1. Standard Brownian Motion & Brownian Bridge Processes
Johar M. Ashfaque
1 The Standard Brownian Motion
A standard Brownian motion is a random process X = {Xt : t ∈ [0, ∞)} with state space R
that satisfies the following properties:
• X0 = 0 with probability 1.
• X has stationary increments. That is for s, t ∈ [0, ∞) with s < t the distribution Xt − Xs
is the same as the distribution of Xt−s.
• X has independent increments. That is for t1, ..., tn ∈ [0, ∞) with
t1 < t2, ... < tn
the random variables
Xt1
, Xt2
− Xt1
, ..., Xtn
− Xtn−1
are independent.
• Xt is normally distributed with mean 0 and variance t for each t ∈ (0, ∞).
• With probability 1, t → Xt is continuous on [0, ∞).
1.1 A Simple Example
Let (Bt) be a 1-dimensional Brownian motion. Prove that
1. for any t > 0, E(Bt) = 0 and E(B2
t ) = t;
2. for any s, t ≥ 0, E(BsBt) = s ∧ t where s ∧ t = min{s, t}.
(1) follows from the definition of the standard Brownian motion.
For (2), we fix any t ≥ s ≥ 0. Then we have
E(BsBt) = E((Bt − Bs)Bs) + E(B2
s ).
Since the Brownian motion has independent increments, the random variables Bt − Bs and Bs
are independent and we find
E((Bt − Bs)Bs) = E(Bt − Bs)E(Bs) = 0.
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2. Furthermore, we know that E(B2
s ) = s. Hence, we conclude that
E(BsBt) = s
which is the required identity.
2 The Brownian Bridge Process
The Brownian bridge process is obtained by taking the standard Brownian motion process X
and restricting it to the interval [0, 1] and conditioning on the event that X1 = 0. Since X0 = 0,
the process is tied down at both ends and so the process in between forms a bridge. The
Brownian bridge turns out to be an interesting stochastic process with surprising applications.
2.1 Definition
A Brownian bridge is a stochastic process
X = {Xt : t ∈ [0, 1]}
with state space R that satisfies the following properties:
• X0 = 0 and X1 = 0 each with probability 1.
• X is a Gaussian process.
• E(Xt) = 0 for t ∈ [0, 1].
• cov(Xs, Xt) = min{s, t} − st for s, t ∈ [0, 1].
• With probability 1, t → Xt, is continuous on [0, 1].
In short, a Brownian bridge X is a continuous Gaussian process with X0 = X1 = 0 and with
mean and covariance functions.
2.2 An Example
Suppose that Z = {Zt, t ∈ [0, ∞)} with Z0 = 0 and E(Zt) = 0 is the standard Brownian motion
and let
Xt = Zt − tZ1
for t ∈ [0, 1]. Then
X = {Xt : t ∈ [0, 1]}
is a Brownian bridge.
• X0 = Z0 = 0 and X1 = Z1 − Z1 = 0.
• Linear combinations of the variables in X reduce to linear combinations of the variables
in Z and hence have normal distributions. Thus X is a Gaussian process.
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3. •
E(Xt) = E(Zt) − tE(Z1) = 0
for t ∈ [0, 1].
•
cov(Xs, Xt) = cov(Zs−sZ1, Zt−tZ1) = cov(Zs, Zt)−tcov(Zs, Z1)−scov(Zt, Z1)+stcov(Z1, Z1) = min{s, t}−st
for s, t ∈ [0, 1].
• t → Xt is continuous on [0, 1] since t → Zt is continuous on [0, 1].
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