Unit 6: Brownian Motion
Brownian motion is a continuous-time process closely related to the simple
random walk (SRW) of unit 3 and to the innovations process {et } of Unit 5.
To construct it, start with a symmetric random walk X1 (t) which jumps
every 1 time unit, with jump height ±1. Observe that IEX1 (t) = 0, VarX1 (t) =
t.
Now define√another random walk X2 (t) jumping every 0.5 time units,
jump height ± 0.5. Note IEX2 (t) = 0, VarX2 (t) = t.
Continue this procedure, doubling jump frequency but reducing jump
height so as to preserve expectation and variance. In the limit we have a
process which jumps all the time but with jump height 0, i.e. it is continuous.
The limiting process is standard Brownian motion (BM), B(t).
Properties of SRW and Brownian motion
BM inherits properties from the SRW:
1. IEB(t) = 0, VarB(t) = t.
2. B(t) has stationary, independent increments.
3. B(t) possesses the Markov property.
4. Cov(B(s), B(t)) = min(s, t).
5. B(t) and B(t)2 − t are both martingales.
6. B(t) returns to 0 infinitely often.
7. B(t) eventually hits any finite value a.
Properties 1–7 are easily proved for the SRW and are preserved by the
limiting process.
Additional properties of BM
BM has other properties as well:
8. B(t) is Normally distributed.
9. Scaling property: if B1 (t) = √1c B(ct) then B1 is also a BM.
10. Time-inversion property: if B2 (t) = tB(1/t) then B2 is also a BM.
11. BM is nowhere differentiable (with prob. 1)
From properties 2 and 8 it follows that, conditional on Fs , the distribution of B(t) is N (B(s), t − s) for t > s. This gives a transition density
(density of B(t) given that B(s) = x) of
1
(y − x)2
p
exp −
2(t − s)
2π(t − s)
1
!
.
Proofs of properties 8–10
Property 8: as B(t) can be represented as the sum of a large number of
i.i.d. variables, the Central Limit Theorem ensures that the distribution is
Normal.
Properties 9 and 10: any Gaussian process with mean 0 and covariance
function min(s, t) is a BM. Both B1 and B2 have mean 0. For covariances,
1
Cov(B(cs), B(ct))
c
min(cs, ct)
=
= min(s, t).
c
Cov(B2 (s), B2 (t)) = stCov(B(1/s), B(1/t))
1 1
= min(s, t).
,
= st min
s t
Cov(B1 (s), B1 (t)) =
Partial proof of property 11
If B is differentiable at 0 with derivative a then given we can find δ such
that
B(h)
a−<
< a + for all 0 < h < δ.
h
Put h = 1/t and use the time-inversion property. We then have
1
a − < B2 (t) < a + for all t > .
δ
But the probability of this is zero. So, for any given t, the probability that
BM is differentiable at t is zero.
Reflection principle
Let τa be the first time that BM hits a, i.e. τa = inf{t : B(t) = a}. This is
a stopping time. Then
IP[τa ≤ t ∩ B(t) ≥ a + y] = IP[τa ≤ t ∩ B(t) ≤ a − y].
Proof. If τa < t then B(t) − B(τa ) is independent of Fτa and has a Normal
distribution with mean 0, symmetric about 0.
The running maximum of BM
Let M (t) = max0≤s≤t B(s). Then, for x > 0,
IP[M (t) ≥ x] = IP[τx ≤ t]
= IP[τx ≤ t ∩ B(t) > x]
+ IP[τx ≤ t ∩ B(t) < x]
= 2IP[τx ≤ t ∩ B(t) > x]
= 2IP[B(t) > x]
2
So the density function
of2 M
(t) is double that of B(t):
m
2
√
fM (t) (m) = 2πt exp − 2t , m > 0.
Density of τx
In addition, we have the density of τx :
d
IP[τx ≤ t]
dt
d
=
2IP[B(t) > x]
dt
√
d 1 − Φ(x/ t))
= 2
dt
!
x
x2
= √
exp −
,
2t
2πt3
fτx (t) =
Note that
Z
t > 0.
∞
0
fτx (t) dt = 1.
This means that the BM is certain to hit x eventually, constituting a direct
proof of property 7.
It also proves property 6, since no matter how often the BM has already
hit 0 by time t, it is certain to return again later.
Wiener process and Geometric BM
A more general form of BM is given by
W (t) = µt + σB(t).
Here µ is the drift, σ the diffusion coefficient. (Sometimes called the Wiener
process.)
A more useful process in many modelling applications is the geometric
BM,
S(t) = S0 eµt+σB(t) .
This is non-negative for all t and indeed has a logNormal distribution. Thus
1
IES(t) = S0 e(µ+ 2 σ
VarS(t) =
2 )t
,
2
2
S02 e(2µ+σ )t (eσ t
3
− 1).
Martingale properties
Property 5 is that B(t) and B(t)2 − t are both martingales. In addition we
have the exponential martingale,
1
2
S(t) = eλB(t)− 2 λ t .
Proof: take t > s. Then
IE[eλB(t) | Fs ] = eλB(s) IE[eλ[B(t)−B(s)] | Fs ]
= eλB(s) IE[eλ[B(t)−B(s)] ]
1
2 (t−s)
= eλB(s)+ 2 λ
Therefore
1
2s
IE[S(t) | Fs ] = eλB(s)− 2 λ
.
= S(s).
First hitting time with drift
Let τa,x be the first time standard BM hits the line at + x, where a > 0.
Then
h i
p
IE[e−λτa,x ] = exp − a + a2 + 2λ x .
1
Proof: Consider the martingale eθB(t)− 2 θ
θ > 0, θ > 2a. We have
2t
stopped at time τa,x . Assume
1
θ(aτa,x +x)− 2 θ2 τa,x
,
1 = IE e
so that
−θx
e
1
(aθ− 2 θ2 )τa,x
= IE e
.
Solving the equation λ = 12 θ2 − aθ gives the answer.
Diffusions
A real-valued stochastic process {X(t) : t ≥ 0} is a time-homogeneous diffusion process if
∗ it is Markov
∗ it has continuous sample paths
∗ conditional on X(t) = x, we have
IE[X(t + h) − X(t)] = µ(x)h + o(h),
h
i
= σ 2 (x)h + o(h),
h
i
= o(h).
IE (X(t + h) − X(t))2
IE |X(t + h) − X(t)|3
4
Example: Geometric Brownian motion S(t) is a continuous Markov
process, and
IE [S(t + h) − S(t) | Ft ] =
σ2
µ+
2
!
S(t)h + o(h),
i
h
IE (S(t + h) − S(t))2 | Ft = σ 2 S(t)2 h + o(h).
This means that GBM is a diffusion with µ(x) = (µ + 21 σ 2 )x, σ 2 = σ 2 x2 .
Example: The continuous-time equivalent of the AR(1) is the OrnsteinUhlenbeck process, a diffusion with µ(x) = −γx, σ(x) = σ.
Example: A generalisation of the O-U process is the Vasicek model for
interest rates, a diffusion with µ(x) = −α(x − r), σ(x) = σ. Here r is the
constant long-term average interest rate.
Behaviour of a diffusion
The transition density of a diffusion satisfies equations analogous to the
Chapman-Kolmogorov equations and the Kolmogorov forward and backward equations.
There may also be an equilibrium distribution Π, in the sense that
lim IP[X(t) ∈ A | X(0) = x] = Π(A) for all x.
t→∞
If Π is continuous, we have an equilibrium density function π(x). Any
equilibrium density must be stationary, implying that
µ(x)π(x) =
1 d
[π(x)σ 2 (x)].
2 dx
Conversely,
if π(x), a ≤ x ≤ b is a solution to this equation such that
Rb
∗ a π(x) dx = 1
∗ π(x)σ 2 (x) → 0 as x → a and as x → b
then π is a stationary distribution for the diffusion.
Example: The O-U process has µ(x) = −γx, σ(x) = σ. Therefore we
need
1
−γxπ(x) = σ 2 π 0 (x),
2
2
an equation whose solution is a Normal density, mean 0, variance σ2γ .
In fact in this example it is possible to show that, given X(0) = x0 , we
would have
!
2
−2γt
−γt σ
)
Xt ∼ N x0 e , (1 − e
2γ
5
Covariance of a diffusion
We can use the defining equation to find the covariance and correlation of a
diffusion. Take the O-U process as an example. For t > s define
m(t) = IE[Ut | Us ].
Then
m(t + dt) = IE[Ut − γUt dt | Us = x] = (1 − γdt)m(t).
The solution is m(t) = Us e−γ(t−s) .
It follows that
IE[Us Ut ] = IE[Us IE[Ut | Us ]] = e−γ(t−s) IE[Us2 ].
Assuming the process is in equilibrium, Var(Ut ) = Var(Us ) =
Corr(Us , Ut ) = e−γ|t−s| .
6
σ2
2γ ,
so that