Arbeitspapier

On Itô's formula for multidimensional Brownian motion

Consider a d-dimensional Brownian motion X (Xl, ... ,Xd ) and a function F which belongs locally to the Sobolev space W 1,2. We prove an extension of Ito's formula where the usual second order terms are replaced by the quadratic covariations [fk(X), Xkj involving the weak first partial derivatives fk of F. In particular we show that for any locally square-integrable function f the quadratic covariations [f(X), Xkj exist as limits in probability for any starting point, except for some polar set. The proof is based on new approximation results for forward and backward stochastic integrals.

Language: Englisch

Bibliographic citation: Series: SFB 373 Discussion Paper ; No. 2001,90

Classification: Wirtschaft

Subject: Ito's formula
Brownian motion
stochastic integrals
quadratic covariation
Dirichlet spaces
polar sets

Event: Geistige Schöpfung

(who): Föllmer, Hans
Protter, Philip E.

Event: Veröffentlichung

(who): Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes

(where): Berlin

(when): 2001

Handle: http://hdl.handle.net/10419/62732

URN: urn:nbn:de:kobv:11-10051072

Last update: 10.03.2025, 11:41 AM CET

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Object type

Arbeitspapier

Associated

Föllmer, Hans
Protter, Philip E.
Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes

Time of origin

2001

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