MATH 5050: Practice
In what follows we will need a multidimensional variant of Ito’s formula. But first,
let us recall Taylor’s expansion in multi dimensions: Let f (y1 , . . . , yd ) be a function of d
variables that has two continuous derivatives in each coordinate. Let (δ1 , . . . , δd ) be d small
increments. Then
d
X
∂f
f (y1 + δ1 , . . . , yd + δd ) = f (y1 , . . . , yd ) +
(y1 , . . . , yd ) × δi
∂yi
i=1
+
1
2
d X
d
X
i=1 j=1
∂2f
× δi δj + errors smaller than any δi δj .
∂yi ∂yj
Now let B1 (t), . . . , Bd (t) be d independent standard Brownian motions. Let f (y1 , . . . , yd )
be a function of d variables that has two continuous derivatives in each coordinate. Let
Z(t) = f (B1 (t), . . . , Bd (t)). Then applying the above Taylor expansion, replacing every
(dBi (t))2 by dt and ignoring anything that is smaller than dt (e.g. (dt)2 or (dt)(dBi (t))),
we arrive at
d
X
1
∂f
dZ(t) =
(B1 (t), . . . , Bd (t)) dBi (t) + ∆f (B1 (t), . . . , Bd (t)) dt .
∂yi
2
i=1
P
2
(Recall that ∆f (y1 , . . . , yd ) = di=1 ∂∂yf2 (the sum of all second derivatives).
i
Make sure you understand the above derivation before you keep going (like where the
2f
∆f came from, and why things like ∂y∂1 ∂y
are not there).
2
Problem 1 Let d ≥ 2 and let B1 (t), . . . , Bd (t) be d independent standard Brownian motions. Let x1 , . . . , xd be numbers not all 0 at once. Abbreviate B(t) = (B1 (t), . . . , Bd (t))
(the d-dimensional standard Brownian motion) and x = (x1 , . . . , xd ) (the d-dimensional
vector). Let
M (t) = kB(t) − xk2−d = ((B1 (t) − x1 )2 + · · · + (Bd (t) − xd )2 )1−d/2 .
(a) Prove that M (t) is a martingale relative to the filtration Ft given by the Brownian
motions B1 (s), . . . , Bd (s), s ≤ t.
Hint: Apply the above Ito formula to the function
f (y1 , . . . , yd ) = ky − xk2−d = ((y1 − x2 )2 + · · · + (yd − xd )2 )1−d/2 .
(b) What is E[kB(t) − xk2−d ]?
1
2
The next problem is a bit long, but the steps are not too difficult. So be patient.
Problem 2 Let W (t) be standard Brownian motion. Consider a process D(t) that is the
solution of the stochastic differential equation
d−1
dD(t) =
(1)
dt + dW (t).
2D(t)
Note that D(t) is a one dimensional process. The number d is just a parameter here. We
will only consider the case of integer d ≥ 2, although the process is well defined for any real
number d (even negative!).
Such a process is called a Bessel process. Note that it has diffusion coefficient 1, but it
also has a drift that is proportional to the value of the process. I.e. the closer is D(t) to 0
the more it is pushed away from 0. Furthermore, the larger d is the larger the drift. We
want to prove that if d ≥ 3 then the drift is large enough to make D(t) transient, while if
d = 2, then the drift is not large enough and D(t) is neighborhood recurrent. I.e. we want
to show that: when d ≥ 3 the process has a positive chance of going above any specified
number R and never going back below it, no matter how large R is chosen, while when
d = 2, the process will get eventually smaller than any number ε > 0, no matter how small
ε is chosen to be.
To achieve the above, we will work on recognizing what D(t) really is.
Let d ≥ 2 (the same d as above) and let B1 (t), . . . , Bd (t) be d independent standard
Brownian motions. Define
q
D(t) = B12 (t) + . . . + Bd2 (t).
This D(t) is the distance of the d-dimensional Brownian motion B(t) = (B1 (t), . . . , Bd (t))
to the origin (0, . . . , 0). For now, it is not clear that this D(t) is the same as the Bessel
process defined in (1). But this is what we will prove in parts (b)-(f).
(a) Prove that if it is true that the Bessel process in (1) is indeed simply the distance of
a standard d-dimensional Brownian motion to the origin 0, then the transience/recurrence
properties we are after do hold.
Now we will prove that the distance of a standard d-dimensional Brownian motion is in
fact a Bessel process with parameter d, as in (1).
(b) For any given t > 0, explain why P {D(t) = 0} = 0.
(c) Using the multidimensional Ito formula prove that
dD(t) =
d−1
B1 (t)
Bd (t)
dt +
dB1 (t) + · · · +
dBd (t) .
2D(t)
D(t)
D(t)
The above equation can be written as
dD(t) =
d−1
dt + dW (t)
2D(t)
3
where
Z
W (t) =
0
t
B1 (s)
dB1 (s) + · · · +
D(s)
Z
0
t
Bd (s)
dBd (s) .
D(s)
To show that D(t) satisfies (1) we need to prove that W (t) is in fact a Brownian motion.
Being a stochastic integral, W (t) is continuous and a martingale with respect to the
filtration Ft of the Brownian motions B1 (s), . . . , Bd (s), s ≤ t.
(d) If s < t, what is E[W (t) − W (s)]?
(e) If s < t, what is E[(W (t) − W (s))2 ]?
Hint: you can use the fact that stochastic integrals with respect to independent Brownian
motions are themselves independent. Hence,
2 i
h Z t B (s)
2 i
h Z t B (s)
1
1
2
dB1 (s)
+ ··· + E
dBd (s)
.
E[(W (t) − W (s)) ] = E
0 D(s)
0 D(s)
(The averages of the cross terms cancel out because they split by independence into products
of averages and each of the averages is that of a martingale that is 0 at time 0. Now write
this down carefully and in details!) Then, recall that stochastic integrals have the following
Rt
Rt
isometry property: E[( 0 Y (s) dB(s))2 ] = 0 E[Y (s)2 ] ds. Apply this property to each of
the above averages of squares, and then combine into one integral of one average of a
quantity. See what that becomes!
(f) [Within the scope of the course, but beyond the level of the exam.] Prove that
W (t) − W (s) is Normal with mean 0 and variance t − s and is independent of the Brownian
motions B1 (r), . . . , Bd (r) for r ≤ s and hence is independent of W (r), r ≤ s.
Hint: Fix a real number λ. Use Ito’s formula to show that eλW (t)−λ
2
This shows that E[eλ(W (t)−W (s)) | Fs ] = eλ (t−s)/2 . Conclude.
2 t/2
is a martingale.
As a conclusion, we see that the distance D(t) of a standard d-dimensional Brownian
motion from the origin is in fact a diffusion process that satisfies the stochastic differential
equation (1). This fact helped us deduce transience/recurrence properties of the solution
of (1), as you have done in (a). However, this connection is also useful when studying the
distance of Brownian motion to the origin: describing it as just the distance to the origin
makes it a complicated quantity to analyze, while describing it as a diffusion process (i.e.
as a solution to (1)) shows that it has many nice properties that diffusion processes have.