I introduce the mean and variance of stochastic process goverend by an SDE (Stochastic Differential Equation). They are described by two oridinary differential equations (ODEs).
Form of SDE
An SDE takes the form
dx_t = f(x_t, t)dt + g(t) dw_t,where f and g are called the drift term and the diffusion term, respectively, and w_t represents Brownian motion. We are interested only in the case where g(t) is real-valued function of t , because we are dealing the SDEs appearing in diffusion model.
ODE for the meand and the covariance of SDE
Let \mu_t :=E[x_t], \sigma_t^2 = E[(x_t-\mu_t)(x_t-\mu_t)^T] be the mean and the covariance of SDE, respectively.
These two quantites satisfy the following two ODEs [1]
d\mu_t = E[f(x_t,t)] dt , \quad \frac{d\sigma^2_t}{dt} = E\left[(f(x_t,t)-E[f(x_t,t)])(x_t-\mu_t)^T\right]+E\left[(x_t-\mu_t)(f(x_t,t)-E[f(x_t,t)])\right]+g(t)^2 \mathbf{I},where \mathbf{I} is the identity matrix.
In this posting, we have presented the two ODEs that describe the meand and the covariance of an SDE. In the next posting, we will explore cases these ODEs can be solved analytically and presents methods for solving them in such cases.
Reference: S. Särkkä and A. Solin, Applied Stochastic Differential Equations, vol. 10. Cambridge, U.K.: Cambridge University Press, 2019.