Standard Brownian Motion: { Bt, t ≥ 0 }
Standard Brownian Motion (also
called the Wiener process) is a stochastic process {Bt , t ≥ 0} with state space S = R (the set of real numbers)
and the following defining properties:
1)
Bt has
independent increments, ie Bt - Bs is independent of {Br ,r £ s} whenever s
< t.
It follows that
the changes in the value of the process over any two non-overlapping periods
are statistically independent.
2)
Bt has stationary
increments, ie the distribution of Bt - Bs depends only on t – s
3)
Bt has Gaussian
increments, ie the distribution of Bt - Bs is N(0,t - s)
“Gaussian”
is the official name for the normal distribution.
4)
Bt has continuous
sample paths tàBt
This means that
the graph of Bt as a function of t doesn’t have any breaks in it.
5)
B0 = 0
Actually, Brownian motion is
the continuous-time analogue of a random walk. Other properties
of Standard Brownian motion include:
² Cov(Bs , Bt ) = min(s,t)
² {Bt ,t ≥ 0} is a Markov
process
² {Bt ,t ≥ 0} is a martingale,
ie E(Bt | Fs ) = Bs
² {Bt ,t ≥ 0}
returns infinitely often to 0, or indeed to any other level
Brownian motion with drift: {Wt, t ≥
0 }
Brownian motion with drift
is related to standard Brownian motion by the equation:
Wt =W0 +σBt +μt,
where σ is the
volatility or diffusion coefficient and μ is the drift.
The Standard Brownian Motion is the special case of Brownian Motion where W0=0,
μ=0 & σ=1.
Geometric Brownian Motion: { St, t ≥
0 }
For modelling
purposes a Brownian motion may have to be transformed, for example by taking
logarithms. A useful model for security prices is geometric Brownian motion:
St = exp(Wt)
where Wt is the Brownian
process Wt = W0 + σBt + μt. Thus St, which is called Geometric Brownian motion,
is lognormally distributed with parameters W0+μt and σ^2t.
