In this post, I will introduce the score function of a stochastic process. Before proceeding, I recommend that reading my previous posts below
Score function of a random vector
stochastic process, random process, 랜덤프로세스, 확률과정의 정의
Stochastic process
A stochastic process is a collection of random vectors x_t \in \mathbf{R}^d , 0\leq t \leq T . For each time index t \in [0, T ] , x_t is a random vector whose probability density function (pdf) is denoted as p_t (x_t) .
Definition of the score function of a stochastic process
The score function of the stochastic process \{ x_t \} is defined for each t as follows
\nabla_{x_t} \log p_t (x_t) \tag{Score function of $\{x_t\}$ for each t}