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Brownian motion

A Brownian motion, also called a Wiener process, is a continuous-time stochastic process $W_t$, $t\ge 0$. It is named in honor of botanist Robert Brown who noted the seemingly random movements of particles suspended in fluid.

Characterization

A Brownian motion is typically characterized in terms of its increments. Given two distinct points in time $t$ and $s$ with $s<t$, the increment is the change in the process between $s$ and $t$, given by $W_t-W_s$.

A Brownian motion has the following three properties:

• The initial condition is known: $W_0=0$ a.s.

• The increments are independent. That is, if $0\le t_0\le t_1\le \cdots \le t_k$, then

  $$\mathrm{Pr}(W_{t_i}-W_{t_{i-1}}\in H_i,i=\mathrm{0,1},\dots ,k)=\prod _{i=0}^k\mathrm{Pr}(W_{t_i}-W_{t_{i-1}}\in H_i).$$\Pr( W_{t_i} - W_{t_{i-1}} \in H_i, i = 0,1,\ldots,k) = \prod _{i=0}^k \Pr( W_{t_i} - W_{t_{i-1}} \in H_i ).

• For $s$ and $t$ with $0\le s<t$,

  $$W_t-W_s\sim N(0,t-s)$$W_t - W_s \sim \mathop{N}(0, t-s)

  where $N$ denotes the Normal distribution.

Note that characterizing the process in terms of increments is especially useful for empirical studies as the process can only feasibly be observed at a finite number of points in time and the observed intervals follow a well-known distribution.

Variants

Geometric Brownian Motion

A geometric Brownian motion with drift $\mu$ and volatility $\sigma$ satisfies

where $W_t$ is a Wiener process. $\mu$ is the percentage drift, the expected percentage change in $X_t$ per unit of time. $\sigma$ is the percentage volatility, the expected standard deviation over one unit of time.

Simulation

A Brownian motion is a continuous phenomenon that we can only sample at a finite number of points. We can then interpolate linearly between these sampled values to create a plot. One can achieve closer approximations by choosing successively smaller sampling intervals.

To simulate a standard Brownian motion on $[\mathrm{0,1}]$ with $N$ intervals of length $1/N$, draw a sequence of independent Normally distributed random variables with mean $0$ and variance $1/N$. Then the value of the Brownian motion at time $t=i/N$ for $i\in \{1,\dots ,N\}$ is the sum of the first $i$ draws.

References

• Billingsley, Patrick (1995): Probability and Measure.