A weak convergence criterion for constructing changes of measure

Abstract

Based on a weak convergence argument, we provide a necessary and sufficient condition that guarantees that a nonnegative local martingale is indeed a martingale. Typically, conditions of this sort are expressed in terms of integrability conditions (such as the well-known Novikov condition). The weak convergence approach that we propose allows to replace integrability conditions by a suitable tightness condition. We then provide several applications of this approach ranging from simplified proofs of classical results to characterizations of processes conditioned on first passage time events and changes of measures for jump processes. * We thank Richard Davis, Paul Embrechts, and Thomas Mikosch for putting together a very interesting Oberwolfach seminar, where this project started. We are grateful to Kay Giesecke, Peter Glynn, Jan Kallsen, Ioannis Karatzas, and Philip Protter for stimulating conversations on topics related to the theme of this project, and to Philippe Charmoy, Zhenyu Cui, Roseline Falafala, and Nicolas Perkowski for their comments on an earlier version of this note. †

The martingale property in the context of stochastic differential equations

“…In the specific setup of "removing the drift," [25] and, in the context of stochastic volatility models, [26] give easily verifiable conditions. [2] describes a methodology to decide on the martingale property of a nonnegative local martingale, based on weak convergence considerations. For further pointers to a huge amount of literature in this area, we refer the reader to [24].…”

“…Let now d = 1, E = R, and x 0 = 0. Moreover, for some fixed T > 0 set a(t, x ) = 1, b(t, x ) = 0, µ(t, x ) = (x (t)) 2 To understand, why the assumption of Corollary 3.4 is not satisfied if we replace Ω by Ω fix the path x ∈ Ω \ Ω with x(t) = tan(tπ/(2T ))1 t<T + ∆1 t≥T for all t ≥ 0. Then, we have T 0 (µ(t, x)) 2 dt = ∞.…”

On the Martingale Property in Stochastic Volatility Models Based on Time‐homogeneous Diffusions

“…In the context of stochastic volatility models, Sin (1998), Andersen and Piterbarg (2007), and Lions and Musiela (2007) provide easily verifiable sufficient conditions. Blanchet and Ruf (2016) describe a method to decide on the martingale property of a nonnegative local martingale based on weak convergence arguments. Through the study of the classical solutions to the valuation partial differential equation associated with the stochastic volatility model, Bayraktar, Kardaras, and Xing (2012) establish a necessary and sufficient condition when the asset price is a martingale.…”

Strict Local Martingales via Filtration Enlargement

“…holds true for all t ≥ 0. We take this opportunity to mention that this idea (discovered independently by the first author) is developed in a beautiful (and more general) way in the recent paper of J. Blanchet and J. Ruf [4].…”