Mimicking an Itô Process by a Solution of a Stochastic Differential Equation

Details Mimicking an Itô Process by a Solution of a Stochastic Differential Equation Mimicking an Itô Process by a Solution of a Stochastic Differential Equation

2010-10-30

arxiv.org/abs/1011.0111v2

Mathematics

Probability

Scientific paper

Given a multi-dimensional It\^o 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 It\^o process at each fixed time. Moreover, we show how to match the distributions at each fixed time of functionals of the It\^o 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 modelled by the original It\^o process or the mimicking process that solves the stochastic differential equation.

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