The stochastic Leibniz formula for Volterra integrals under enlarged filtrations

Abstract

In this paper, we derive stochastic Leibniz formulas for Volterra integrals under enlarged filtrations. We investigate both pure-jump and Brownian Volterra processes under diverse initially enlarged filtration approaches. In these setups, we compare the ordinary with the stochastic (Doléans-Dade) exponential of a Volterra process and provide the corresponding martingale conditions. We also consider backward stochastic Volterra integral equations (BSVIEs) under enlarged filtrations and obtain the related solution formulas. We finally propose an anticipative Heath Jarrow Morton (HJM) forward rate model of Volterra-type and infer the associated bond price representation. In an introductory section, we compile various facts on deterministic and stochastic Leibniz formulas for parameter integrals.

KEYWORDS:

• Backward stochastic Volterra integral equation (BSVIE)
• Doléans-Dade exponential
• drift restriction
• enlarged filtration
• Lévy process
• parameter integral
• stochastic differential equation (SDE)
• stochastic Leibniz formula
• Volterra process/integral/equation

MSC 2010 codes:

• primary: 60H05
• 60H20
• 60G20
• 60G44
• secondary: 60H10
• 60G51
• 60G57