Semimartingale decomposition of convex functions of continuous semimartingales by brownian perturbation

Nastasiya F. Grinberg

ESAIM: Probability and Statistics (2013)

  • Volume: 17, page 293-306
  • ISSN: 1292-8100

Abstract

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In this note we prove that the local martingale part of a convex function f of a d-dimensional semimartingale X = M + A can be written in terms of an Itô stochastic integral ∫H(X)dM, where H(x) is some particular measurable choice of subgradient ∇ f ( x ) off at x, and M is the martingale part of X. This result was first proved by Bouleau in [N. Bouleau, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981) 87–90]. Here we present a new treatment of the problem. We first prove the result for X ˜ = X + ϵ B x10ff65; X = X + ϵB ,ϵ > 0, where B is a standard Brownian motion, and then pass to the limit as ϵ → 0, using results in [M.T. Barlow and P. Protter, On convergence of semimartingales. In Séminaire de Probabilités, XXIV, 1988/89, Lect. Notes Math., vol. 1426. Springer, Berlin (1990) 188–193; E. Carlen and P. Protter, Illinois J. Math. 36 (1992) 420–427]. The former paper concerns convergence of semimartingale decompositions of semimartingales, while the latter studies a special case of converging convex functions of semimartingales.

How to cite

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Grinberg, Nastasiya F.. "Semimartingale decomposition of convex functions of continuous semimartingales by brownian perturbation." ESAIM: Probability and Statistics 17 (2013): 293-306. <http://eudml.org/doc/274387>.

@article{Grinberg2013,
abstract = {In this note we prove that the local martingale part of a convex function f of a d-dimensional semimartingale X = M + A can be written in terms of an Itô stochastic integral ∫H(X)dM, where H(x) is some particular measurable choice of subgradient $$ ∇ f ( x ) off at x, and M is the martingale part of X. This result was first proved by Bouleau in [N. Bouleau, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981) 87–90]. Here we present a new treatment of the problem. We first prove the result for $\widetilde\{X\}=X+\epsilon B$ x10ff65; X = X + ϵB ,ϵ &gt; 0, where B is a standard Brownian motion, and then pass to the limit as ϵ → 0, using results in [M.T. Barlow and P. Protter, On convergence of semimartingales. In Séminaire de Probabilités, XXIV, 1988/89, Lect. Notes Math., vol. 1426. Springer, Berlin (1990) 188–193; E. Carlen and P. Protter, Illinois J. Math. 36 (1992) 420–427]. The former paper concerns convergence of semimartingale decompositions of semimartingales, while the latter studies a special case of converging convex functions of semimartingales.},
author = {Grinberg, Nastasiya F.},
journal = {ESAIM: Probability and Statistics},
keywords = {Itô’s lemma; continuous semimartingales; convex functions; semimartingales; Itō’s lemma; Brownian perturbation},
language = {eng},
pages = {293-306},
publisher = {EDP-Sciences},
title = {Semimartingale decomposition of convex functions of continuous semimartingales by brownian perturbation},
url = {http://eudml.org/doc/274387},
volume = {17},
year = {2013},
}

TY - JOUR
AU - Grinberg, Nastasiya F.
TI - Semimartingale decomposition of convex functions of continuous semimartingales by brownian perturbation
JO - ESAIM: Probability and Statistics
PY - 2013
PB - EDP-Sciences
VL - 17
SP - 293
EP - 306
AB - In this note we prove that the local martingale part of a convex function f of a d-dimensional semimartingale X = M + A can be written in terms of an Itô stochastic integral ∫H(X)dM, where H(x) is some particular measurable choice of subgradient $$ ∇ f ( x ) off at x, and M is the martingale part of X. This result was first proved by Bouleau in [N. Bouleau, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981) 87–90]. Here we present a new treatment of the problem. We first prove the result for $\widetilde{X}=X+\epsilon B$ x10ff65; X = X + ϵB ,ϵ &gt; 0, where B is a standard Brownian motion, and then pass to the limit as ϵ → 0, using results in [M.T. Barlow and P. Protter, On convergence of semimartingales. In Séminaire de Probabilités, XXIV, 1988/89, Lect. Notes Math., vol. 1426. Springer, Berlin (1990) 188–193; E. Carlen and P. Protter, Illinois J. Math. 36 (1992) 420–427]. The former paper concerns convergence of semimartingale decompositions of semimartingales, while the latter studies a special case of converging convex functions of semimartingales.
LA - eng
KW - Itô’s lemma; continuous semimartingales; convex functions; semimartingales; Itō’s lemma; Brownian perturbation
UR - http://eudml.org/doc/274387
ER -

References

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