Martingales Again As in discrete time, we need a probability space ( - - PowerPoint PPT Presentation

Martingales Again

• As in discrete time, we need a probability space (Ω, F, P) and

a filtration {Ft}t≥0: Ft is a sub-σ-field of F and for 0 ≤ s < t, Fs ⊆ Ft.

• The natural filtration generated by a stochastic process {Xt}t≥0

is {FX

t }t≥0;

– Here FX

t

is the smallest σ-field with respect to which all Xs are measurable, 0 ≤ s ≤ t.

1

• {Mt}t≥0 is a
• P, {Ft}t≥0
• martingale if

EP[|Mt|] < ∞ for all t ≥ 0, and for any 0 ≤ s ≤ t, EP[Mt|Fs] = Ms.

• More generally, {Mt}t≥0 is a local
• P, {Ft}t≥0
• martingale if

there exists a sequence of stopping times {Tn}n≥1 such that: – {Mt∧Tn}t≥0 is a

• P, {Ft}t≥0
• martingale for each n, and

– P[Tn → ∞ as n → ∞] = 1.

2

• Suppose that {Wt}t≥0 is standard Brownian motion, and

{Ft}t≥0 is the natural filtration. Then: – {Wt}t≥0 is a

• P, {Ft}t≥0
• martingale;

–

• W 2

t − t

• t≥0 is a
• P, {Ft}t≥0
• martingale;

–

• exp
• σWt − σ2

2 t

• t≥0

is a

• P, {Ft}t≥0
• martingale.

3

• Optional Stopping Theorem: if {Mt}t≥0 is a continuous
• P, {Ft}t≥0
• martingale (or more generally, almost surely c`

adl` ag), and if τ1 ≤ τ2 are two bounded stopping times, then E

|Mτ2| < ∞,

and E

Mτ2| Fτ1 = Mτ1,

with P-probability 1.

4

• We can use this theorem for instance to find the moment

generating function (Laplace transform) of the distribution

• f the hitting time Ta,

E

• e−θTa

.

• Suppose that a > 0; recall that

Mt = exp

• σWt − σ2

2 t

• is a martingale.
• Take τ1 = 0 and τ2 = Ta.

5

• So

1 = M0

?

= E

• MTa
• = eσaE
• e−σ2

2 Ta

• .
• The argument fails at “ ?

=”, because Ta is not bounded.

• It can be fixed by taking τ2 = Ta ∧ n, and letting n → ∞.
• Correct:

1 = M0 = E

• MTa∧n
• = E
• MTa∧n1{Ta≤n}
• + E
• MTa∧n1{Ta>n}
• = E
• MTa1{Ta≤n}
• + E
• Mn1{Ta>n}
• → E
• MTa
• + 0.

6

• So

E

• e−σ2

2 Ta

• = e−σa,
• r

E

• e−θTa

= e−

√ 2θa.

• For a < 0, we use

Mt = exp

• −σWt − σ2

2 t

• as the martingale, and find the general result

E

• e−θTa

= e−

√ 2θ|a|.

7