[Paper Review] Backward Stochastic Differential Equations with no driving martingale, Markov processes and associated Pseudo Partial Differential Equations

Adrien Barrasso, Francesco Russo|arXiv (Cornell University)|Jan 11, 2017

Stochastic processes and financial applications14 references8 citations

TL;DR

This paper introduces a new class of backward stochastic differential equations (BSDEs) without a driving martingale, driven by a general Markov process and associated with a pseudo-PDE that generalizes the classical semilinear PDE. The key contribution is establishing the equivalence between solutions of these Markovian BSDEs and both classical and martingale solutions of the corresponding pseudo-PDE, extending the classical Pardoux-Peng framework beyond Brownian motion to general Markov processes.

ABSTRACT

We discuss a class of Backward Stochastic Differential Equations(BSDEs) with no driving martingale. When the randomness of the driver depends on a general Markov process $X$, those BSDEs are denominated Markovian BSDEs and can be associated to a deterministic problem,called Pseudo-PDE which constitute the natural generalization of a parabolicsemilinear PDE which naturally appears when the underlying filtration is Brownian. We consider two aspects of well-posedness forthe Pseudo-PDEs: "classical" and "martingale" solutions.

Motivation & Objective

• To develop a new class of backward stochastic differential equations (BSDEs) without a driving martingale, applicable to general Markov processes.
• To establish a deterministic PDE-like problem, termed a pseudo-PDE, that generalizes the classical semilinear PDE arising in the Brownian motion case.
• To prove the equivalence between solutions of the Markovian BSDE and both classical and martingale solutions of the associated pseudo-PDE.
• To extend the classical Pardoux-Peng BSDE-PDE duality to non-Brownian, general Markovian settings using the carré du champ operator and martingale problem formulation.

Proposed method

• Formulates a Markovian BSDE of the form $ Y^{s,x}_t = g(X_T) + \int_t^T f(r,X_r,Y^{s,x}_r, \sqrt{d\langle M^{s,x}\rangle/dV}(r)) dV_r - (M^{s,x}_T - M^{s,x}_t) $, where $ X $ is a general Markov process.
• Associates the BSDE with a pseudo-PDE of the form $ a(u)(t,x) + f(t,x,u(t,x), \sqrt{\Gamma(u,u)}(t,x)) = 0 $, with terminal condition $ u(T,\cdot) = g $, where $ \Gamma $ is the carré du champ operator.
• Uses the martingale problem formulation to define the underlying Markov process $ X $ via a deterministic generator $ a $, generalizing SDEs in law.
• Applies Kunita-Watanabe decomposition and angular bracket decomposition to decompose the local martingale $ M $ into components $ M^V $ and $ M^{\perp V} $, orthogonal with respect to the measure $ V $.
• Establishes that $ \mathcal{H}^{2,V} $ and $ \mathcal{H}^{2,\perp V} $ are orthogonal sub-Hilbert spaces of $ \mathcal{H}^2_0 $, enabling the decomposition of the martingale component.
• Proves existence and uniqueness of solutions via the notion of classical and martingale solutions to the pseudo-PDE, under appropriate regularity and integrability conditions.

Experimental results

Research questions

• RQ1 How can backward stochastic differential equations be formulated without a driving martingale when the underlying process is a general Markov process?
• RQ2 What is the natural generalization of the classical semilinear PDE in the context of non-Brownian, Markovian forward processes?
• RQ3 How can the solution of a Markovian BSDE be linked to a deterministic problem (pseudo-PDE) in the absence of a Brownian filtration?
• RQ4 What conditions ensure the existence and uniqueness of solutions to the associated pseudo-PDE in the classical and martingale sense?
• RQ5 How does the carré du champ operator $ \Gamma(u,u) $ emerge naturally in the context of Markovian BSDEs and their deterministic counterparts?

Key findings

• The paper establishes a duality between Markovian BSDEs and pseudo-PDEs, generalizing the classical Pardoux-Peng result beyond Brownian motion.
• Solutions to the Markovian BSDE are shown to correspond to classical solutions of the pseudo-PDE when sufficient regularity is assumed.
• The existence of a martingale solution to the pseudo-PDE is proven under weaker conditions, extending the classical framework to more general Markov processes.
• The decomposition of the local martingale $ M $ into $ M^V $ and $ M^{\perp V} $, orthogonal with respect to $ V $, is fundamental to the analysis and ensures the well-posedness of the BSDE.
• The carré du champ operator $ \Gamma(u,u) = a(u^2) - 2ua(u) $ naturally arises as the quadratic variation term in the pseudo-PDE, generalizing the role of $ \sigma \nabla u $ in the classical PDE.
• The paper proves that $ \mathcal{H}^{2,V} $ and $ \mathcal{H}^{2,\perp V} $ are closed, orthogonal sub-Hilbert spaces of $ \mathcal{H}^2_0 $, ensuring the uniqueness of the decomposition and the stability of the solution framework.