Open Access
August 2013 Mimicking an Itô process by a solution of a stochastic differential equation
Gerard Brunick, Steven Shreve
Ann. Appl. Probab. 23(4): 1584-1628 (August 2013). DOI: 10.1214/12-AAP881


Given a multi-dimensional Itô process whose drift and diffusion terms are adapted processes, we construct a weak solution to a stochastic differential equation that matches the distribution of the Itô process at each fixed time. Moreover, we show how to match the distributions at each fixed time of functionals of the Itô process, including the running maximum and running average of one of the components of the process. A consequence of this result is that a wide variety of exotic derivative securities have the same prices when the underlying asset price is modeled by the original Itô process or the mimicking process that solves the stochastic differential equation.


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Gerard Brunick. Steven Shreve. "Mimicking an Itô process by a solution of a stochastic differential equation." Ann. Appl. Probab. 23 (4) 1584 - 1628, August 2013.


Published: August 2013
First available in Project Euclid: 21 June 2013

zbMATH: 1284.60109
MathSciNet: MR3098443
Digital Object Identifier: 10.1214/12-AAP881

Primary: 60G99 , 60H10 , 91G20

Keywords: derivative security pricing , Itô process , Stochastic differential equation , stochastic volatility models

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.23 • No. 4 • August 2013
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