[Paper Review] Multidimensional Markov FBSDEs with superquadratic growth

Michael Kupper, Peng Luo+1 more|arXiv (Cornell University)|May 5, 2015

Stochastic processes and financial applications18 references3 citations

TL;DR

This paper establishes local and global existence and uniqueness for multidimensional Markovian forward-backward stochastic differential equations (FBSDEs) with generators exhibiting superquadratic growth in the control variable Z. The approach combines Malliavin calculus to bound the control process and a pasting technique to extend local solutions to global ones, under non-degeneracy and regularity conditions on the coefficients, extending prior results to the coupled, multidimensional case with arbitrary growth in Z.

ABSTRACT

We give local and global existence and uniqueness results for systems of coupled FBSDEs in the multidimensional setting and with generators allowed to grow arbitrarily fast in the control variable. Our results are based on Malliavin calculus arguments and pasting techniques.

Motivation & Objective

• To address the lack of well-posedness results for coupled, multidimensional FBSDEs with generators growing faster than quadratic in the control variable Z.
• To extend existing local solvability results for one-dimensional FBSDEs with superquadratic growth to the multidimensional case.
• To construct a global solution for such systems under non-degeneracy and regularity conditions on the coefficients.
• To provide a framework that can be extended to non-Markovian settings and random diffusion coefficients under stronger Malliavin derivative assumptions.

Proposed method

• Utilizes Malliavin calculus to derive uniform bounds on the control process Z, enabling truncation arguments for locally Lipschitz generators.
• Employs a Picard iteration scheme in a suitable Banach space to prove local existence and uniqueness, relying on boundedness of Z derived via Malliavin calculus.
• Applies a pasting procedure across small time intervals to extend local solutions to a global solution over the entire time horizon T.
• Uses the PDE representation of Markovian FBSDEs (as in Delarue [7]) to link the stochastic system to a parabolic PDE, facilitating the construction of global solutions.
• Imposes conditions on the generator g to grow at most quadratically on the diagonal and assumes non-degeneracy of the volatility σ to ensure solvability.
• Relies on the fact that the trace of the Malliavin derivative of Y is a version of Z in the Lipschitz case, which is extended to handle superquadratic growth via boundedness arguments.

Experimental results

Research questions

• RQ1 Under what conditions does a coupled system of multidimensional FBSDEs with superquadratic growth in the control variable Z admit a unique solution?
• RQ2 Can Malliavin calculus techniques used for one-dimensional FBSDEs be extended to prove boundedness of the control process Z in the multidimensional case?
• RQ3 How can local solutions be patched together to yield a global solution for FBSDEs with superquadratic generators?
• RQ4 What role does the non-degeneracy of the volatility σ play in ensuring global solvability of such systems?
• RQ5 To what extent can the Markovian framework be generalized to non-Markovian or random diffusion coefficient settings?

Key findings

• A unique local solution exists for multidimensional Markovian FBSDEs with locally Lipschitz generators in (X,Y) and locally Lipschitz in Z, provided the time horizon is small enough and Z is uniformly bounded via Malliavin calculus.
• Global existence and uniqueness are established under the assumption that the generator g grows at most quadratically on the diagonal and that the volatility σ is non-degenerate.
• The control process Z is uniformly bounded by a constant M̄ depending on the problem parameters, which enables the use of truncation and Picard iteration in a Banach space framework.
• The solution lies in the space S²(ℝᵐ) × S∞(ℝˡ) × S∞(ℝˡ×ᵈ), ensuring pathwise regularity and integrability.
• The pasting technique allows construction of a global solution by concatenating local solutions over intervals of length bounded by a constant depending on the problem’s Lipschitz and growth parameters.
• The results can be extended to non-Markovian settings and random diffusion coefficients under stronger Malliavin derivative conditions on g and h, as detailed in Luo’s Ph.D. thesis.