Created: August 27, 2022
Modified: August 27, 2022

Itô process

This page is from my personal notes, and has not been specifically reviewed for public consumption. It might be incomplete, wrong, outdated, or stupid. Caveat lector.

A Itô process is a stochastic process satisfying a stochastic differential equation of the form

dX_t = \mu_t(X_t)dt + \sigma_t(X_t)dW_t

where $W_t$ is Brownian motion. This can be thought of as the continuous-time analogue of the conditionally Gaussian driftNote that the $x_t$ 's are not jointly Gaussian unless we have linear $\mu_t(x) = ax$ and constant $\sigma_t(x) = \sigma$ .

x_{t + 1} \sim \mathcal{N}\left(x_t + \mu_t(x_t), \sigma_t(x_t)\right).

Itô process

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