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.
Highlights
We recover the pathwise Itô solution by concatenating averaged Stratonovich solutions on small intervals and by letting the mesh of the partition in the approximations tend to zero
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
Itô calculus [11, 12] can be seen as a transformation between semi-martingales and is widely used in various mathematical models
Summary
Itô calculus [11, 12] can be seen as a transformation between semi-martingales (i.e. the map which sends the driving process to the solution of a stochastic differential equation) and is widely used in various mathematical models. By taking the expectation of the second Stratonovich solution with respect to the selected noise, we recover the bracket process, and by working with a chosen functional of these two Stratonovich solutions, we get the leading order approximation of the increment of the pathwise Itô solution with a small error. By taking the expectation with respect to the selected independent noise W and by working with a chosen functional of the Stratonovich solutions on a small interval, we obtain the leading order approximation of the increment of the Itô solution y, and recover y as the limit of discrete concatenations when the mesh tends to zero. While the idea is similar and captured in this example
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