A property of two-parameter martingales with path-independent variation

David Nualart; Frederic Utzet

doi:https://doi.org/10.1016/0304-4149(87)90026-3

A property of two-parameter martingales with path-independent variation

David Nualart, Frederic Utzet

Department of Mathematics

Research output: Contribution to journal › Article › Research › peer-review

Abstract

Let M be a continuous two-parameter L4-martingale, vanishing on the axes, and f a C-function. In Itô's formula for f(M2) a new martingale M̃ is involved. This martingale can be interpreted formally as the stochastic integral ∫∂1M∂2M and it coincides with the martingale JM introduced by Cairoli and Walsh when M is strong. In this paper we prove that if M has path-independent variation, then M and M̃ are orthogonal. Also. we give some counter-examples to the reciprocal implication. © 1987.

Original language	English
Pages (from-to)	31-49
Journal	Stochastic Processes and their Applications
Volume	24
DOIs	https://doi.org/10.1016/0304-4149(87)90026-3
Publication status	Published - 1 Jan 1987

Keywords

path-independent variation
quadratic variation
two-parameter martingales

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abstract = "Let M be a continuous two-parameter L4-martingale, vanishing on the axes, and f a C-function. In It{\^o}'s formula for f(M2) a new martingale {\~M} is involved. This martingale can be interpreted formally as the stochastic integral ∫∂1M∂2M and it coincides with the martingale JM introduced by Cairoli and Walsh when M is strong. In this paper we prove that if M has path-independent variation, then M and {\~M} are orthogonal. Also. we give some counter-examples to the reciprocal implication. {\textcopyright} 1987.",

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AU - Nualart, David

AU - Utzet, Frederic

PY - 1987/1/1

Y1 - 1987/1/1

N2 - Let M be a continuous two-parameter L4-martingale, vanishing on the axes, and f a C-function. In Itô's formula for f(M2) a new martingale M̃ is involved. This martingale can be interpreted formally as the stochastic integral ∫∂1M∂2M and it coincides with the martingale JM introduced by Cairoli and Walsh when M is strong. In this paper we prove that if M has path-independent variation, then M and M̃ are orthogonal. Also. we give some counter-examples to the reciprocal implication. © 1987.

AB - Let M be a continuous two-parameter L4-martingale, vanishing on the axes, and f a C-function. In Itô's formula for f(M2) a new martingale M̃ is involved. This martingale can be interpreted formally as the stochastic integral ∫∂1M∂2M and it coincides with the martingale JM introduced by Cairoli and Walsh when M is strong. In this paper we prove that if M has path-independent variation, then M and M̃ are orthogonal. Also. we give some counter-examples to the reciprocal implication. © 1987.

KW - path-independent variation

KW - quadratic variation

KW - two-parameter martingales

U2 - https://doi.org/10.1016/0304-4149(87)90026-3

DO - https://doi.org/10.1016/0304-4149(87)90026-3

M3 - Article

SN - 0304-4149

VL - 24

SP - 31

EP - 49

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

ER -

A property of two-parameter martingales with path-independent variation

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Keywords

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