Brownian Motion

Variation

For a partition $\Pi = \{t_0 = 0 < t_1 < \cdots < t_n = T\}$ with mesh $\|\Pi\| = \max_j (t_{j+1} - t_j)$, the first variation and quadratic variation of $f$ on $[0, T]$ are

$$FV_T(f) = \lim_{\|\Pi\| \to 0} \sum_{j=0}^{n-1} |f(t_{j+1}) - f(t_j)|, \qquad [f, f](T) = \lim_{\|\Pi\| \to 0} \sum_{j=0}^{n-1} \big(f(t_{j+1}) - f(t_j)\big)^2$$

If the first variation of $g$ is bounded, $\int_a^b f\, dg$ is defined.

If $f$ is differentiable, $FV_T(f) = \int_0^T |f'(t)|\, dt$.

Random Walks

Symmetric random walk With coin tosses $\omega_j$, $X_j = 1$ if $\omega_j = H$ and $-1$ if $\omega_j = T$, $M_0 = 0$ and $M_k = \sum_{j=1}^k X_j$. The increments $M_l - M_k = \sum_{j=k+1}^l X_j$ over disjoint intervals are independent, with mean 0 and variance $l - k$, and $[M, M]_k = \sum_{j=1}^k (M_j - M_{j-1})^2 = k$.

Scaled symmetric random walk $W^{(n)}(t) = \frac{1}{\sqrt{n}}M_{nt}$ (for $nt$ an integer). Each step is $\pm\frac{1}{\sqrt{n}}$, so

$$[W^{(n)}, W^{(n)}](t) = \sum_{j=1}^{nt}\Big[ W^{(n)}\Big(\frac{j}{n}\Big) - W^{(n)}\Big(\frac{j - 1}{n}\Big) \Big]^2 = \sum_{j=1}^{nt}\Big( \frac{\pm 1}{\sqrt{n}} \Big)^2 = t$$

and as $n \to \infty$, $W^{(n)}(t) - W^{(n)}(s) \to \mathcal{N}(0, t - s)$ in distribution.

Brownian Motion

$W(t)$, $t \ge 0$, is a Brownian motion if for each $\omega$ the path $t \mapsto W(t)$ is continuous with $W(0) = 0$, and for $0 = t_0 < t_1 < \cdots < t_m$ the increments $W(t_{i+1}) - W(t_i)$ are independent and $\mathcal{N}(0, t_{i+1} - t_i)$.

A filtration for Brownian motion is $(\mathcal{F}_t)_{t \ge 0}$ with

• $\mathcal{F}_s \subseteq \mathcal{F}_t$ for $s \le t$,
• $W(t)$ is $\mathcal{F}_t$-measurable (adapted),
• $W(u) - W(t)$ is independent of $\mathcal{F}_t$ for $0 \le t < u$ (independent future increments).

Properties

• $W$ is a martingale and a Markov process.
• Its paths are almost surely nowhere differentiable.
• $[W, W](T) = T$ almost surely (the sums converge in $L^2$).
• In differential form, $dW\, dW = dt$, $dW\, dt = 0$, $dt\, dt = 0$.

Exponential Martingale and Geometric Brownian Motion

If $X \sim \mathcal{N}(\mu, \sigma^2)$ then $e^X$ is log-normal with $E[e^X] = e^{\mu + \sigma^2/2}$. Hence

$$Z(t) = \exp\left(\sigma W(t) - \frac{1}{2}\sigma^2 t\right)$$

is a martingale (the exponential martingale). Geometric Brownian motion

$$S(t) = S(0)\exp\left(\Big(\mu - \frac{\sigma^2}{2}\Big)t + \sigma W(t)\right)$$

solves $dS = \mu S\, dt + \sigma S\, dW$, has $d(\log S) = (\mu - \frac{\sigma^2}{2})\, dt + \sigma\, dW$, $E[S(t)] = S(0)e^{\mu t}$, and $e^{-\mu t}S(t)$ is a martingale.

Reflection Principle

For $B > 0$,

$$\mathbb{P}\Big(\max_{0 \le t \le T} W(t) \ge B\Big) = 2\,\mathbb{P}\big(W(T) \ge B\big) = 2N\left(-\frac{B}{\sqrt{T}}\right)$$

where $N$ is the standard normal CDF.