E F s , s t and, after some rearranging, 1 2 ( t s ) 2 . - - PowerPoint PPT Presentation

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Apr 01, 2023 212 likes •351 views

L evys Characterization of Brownian Motion A continuous P , {F t } t 0 -martingale { W t } t 0 with W 0 = 0 and quadratic variation [ W ] t = t is -Brownian motion. P , {F t } t 0 1 Write W t

SLIDE 1

L´ evy’s Characterization of Brownian Motion

A continuous
P, {Ft}t≥0
martingale {Wt}t≥0 with W0 = 0

and quadratic variation [W]t = t is

P, {Ft}t≥0
Brownian motion.

1

SLIDE 2

Write

M(θ)

t △

= exp

θWt − θ2

2 t

.
Then by Itˆ
’s formula (generalized!),

dM(θ)

t

= θ exp

θWt − θ2

2 t

dWt − 1

2θ2M(θ)

t

dt + 1 2θ2M(θ)

t

dt = θ exp

θWt − θ2

2 t

dWt.
So {M(θ)

t

}t≥0 is a

P, {Ft}t≥0
martingale.

2

SLIDE 3

So for 0 ≤ s < t,

E

M(θ)

t

Fs
= M(θ)

s

, and, after some rearranging,

E

eθ(Wt−Ws)
Fs
= e

1 2(t−s)θ2.

That is, Wt − Ws ∼ N(0, t − s) and is independent of Fs.

3

SLIDE 4

Stochastic Differential Equation

Suppose that {Wt}t≥0 is
P, {Ft}t≥0
Brownian motion, and

Zt = f(t, Wt).

Then Itˆ
’s formula is

dZt =

f(t, Wt) + 1 2f′′(t, Wt)

dt + f′(t, Wt)dWt.
If f is invertible, this may be written

dZt = µ(t, Zt)dt + σ(t, Zt)dWt.

4

SLIDE 5

An equation of this form with deterministic functions µ(t, x)

and σ(t, x) is a stochastic differential equation for {Zt}t≥0. – Note that for given µ(t, x) and σ(t, x), there may or may not exist a solution; that is, a function f such that Zt = f(t, Wt); – And if a solution exists, it may or may not be unique. – If µ(t, x) and σ(t, x) are bounded and uniformly continu-

us in x, a unique solution exists; these conditions are

therefore sufficient, but far from necessary.

5

SLIDE 6

The quadratic variation of {Zt}t≥0 is

[Z]t =

t

0 σ(s, Zs)2ds.

In terms of differentials:

d[Z]t = σ(t, Zt)2dt.

6

SLIDE 7

Integration By Parts

The classical “integration by parts” formula

b

a f(x)g′(x)dx = [f(x)g(x)]b a −

b

a f′(x)g(x)dx

is a consequence of the rule for the derivative of a product, [f(x)g(x)]′ = f′(x)g(x) + f(x)g′(x).

If f and g are replaced by (semi-)martingales, their covaria-

tion must be included.

7

SLIDE 8

Suppose that {Yt}t≥0 is a
P, {Ft}t≥0
semi-martingale:

Yt = M(Y )

t

+ A(Y )

t

where: – {M(Y )

t

}t≥0 is a

P, {Ft}t≥0
martingale:

– {A(Y )

t

}t≥0 is a continuous process of bounded variation.

Similarly Zt = M(Z)

t

+ A(Z)

t

.

Then

d(YtZt) = YtdZt + ZtdYt + d

M(Y ), M(Z)

t .

8

SLIDE 9

Here
MY , MZ

t is the covariation (or mutual variation) of

{M(Y )

t

}t≥0 and {M(Z)

t

}t≥0:

M(Y ), M(Z)

t =

lim

δ(π)→0 N(π)−1

j=0
M(Y )

tj+1 − M(Y ) tj

M(Z)

tj+1 − M(Z) tj

where π is a partition of [0, t].
Note that
M(Y ), M(Y )

t =

M(Y )

t ;

– that is, the covariation of a process with itself is just its quadratic variation.

9

SLIDE 10

For example, if

dYt = µ(t, Yt)dt + σ(t, Yt)dWt and dZt = ˜ µ(t, Zt)dt + ˜ σ(t, Zt)dWt and {M(Y )

t

}t≥0 and {M(Z)

t

}t≥0 are the

P, {Ft}t≥0
martingales

M(Y )

t

=

t

0 σ(s, Ys)dWs,

M(Z)

t

=

t

0 ˜

σ(s, Zs)dWs, then

M(Y ), M(Z)

t =

t

0 σ(s, Ys)˜

σ(s, Zs)ds.

10

SLIDE 11

In that example, {Yt}t≥0 and {Zt}t≥0 are driven by the same

Brownian motion {Wt}t≥0.

More generally, if instead

dZt = ˜ µ(t, Zt)dt + ˜ σ(t, Zt)d ˜ Wt and {M(Z)

t

}t≥0 is the martingale M(Z)

t

=

t

0 ˜

σ(s, Zs)d ˜ Ws, where Wt and ˜ Wt have correlation ρ, then

M(Y ), M(Z)

t =

t

0 σ(s, Ys)˜

σ(s, Zs)ρds.

11