## Electronic Communications in Probability

### Recovering the pathwise Itô solution from averaged Stratonovich solutions

#### Abstract

We recover the pathwise Itô solution (the solution to a rough differential equation driven by the Itô signature) by concatenating averaged Stratonovich solutions on small intervals and by letting the mesh of the partition in the approximations tend to zero. More specifically, on a fixed small interval, we consider two Stratonovich solutions: one is driven by the original process and the other is driven by the original process plus a selected independent noise. Then by taking the expectation with respect to the selected noise, we can recover the increment of the bracket process and so recover the leading order approximation of the Itô solution up to a small error. By concatenating averaged increments and by letting the mesh tend to zero, the error tends to zero and we recover the Itô solution.

#### Article information

Source
Electron. Commun. Probab., Volume 21 (2016), paper no. 7, 18 pp.

Dates
Accepted: 10 August 2015
First available in Project Euclid: 5 February 2016

https://projecteuclid.org/euclid.ecp/1454682823

Digital Object Identifier
doi:10.1214/16-ECP3795

Mathematical Reviews number (MathSciNet)
MR3485376

Zentralblatt MATH identifier
1338.60148

#### Citation

Lyons, Terry; Yang, Danyu. Recovering the pathwise Itô solution from averaged Stratonovich solutions. Electron. Commun. Probab. 21 (2016), paper no. 7, 18 pp. doi:10.1214/16-ECP3795. https://projecteuclid.org/euclid.ecp/1454682823

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