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Introduction to Ito’s Lemma
Wenyu Zhang
Cornell UniversityDepartment of Statistical Sciences
May 6, 2015
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Overview
1 Background
2 Ito Processes
3 Ito’s Lemma
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Background
Proved by Kiyoshi Ito (not Ito’s theorem on group theory by NoboruIto)
Used in Ito’s calculus, which extends the methods of calculus tostochastic processes
Applications in mathematical finance e.g. derivation of theBlack-Scholes equation for option values
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Ito Processes
Question
Want to model the dynamics of process X (t) driven by Brownian motionW (t).
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Ito Processes: Discrete-time Construction
Partition time interval [0,T] into N periods, each of length ∆t = TN ;
tn = n∆t
Xtn+1 = Xtn + µtn∆t + σtn∆Wtn
I drift µI volatility σI fluctuations ∆Wtn = Wtn+1 −Wtn ∼ N(0,∆t)
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Ito Processes: Discrete-time Construction
Summing the increments,
XT = X0 +N−1∑n=0
µtn∆t +N−1∑n=0
σtn∆Wtn
Continuous-time analogue as N →∞,
XT?−→ X0 +
∫ T
0µtdt +
∫ T
0σtdWt
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Ito Processes: Discrete-time Construction
Regularity conditions for µt and σt
adapted to FWt
continuous in t
integrability conditions (σt ∈M2 i.e. E(∫ T
0 σ2t dt)<∞)
Riemann-Stieltjes integral
N−1∑n=0
µtn∆t →∫ T
0µtdt
Ito integral
N−1∑n=0
σtn∆WtnL2−→∫ T
0σtdWt
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Ito Integrals
Question
What is∫ T0 σtdWt?
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Ito Integrals
Theorem (Existence and Uniqueness of Ito Integral)
Suppose that vt ∈M2 satisfies the following: For all t ≥ 0,A1) vt is a.s. continuousA2) vt is adapted to FW
t
Then, for any T > 0, the Ito integral IT (v) =∫ T0 vtdWt exists and is
unique a.e.
Steps for proof
1 Construct a sequence of adapted stochastic processes vn such that
‖v − vn‖M2 =
√E(∫ T
0 |vn(t)− v(t)|2dt)→ 0
2 Show that ‖IT (vn)− IT (v)‖L2 → 0
3 Show the a.s. uniqueness of the limit IT (v)
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Ito Integrals: Example
Example (Ito Integral)∫ T
0WtdWt with approximating sums
N−1∑n=0
Wtn∆Wtn
N−1∑n=0
Wtn(Wtn+1 −Wtn) =N−1∑n=0
[1
2(W 2
tn+1−W 2
tn)− 1
2(Wtn+1 −Wtn)2
]
=1
2W 2
T −1
2
N−1∑n=0
(Wtn+1 −Wtn)2
L2−→ 1
2W 2
T −1
2T as N →∞
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Ito Integrals: Example
Example (Riemann-Stieltjes Integral)∫ T
0GtdGt with G ∈ C1,G (0) = 0
∫ T
0GtdGt =
∫ T
0GtG
′tdt
=1
2G 2T
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Ito Integral: Properties
Linear in the integrandTime-additiveMartingaleProofFor t < T , increase the partition by an extra point tk = t.
E [IT (vn)− It(vn)|Ft ] = E
[N−1∑n=k
σtn∆Wtn |Ft
]
= E
[N−1∑n=k
E (σtn∆Wtn |Ftn)|Ft
]
= E
[N−1∑n=k
σtnE (∆Wtn |Ftn)|Ft
]= 0
IT (vn) is a martingale. Martingales are preserved under L2 limits.
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Ito Processes
Xt − X0 =
∫ t
0µsds +
∫ t
0σsdWs
SDE notation:
dXt = µtdt + σtdWt
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Ito’s Lemma
Theorem (Ito’s Lemma)
Suppose that f ∈ C2. Then with probability one, for all t ≥ 0,
df (Xt) =∂f
∂x(Xt)dXt +
1
2
∂2f
∂x2(Xt)(dXt)
2
f (Xt)− f (X0) =
∫ t
0f ′(Xs)dXs +
1
2
∫ t
0f ′′(Xs)ds
Explicit statement:
df (Xt) =
(µt∂f
∂x(Xt) +
1
2σ2t∂2f
∂x2(Xt)
)dt + σt
∂f
∂x(Xt)dWt
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Ito’s Lemma: Idea
Can be obtained heuristically by second order Taylor expansion of fabout Xt
(dXt)2 = (µtdt + σtdWt)
2 term cannot be dropped
I (dWt)2 = dt
drop terms � dt∫ T
0(dt)p = lim
N→∞
N−1∑n=0
(∆t)p
= limN→∞
N
(T
N
)p
= 0 as N →∞ if p > 1
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Ito’s Lemma: Idea
(dWt)2 = dt since
∫ T
0(dWt)
2 = limN→∞
N−1∑n=0
(∆Wtn)2L2= T =
∫ T
0dt
E
[N−1∑n=0
(∆Wtn)2
]=
N−1∑n=0
E (∆Wtn)2 =N−1∑n=0
∆t = T
E
[N−1∑n=0
(∆Wtn)2 − T
]2= Var
[N−1∑n=0
(∆Wtn)2
]=
N−1∑n=0
Var(∆Wtn)2
=N−1∑n=0
(E (∆W )4 − [E (∆W )2]2) =N−1∑n=0
(3(∆t)2 − (∆t)2)
=2T 2
N→ 0 as N →∞
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Ito’s Lemma: More details
More rigorously, for bounded continuous g ,
N−1∑n=0
g(Wtn)(∆Wtn)2L2−→∫ T
0g(Wt)dt as N →∞
ProofSince t 7→ g(Wt) is a.s. continuous,
∑N−1n=0 g(Wtn)∆t →
∫ T0 g(Wt)dt.
WTS:
IN =N−1∑n=0
g(Wtn)[(∆Wtn)2 −∆t
] L2−→ 0
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Ito’s Lemma: More details
1. E (I 2N) = E{∑N−1
n=0 (g(Wtn))2[(∆Wtn)2 −∆t]2}
E{
g(Wtn)g(Wtm)[(∆Wtn)2 −∆t][(∆Wtm)2 −∆t]}
= E{
E(g(Wtn)g(Wtm)[(∆Wtn)2 −∆t][(∆Wtm)2 −∆t]|Ftm
)}= E
{g(Wtn)g(Wtm)[(∆Wtn)2 −∆t]E
[(∆Wtm)2 −∆t|Ftm
]}= 0 since {W 2
t − t} is martingale
2. E (I 2N) ≤ CT 2
N
E (I 2N) ≤ ‖g‖2∞N−1∑n=0
E [(∆Wtn)2 −∆t]2 since g is bounded
= ‖g‖2∞N−1∑n=0
(∆t)2E (W 21 − 1)2 since ∆Wtn ∼
√∆tW1
= CN−1∑n=0
(T
N
)2
=CT 2
N
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Ito’s Lemma: Example
Example (Ito’s Lemma)
Use Ito’s Lemma, write Zt = W 2t as a sum of drift and diffusion terms.
Zt = f (Xt) with µt = 0, σt = 1,X0 = 0, f (x) = x2
dZt = df (Xt)
= f ′(Xt)dXt +1
2f ′′(Xt)(dXt)
2
= 2WtdWt +1
22(dWt)
2
= 2WtdWt + dt
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Ito’s Lemma: Higher dimensions
Ito’s Lemma
If Xt and Yt are Ito processes and f : R2 7→ R is sufficiently smooth, then
df (Xt ,Yt) =∂f
∂xdXt +
∂f
∂ydYt
+1
2
∂2f
∂x2(dXt)
2 +1
2
∂2f
∂y2(dYt)
2 +∂2f
∂x∂y(dXt)(dYt)
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References
Rene L. Schilling/Lothar Partzsch
Brownian Motion - An Introduction to Stochastic Processes (2012)
CUHK course notes (2013)
Chapter 6: Ito’s Stochastic Calculus
Karl Sigman
Columbia course notes (2007)
Introduction to Stochastic Integration
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