[Paper Review] Backward stochastic Volterra integral equations with jumps in a general filtration

Alexandre Popier|arXiv (Cornell University)|Feb 17, 2020

Stochastic processes and financial applications32 references4 citations

TL;DR

This paper establishes existence, uniqueness, and time regularity of $L^p$-solutions for Type-I and Type-II backward stochastic Volterra integral equations (BSVIEs) with jumps in a general filtration, extending prior results beyond Brownian-Poisson settings. The approach uses fixed-point arguments and BDG-type inequalities to prove path regularity and integrability under minimal filtration assumptions, including càdlàg martingales and random measures with general compensators.

ABSTRACT

In this paper, we study backward stochastic Volterra integral equations introduced in [26, 45] and extend the existence, uniqueness or comparison results for general filtration as in [31] (not only Brownian-Poisson setting). We also consider Lp-data and explore the time regularity of the solution in the It{\\^o} setting, which is also new in this jump setting.

Motivation & Objective

• To extend existence and uniqueness results for backward stochastic Volterra integral equations (BSVIEs) beyond the Brownian-Poisson filtration setting.
• To establish $L^p$-solution theory for Type-I and Type-II BSVIEs with jumps under general filtration assumptions (complete and right-continuous).
• To analyze the time regularity of the solution process $Y$ in the Itô setting with jumps, generalizing prior results in the continuous case.
• To incorporate general semimartingale components, including càdlàg martingales and random measures with general compensators, into the BSVIE framework.

Proposed method

• Uses a fixed-point argument in an appropriate $L^p$-space to prove existence and uniqueness of the solution quadruplet $(Y, Z, U, M)$ for Type-I BSVIEs.
• Applies BDG-type inequalities to control the quadratic variation of martingale terms and derive path regularity estimates for $Y$.
• Employs Itô’s formula on a semimartingale $X_t(u)$ constructed from the solution to derive a priori estimates involving $Y(t)$, $Z(t,s)$, $U(t,s,x)$, and $M(t,s)$.
• Imposes minimal assumptions on the filtration: only completeness and right-continuity, allowing general semimartingale components $B$, $X^ullet$, $ ilde{ u}^ atural$, and a martingale $M$.
• Derives $L^p$-estimates for the solution components by bounding the generator $f$ and using Young’s inequality on the drift and diffusion terms.
• Uses the M-solution concept to ensure the solution satisfies a martingale representation property, crucial for deriving path regularity.

Experimental results

Research questions

• RQ1 Can existence and uniqueness of $L^p$-solutions for BSVIEs with jumps be established in a general filtration, not restricted to Brownian-Poisson settings?
• RQ2 What is the time regularity of the solution $Y$ in the $L^p$-setting when the filtration is general and includes jumps and semimartingale components?
• RQ3 How do the components $Z$, $U$, and $M$ contribute to the regularity of $Y$ in the absence of Brownian motion or Poisson processes?
• RQ4 Can the fixed-point method be adapted to handle the non-Markovian, non-McKean structure of BSVIEs with general semimartingale noise and jump measures?
• RQ5 What are the precise $L^p$-estimates for the solution that ensure the pathwise regularity of $Y$ in time?

Key findings

• The paper proves existence and uniqueness of $L^2$-solutions for Type-I BSVIEs in a general filtration, generalizing prior results that required Brownian or Poisson filtration.
• It establishes $L^p$-regularity of the solution $Y$ in time under mild assumptions on the generator $f$ and terminal condition $ ilde{ u}^\natural$, extending the path regularity theory to jump settings.
• The solution $Y$ belongs to the space $D([0,T]; \mathbb{L}^p_{\mathbb{F}}(\Omega))$, meaning it has càdlàg paths in $L^p$, under appropriate integrability and continuity conditions on $f$ and $\Phi$.
• The paper derives sharp $L^p$-estimates for the solution components $Z$, $U$, and $M$, using BDG and Young’s inequalities, which are essential for proving regularity.
• The M-solution concept ensures that the solution satisfies a martingale representation property, enabling control of the time increment $Y(t) - Y(t')$ via stochastic integrals.
• The analysis confirms that the solution $Y$ is not necessarily càdlàg almost surely, but its $L^p$-paths are càdlàg, which is sufficient for many applications in stochastic control and PDEs.