[Paper Review] Backward Stochastic Differential Equations with Non-Markovian Singular Terminal Conditions with General Driver and Filtration

Mahdi Ahmadi, Alexandre Popier+1 more|arXiv (Cornell University)|Nov 16, 2019

Stochastic processes and financial applications31 references4 citations

TL;DR

This paper establishes the existence of solutions to backward stochastic differential equations (BSDEs) with non-Markovian singular terminal conditions, generalizing prior results to include multidimensional diffusions, jump measures, and superlinear drivers. The key contribution is proving that minimal supersolutions are indeed solutions—i.e., they attain their terminal values almost surely—under mild regularity conditions on stopping times, such as those arising from exit times of diffusions with strongly elliptic diffusions, which are shown to possess continuous densities near the terminal time.

ABSTRACT

We consider a class of Backward Stochastic Differential Equations with superlinear driver process $f$ adapted to a filtration supporting at least a $d$ dimensional Brownian motion and a Poisson random measure on ${\\mathbb R}^m- \\{0\\}.$ We consider the following class of terminal conditions $\\xi_1 = \\infty \\cdot 1_{\\{\ au_1 \\le T\\}}$ where $\ au_1$ is any stopping time with a bounded density in a neighborhood of $T$ and $\\xi_2 = \\infty \\cdot 1_{A_T}$ where $A_t$, $t \\in [0,T]$ is a decreasing sequence of events adapted to the filtration ${\\mathcal F}_t$ that is continuous in probability at $T$. A special case for $\\xi_2$ is $A_T = \\{\ au_2 > T\\}$ where $\ au_2$ is any stopping time such that $P(\ au_2 =T) =0.$ In this setting we prove that the minimal supersolutions of the BSDE are in fact solutions, i.e., they attain almost surely their terminal values. We further show that the first exit time from a time varying domain of a $d$-dimensional diffusion process driven by the Brownian motion with strongly elliptic covariance matrix does have a continuous density; therefore such exit times can be used as $\ au_1$ and $\ au_2$ to define the terminal conditions $\\xi_1$ and $\\xi_2.$ The proof of existence of the density is based on the classical Green's functions for the associated PDE.

Motivation & Objective

• To extend existing results on BSDEs with singular terminal conditions beyond the Markovian case to non-Markovian settings.
• To establish the existence of solutions (not just minimal supersolutions) for BSDEs with terminal conditions of the form ∞·1_{τ≤T} and ∞·1_{A_T}, where τ is a stopping time with bounded density near T and A_t is a left-continuous decreasing sequence of events.
• To generalize the framework to include multidimensional Brownian motion, Poisson random measures, and general driver processes f that are superlinear and adapted.
• To prove that exit times of d-dimensional diffusions with strongly elliptic covariance matrices have continuous densities near T, enabling their use as τ in defining singular terminal conditions.
• To resolve the 'continuity problem'—i.e., showing that the limit of the solution process at T equals the terminal condition almost surely—under general assumptions on the driver and filtration.

Proposed method

• The authors consider a general BSDE with a driver f that is superlinear and adapted to a filtration supporting a d-dimensional Brownian motion and a Poisson random measure.
• They define two classes of singular terminal conditions: ξ₁ = ∞·1_{τ≤T} for stopping times τ with bounded density near T, and ξ₂ = ∞·1_{A_T} for decreasing, left-continuous-in-probability adapted sets A_t.
• The proof relies on constructing a sequence of approximating solutions (Y^k, Z^k, ψ^k, M^k) and using comparison theorems via a transformed process Y^ε,k to control growth.
• A key technical step involves deriving a priori bounds using Hölder’s inequality and the integrability of the process V^ε,k_t,s, which captures the stochastic exponential of the driver's dependence on Z and ψ.
• The authors use the fact that exit times of diffusions with strongly elliptic diffusion matrices have continuous densities, proven via Green’s functions for the associated Fokker-Planck PDE.
• By passing to the limit as ε↓0 and applying dominated convergence and monotone convergence theorems, they show that the limit process satisfies lim_{t→T} Y_t = ξ a.s., thus proving it is a solution.

Experimental results

Research questions

• RQ1 Under what conditions does a minimal supersolution of a BSDE with a non-Markovian singular terminal condition actually solve the BSDE, i.e., attain its terminal value almost surely?
• RQ2 Can the class of admissible singular terminal conditions be extended beyond Markovian exit times to include general stopping times τ with bounded density near T?
• RQ3 Do exit times of d-dimensional diffusions with strongly elliptic diffusion matrices possess continuous densities near the terminal time T, enabling their use in defining singular terminal conditions?
• RQ4 How can the solution theory for BSDEs with superlinear drivers be extended to include jump measures and general adapted drivers f?
• RQ5 What conditions ensure that the solution process of a BSDE is continuous at the terminal time T, thereby resolving the 'continuity problem'?

Key findings

• The minimal supersolution of the BSDE with singular terminal condition ξ₁ = ∞·1_{τ≤T} is shown to be a true solution if τ has a bounded density in a neighborhood of T.
• For the generalized terminal condition ξ₂ = ∞·1_{A_T}, where A_t is a decreasing, left-continuous-in-probability adapted set, the minimal supersolution is a solution if A_T is continuous in probability at T.
• The first exit time τ of a d-dimensional diffusion process with a strongly elliptic covariance matrix from a time-varying domain has a continuous density near T, which justifies its use as τ in defining ξ₁.
• The solution process (Y,Z,ψ,M) is shown to satisfy lim_{t→T} Y_t = ξ almost surely, resolving the continuity problem for non-Markovian singular terminal conditions.
• A priori bounds are derived using Hölder’s inequality and the integrability of the stochastic exponential V^ε,k_t,s, leading to the conclusion that the solution remains finite and continuous at T.
• The proof establishes that the solution process is continuous at T by showing that the conditional expectation of the product ε^p V^ε,k_t,T−ε Y^k_T−ε vanishes as ε↓0, under the given integrability and boundedness assumptions.