Computational Biology Research by Xiaolu Guo

Notes example

This at UCLA.

In following property:

For general stochastic differential equations in math community, such notations are not used. Instead, one uses the abstract symbol $\mathrm{d}W_t$, where $W_t$ represents a Wiener process, also known as the standard Brownian motion. $W_t$ is defined by the following properties:

• The trajectory of $W_t$ is continuous with probability 1.
• For given $t$, $W_t \sim \mathcal{N}(0,t)$, i.e., $W_t$ follows a Gaussian distribution with mean 0, variance $t$ (1d version).
• For any countable set ${t_0,…,t_n}$ with $t_0 < t_1 < \cdots < t_n$, ${W_{t_i}-W_{t_{i-1}}: i\in { 1 ,…, N}}$ are independent of each other.

Implications of $\langle F(t)F(t’) \rangle= 2B \delta(t-t’)$

Suppose that

Consider the $N \rightarrow +\infty$ limit of the following averaged sum: