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?


$c$ 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.


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