LINKÖPING UNIVERSITY
Department of Mathematics
Mathematical Statistics
John Karlsson
TAMS29
Stochastic Processes with
Applications in Finance
11. More on Itô processes
Theorem 11.1. (Optional stopping theorem) Let Xt be a martingale and τ be a
stopping time. Assume that any of the following holds
(a) τ is almost surely bounded, i.e. there exists c ∈ R such that
τ ≤ c a.s. (b) E[τ ] < ∞ and for each h > 0 there exists c ∈ R such that E |Xt+h −Xt |Ft ≤ c.
+
(c) There exists c ∈ R such that Xτ ∧t ≤ c for all
t ∈ R .
(d) τ < ∞ almost surely, E[|Xτ |] < ∞, and E Xt 1{τ >t} → 0 as n → ∞.
Then
E[Xτ ] = E[X0 ].
Remark 11.2. The optional stopping theorem states that under well behaved
stopping times τ we have
E[Mτ ] = E[M0 ],
for a martingale Mt .
Theorem 11.3. (Monotone convergence) Let fn be a pointwise non-decreasing
sequence of [0, ∞]-valued functions i.e. 0 ≤ f1 (x) ≤ f2 (x) ≤ . . .. Let f be the
pointwise limit i.e. f (x) = limn→∞ fn (x). Then
Z
Z
lim
fn (x) dµ = f (x) dµ.
n→∞
Theorem 11.4. (Dominated convergence) Let fn be a sequence of measurable
functions with pointwise limit f i.e. limn→∞ fn (x) = f (x). Suppose that the
sequence is dominated by an integrable function g, i.e. |fn (x)| ≤ g. Then f is
integrable and
Z
Z
lim
n→∞
fn (x) dµ =
f (x) dµ.
Theorem 11.5. (Fatou’s lemma) Let fn be a sequence of [0, ∞]-valued functions
and defined f (x) := lim inf n→∞ fn (x). Then f is measurable and
Z
Z
Z
f (x) dµ = lim inf fn (x) dµ ≤ lim inf fn (x) dµ.
n→∞
n→∞
Example 11.6. Let
(
fn (x) =
Then limn→∞ fn (x) = 0 and
1
0
n ≤ x ≤ n + 1,
otherwise.
R
fn (x) dx = 1, n = 1, 2, . . .. We have
Z
f (x) dµ = 0,
Z
lim inf fn (x) dµ = 1.
n→∞
Remark 11.7. If limn→∞ fn (x) =: f (x) exists then
f (x) = lim fn (x) = lim inf fn (x) = lim sup fn (x).
n→∞
n→∞
1/2
n→∞
Definition 11.8. A continuous time martingale Xt , t ≥ 0 is a stochastic process
that satisfy
(i) E[|Xt |] < ∞,
(ii) E [Xt |Fs ] = Xs , t ≥ s.
Theorem 11.9. (Itô formula) Let Xt be given by dXt = At dt + Bt dWt . Then
∂f
∂f
Bt2 ∂ 2 f
∂f
dt + Bt dWt .
+ At
+
df (t, Xt ) =
∂t
∂x
2 ∂x2
∂x
Definition 11.10. The exponential martingale Yt associated with the martingale
Mt is given by
[M ]
Yt = exp Mt −
,
2
i.e. under this construction Yt is a martingale.
Theorem 11.11. (Doob’s martingale convergence theorem) Let Mt be a martingale. If
sup E[|Mt |] < ∞,
t
then
M (ω) := lim M (ω)t exists P -almost surely.
t→∞
2/2