Let $B_t$ denote a standard Brownian motion (or Wiener process). It is obtained as the continuous limit of a discrete [[Random Walk Model|Random Walk]]. ## Brownian Motion Property We can describe the change of a Brownian motion over the interval $[t_1, t_2]$ as follows: $ X(t_1, t_2)= B(t_2)-B(t_1)$ The key [[Brownian Motion Property|Brownian Motion Property]] says that any interval follows a [[Gaussian Distribution]], where the [[Variance]] is only dependent on the length of the interval. This is regardless of the starting time. $ X(t_1, t_2) \sim \mathcal N(0, t_2-t_1) \quad \text{when} \quad t_1 \le t_2$ ## Constructing the Continuous Path In a discrete [[Random Walk Model|Random Walk]], each step is the fundamental building block and the path is built by summing these steps. In continuous time, the infinitesimal step $dB_t$ plays the analogous role. The full path of $B_t$ is obtained by integrating these small increments. $ B_T = B_0+\int_0^T dB_t \quad \text{where} \quad dB_t \sim\mathcal N(0, dt)$ Since each $dB_t$ is independent, the [[Covariance]] between increments is given by: $ \mathrm{Cov}(dB_t, dB_{t^\prime})= \begin{cases} 0, & t \neq t^\prime \\dt, & t=t^\prime\end{cases}$ ## Incorporating Drift and Volatility We can generalize the Brownian motion by adding a drift term $\mu$ and a volatility term $\sigma$. Thus can be useful to model assets like a stock price. $ B_T=\mu T+ \sigma \int_0^T dB_t$ **Expectation:** Since $\mathbb E[dB_t]=0$, the [[Expectation]] of $B_T$ is solely determined by the drift. $ \mathbb E[B_T] = \mu T$ **Variance:** The deterministic drift $\mu T$ does not contribute to [[Variance]]. By the key [[Brownian Motion Property]] the variance of the integral $\int_0^T dB_t$ is $T$. $ \mathrm{Var}(B_T)= \mathrm{Var}\left(\mu T+ \sigma \int_0^T dB_t \right) = \sigma^2 \mathrm{Var}\left(\int_0^T dB_t\right) = \sigma^2 T $ **Distribution:** Since the [[Linear Functions of Random Variables#Gaussian Distribution|linear function of a Gaussian]] is still a Gaussian, we can assume that the process $B_T$ is distributed as follows: $ B_T \sim\mathcal N(\mu T, \sigma^2 T)$ ## Itô Process This generalized process $B_t$ is an example of a simple [[Itô Processes and Itô's Lemma]]. Further flexibility can be added by expressing $\mu$ and $\sigma^2$ not as constants, but as functions of time $t$ and the process itself: $ X_t = X_0+ \int_0^T a(t, X_t)\, dt + \int_0^T b(t, X_t)\, dB_t$ where: - $a(t, X_t)$ generalizes the drift $\mu$ - $b(t, X_t)$ generalizes the volatility $\sigma$.