Mimicking an Itô process by a solution of a stochastic differential equation

By Gerard P. Brunick, Steven E. Shreve. 2010

121 citations recorded by the source corpus.

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 modeled by the original It\^{o} process or the mimicking process that solves the stochastic differential equation.

Subfields: Finance, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/d9120fedeb7b59589376e67477dcb7936e4b6261

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Citations in the corpus

Superposition and mimicking theorems for conditional McKean–Vlasov equations (2020, methodology): “Application to controlled McKean-Vlasov dynamics The original mimicking theorem of [37, 14] can be used to prove both (1) the existence of optimal Markovian controls in classical stochastic optimal control problems and (2) the equivalence of open-loop and Markovian formulations of said control probl”
McKean–Vlasov optimal control: The dynamic programming principle (2019, methodology): “This technique was followed by Dellacherie [32], by Bertsekas and Shreve in [13; 72; 73; 74], and by Shreve [69; 70; 71] for discrete–time stochastic control problems.”
Mean field games via controlled martingale problems: Existence of Markovian equilibria (2014, methodology): “The argument in [31, 26] for producing a Markovian control from a relaxed control follows Krylov’s ideas (explained in [40, Chapter 12]), but we prefer to exploit the recent “mimicking theorem” of Brunick and Shreve [7], which generalizes a well-known result of Gyöngy [25].”
Hierarchies, entropy, and quantitative propagation of chaos for mean field diffusions (2021, methodology): “The drift b̂ k can be derived via PDE computations, or from a stochastic perspective using the mimicking theorem for Itô processes [26, 9].”