Brownian Motion and Stochastic Calculus: Moments, Integrals, and Itô’s Formula

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


# 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.