Strict local martingales with jumps

Abstract

3 A strict local martingale is a local martingale which is not a martingale. There are few explicit examples 4 of "naturally occurring" strict local martingales with jumps available in the literature. The purpose of 5 this paper is to provide such examples, and to illustrate how they might arise via filtration shrinkage, a 6 phenomenon we would contend is common in applications such as filtering, control, and especially in 7 mathematical finance. We give a method for constructing such examples and analyze one particular method 8 in detail.9

Strict local martingales: Examples

“…The sample continuity assumption is hence essential for the statement below, on the other hand a cádlág local martingale with a.s. finite total jumps is the sum of a continuous local martingale and a finite variation process, and one can construct strict local martingales with jump by adding such a jump process. Our method does not seem to apply to stochastic integrals driven by a Lévy process, for which please see the papers by Philip Protter [Pro15] and that by Constantinos Kardaras, Dörte Kreher, and Ashkan Nikeghbali [KKN15]. = ∞ for some number α ∈ (0, 1) then t 0 f (s), dB s is a strict local martingale.…”

“…A type three financial bubble models a non-zero asset price with bounded life time bursting at on or before the bubble time and is a strict local martingale. The two are linked as following: birth of bubbles are impossible in complete market, and possible in incomplete markets, see Philip Protter [Pro15,section3].…”

Conditioning the Information in Portfolio Optimization

“…is a positive true martingale 17 . Given that some families of semimartingales-namely: local and strict local martingales-may change the nature of their process (from absolutely continuous to mixed) once projected onto smaller filtration sets (Protter (2015) [26], Sala and Barone Adesi (2015) [27]), the same may not be true under the suboptimal scenario. From the previous sections and justified by the rational behaviours of the investors under the neoclassical theory, it follows that the optimal quantities are always larger with respect to the suboptimal ones.…”

Simple examples of pure-jump strict local martingales

“…3.11], [MU14,Sec. 4] and the more recent construction of [Pro14] through filtration shrinkage. In this note we present several simple new examples of pure-jump strict local martingales.…”