;DR: In this paper, the authors considered the problem of finding an adapted pair of processes with values in Rd and Rd×k, respectively, which solves an equation of the form: x(t) + ∫ t 1 f(s, x(s), y(s)) ds + ∪ t 1 [g(m, x, s, g(m)) + y(m)] dW s = X.
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Multidimensional Diffusion Processes
TL;DR: In this paper, the authors propose extension theorems, Martingales, and Compactness, as well as the non-unique case of the Martingale problem, and some estimates on the transition probability functions.
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