# Summary of Stochastic Derivatives, Integrals, Expectations, and Variances

I wanted to make a summary table of stochastic functions to improve my understanding. Maybe the following should be a wiki page on this site so others can add functions and examples? Does the following look right?

Let,

$$c$$ be a constant
$$k$$ be a natural number
$$t$$ be a variable representing time
$$f(t)$$ be a deterministic function of time $$t$$
$$B_t$$ be Brownian motion
$$X_t$$ be a random process driven by $$B_t$$ in some way

Function Derivative Integral Expectation Variance
$$f(t)$$ $$\frac{d}{dt}f(t)=f'(t)$$ $$\int_0^t f(s)dB_s=X_t$$ $$\mathbb{E}[f(t)]=f(t)$$ $$\mbox{Var}[f(t)]=0$$
$$B_t$$ $$\frac{d}{dt} B_t=\mbox{undefined}$$ $$\int_0^t B_s ds=\frac{1}{2}B_t^2 - \frac{1}{2}tB_t$$

$$\int_0^t B_s dB_s=\frac{1}{2}B_t^2-\frac{1}{2}t$$
$$\mathbb{E}[B_t]=0$$ $$\mbox{Var}[B_t]=t$$
$$cB_t$$ $$\frac{d}{dt} cB_t=\mbox{undefined}$$ $$\int_0^t cB_s ds=ctB_t$$

$$\int_0^t B_s dB_s=c(\frac{1}{2}B_t^2-\frac{1}{2})t$$
$$\mathbb{E}[cB_t]=0$$ $$\mbox{Var}[cB_t]=c^2t$$
$$tB_t$$ $$\frac{d}{dt} tB_t =B_t$$ $$\int_0^t tB_s ds=\frac{t^2}{2}B_t$$

$$\int_0^t B_s dB_s=\frac{1}{2}B_t^2 t-\frac{1}{2}t^2$$
$$\mathbb{E}[tB_t]=0$$ $$\mbox{Var}[tB_t]=t^2 \cdot t$$
$$B^2_t$$ $$\frac{d}{dt} B^2_t=\mbox{undefined}$$ $$\int_0^t B^2_tds=?$$

$$\int_0^t B^2_t dB_t = \frac{1}{3} B^3_t-tB_t$$
$$\mathbb{E}[B^2_t]=t$$ $$\mbox{Var}[B_t^2]=t^2$$
$$B^k_t$$ $$\frac{d}{dt} B^k_t=\mbox{undefined}$$ - $$\mathbb{E}[B_t^{2k}]=t^k(2k-1)!!$$

See note 1
-
$$X_t = x + \mu dt+\sigma B_t$$ $$dX_t = a_tdt+b_tdB_t$$ $$\int_0^t X_s ds = t X_s$$

$$\int_0^t X_s dX_s = \frac{1}{2}X^2_s - \frac{1}{2}t$$
- -
$$f(B_t) = f(0) + \frac{1}{2} \int_0^t f''(B_s)ds + \int_0^t f'(B_s)ds$$ $$df(B_t) = f'(B_t)dB_t + \frac{1}{2} f''(B_t)dt$$ - - -

Note 1 - All odd moments are 0. $$(2k-1)!!$$ is the product of all odd integers between 1 and $$2k-1$$.

Note that $$\frac{d}{dt} (B_t)$$ is undefined because $$B_t$$ is non-smooth.

Note that taking the derivative with respect to Brownian motion $$\frac{d}{dB_t}$$ is undefined because $$B_t$$ is a function. This would be similar to taking the derivative with respect to another function $$f(\cdot)$$ such as $$\frac{d}{df}$$ which doesn't make sense.