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Well-Posedness Of High-Dimensional Forward-Backward Stochastic Differential Equations (FBSDEs).pdf

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Preview of Well-Posedness of High-Dimensional Forward-Backward Stochastic Differential Equations (FBSDEs)
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Summary

## Summary of arXiv:1708.05785v1

This paper extends the well-posedness result for Forward-Backward SDEs (FBSDEs) to high dimensions, relaxing key conditions from previous literature. Key features include:

Arbitrary time duration: The result holds for any positive time T.
 Random coefficients: Coefficients can be random.
Degenerate forward diffusion: Allows for a degenerate (i.e., singular) forward diffusion term σ.
 No monotonicity condition: Removes the need for a monotonicity assumption on the coefficients.

Main Results:

1. Well-posedness (Theorem 1.2): For FBSDEs with uniformly Lipschitz continuous coefficients, if a certain inequality (1.6) holds, then there exists a unique solution Θ satisfying a boundedness estimate in terms of an initial condition norm (I2₀).

2. Small Time Duration (Lemma 2.2): Provides an estimate on the size of solutions to linear FBSDEs with small time horizon T, expressed in terms of coefficient bounds and a constant CK depending only on the Lipschitz constants.

Improvements:

General Dimensions: Extends results to higher-dimensional problems (d > 1).
 Weaker Conditions: Significantly weakens key conditions from previous work, including removing differentiation requirements and a monotonicity condition.

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Description

This paper extends the well-posedness result for Forward-Backward Stochastic Differential Equations (FBSDEs) to high dimensions, significantly weakening existing key conditions, allowing for arbitrary time durations, random coefficients, degenerate forward diffusion, and no monotonicity requirements.

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  • File Size: 125 KB
  • Pages: 12
  • Language: EN
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