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March 2019 Differential subordination under change of law
Komla Domelevo, Stefanie Petermichl
Ann. Probab. 47(2): 896-925 (March 2019). DOI: 10.1214/18-AOP1274


We prove optimal $L^{2}$ bounds for a pair of Hilbert space valued differentially subordinate martingales under a change of law. The change of law is given by a process called a weight and sharpness, and in this context refers to the optimal growth with respect to the characteristic of the weight. The pair of martingales is adapted, uniformly integrable and càdlàg. Differential subordination is in the sense of Burkholder, defined through the use of the square bracket. In the scalar dyadic setting with underlying Lebesgue measure, this was proved by Wittwer [Math. Res. Lett. 7 (2000) 1–12], where homogeneity was heavily used. Recent progress by Thiele–Treil–Volberg [Adv. Math. 285 (2015) 1155–1188] and Lacey [Israel J. Math. 217 (2017) 181–195] independently resolved the so-called nonhomogenous case using discrete in time filtrations, where one martingale is a predictable multiplier of the other. The general case for continuous-in-time filtrations and pairs of martingales that are not necessarily predictable multipliers, remained open and is addressed here. As a very useful second main result, we give the explicit expression of a Bellman function of four variables for the weighted estimate of subordinate martingales with jumps. This construction includes an analysis of the regularity of this function as well as a very precise convexity, needed to deal with the jump part.


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Komla Domelevo. Stefanie Petermichl. "Differential subordination under change of law." Ann. Probab. 47 (2) 896 - 925, March 2019.


Received: 1 February 2017; Revised: 1 November 2017; Published: March 2019
First available in Project Euclid: 26 February 2019

zbMATH: 07053559
MathSciNet: MR3916937
Digital Object Identifier: 10.1214/18-AOP1274

Primary: 60G44
Secondary: 60G46

Keywords: Bellman function , Differential subordination , weight

Rights: Copyright © 2019 Institute of Mathematical Statistics


Vol.47 • No. 2 • March 2019
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