Pathwise solutions of stochastic differential equations

1 Wong-Zakai approximation

The Wong-Zakai approach seems to be about approximating the “rough” paths of Brownian motion with smooth paths, then analyzing the resulting ODEs instead of dealing with stochastic integrals directly. The idea is that if we approximate a Wiener process by a family of smooth functions (e.g. via piecewise linear interpolation), the solutions of the corresponding ordinary differential equations will converge—in an “appropriate” sense—to the solution of a stochastic differential equation (SDE).

This much I know from reading abstracts; but take my words with a grain of salt, because I have not used such results in practice.

Key points, as far as an LLM is concerned:

• Smooth Approximations: We replace the Brownian path with smooth approximations, so you can use classical calculus to solve the ODEs driven by these paths.
• Convergence to Stratonovich SDE: The limit of these approximations tends to yield the solution of the SDE interpreted in the Stratonovich sense. This is because the correction term that naturally arises (and is built into the Stratonovich formulation) accounts for the “noise” that would be missing in a naive limit.
• Pathwise Handling: The formulation allows us to handle things pathwise. Instead of working with the full machinery of Itô calculus and the Wiener measure, we approximate the paths and then apply deterministic techniques. This makes the analysis somewhat more intuitive for our monkey brains.
• Regularization: The Wong-Zakai approximation can be seen as a regularization technique. By smoothing out the noise, we can analyze the system more easily and then take the limit to recover the stochastic behaviour.

2 Random ODEs

Random ODEs is what Bongers and Mooij (2018) refers to. Is that a Wong-Zakai-type approach?

TBD

3 Föllmer’s pathwise Itô calculus

TBD

4 Rough path theory

See rough paths.

5 Incoming

For a classical setting, see (Teye 2010; Wedig 1984) and an interesting application to Feynman-Kac formula in Ezawa, Klauder, and Shepp (1974).

6 References