Itô’s lemma and applications
Hideo NAGAI
Osaka University
1 Itô’s early works on stochastic differential
equations
• Differential equations determining a Markoff processes, Journal of
Pan-Japan mathematical colloquium No.1077(1942) (original Japanese:
Zenkoku Sizyo Sugaku Danwakai-si)
• Stochastic differential equations in a differentiable manifold, Nagoya
Mathematical Journal vol.1, 35-47(1950)
• On stochastic differential equations, Memoirs of American
Mathematical Society vol. 4,1-51 (1951)
• On a formula concerning stochastic differentials, Nagoya math.
Journal vol.3, 55-65 (1951)
This is the cover sheet of
Pan - Japan mathematical colloquium
244 (1942)
This is the first page of
Ito's paper entitled "differential
equations which determine
Markov processes"
2 Chain rule for ODE
• ODE (Ordinary Differential Equation):
dXt
= b(Xt )
dt
2 Chain rule for ODE
• ODE:
dXt
= b(Xt )
dt
• Chain rule for ODE
df (Xt )
= f 0 (Xt )b(Xt )
dt
2 Chain rule for ODE
• ODE:
dXt
= b(Xt )
dt
• Chain rule for ODE
df (Xt )
= f 0 (Xt )b(Xt )
dt
Namely,
df (Xt ) = f 0 (Xt )b(Xt )dt
2 Chain rule for ODE
• ODE:
dXt
= b(Xt )
dt
• Chain rule for ODE
df (Xt )
= f 0 (Xt )b(Xt )
dt
Namely,
f (XT ) − f (X0 )
df (Xt ) = f 0 (Xt )b(Xt )dt
=
RT
0
f 0 (Xt )b(Xt )dt
= lim|∆(T )|→0
P
0
f
(Xtki )b(Xtki )(ti+1 − ti )
i
∆(T ) := {0 = t0 < t1 , .. < tN = T, N = 1, 2, ..}, ti ≤ tki ≤ ti+1
3 Itô’s differential rule
• SDE (Stochastic Differential Equation):
dXt = b(Xt )dt + σ(Xt )dWt
Wt :a standard Brownian motion
3 Itô’s differential rule
• SDE :
dXt = b(Xt )dt + σ(Xt )dWt
Wt :a standard Brownian motion
•Itô’s differential rule
1 00
df (Xt ) = {f (Xt )b(Xt ) + f (Xt )σ(Xt )2 }dt + f 0 (Xt )σ(Xt )dWt
2
0
3 Itô’s differential rule
• SDE :
dXt = b(Xt )dt + σ(Xt )dWt
Wt :a standard Brownian motion
•Itô’s differential rule
1 00
df (Xt ) = {f (Xt )b(Xt ) + f (Xt )σ(Xt )2 }dt + f 0 (Xt )σ(Xt )dWt
2
0
Why
1 00
” f (Xt )σ(Xt )2 ”
2
comes in ?
3 Itô’s differential rule
• SDE
dXt = b(Xt )dt + σ(Xt )dWt
Wt :a standard Brownian motion
•Itô’s differential rule
1 00
df (Xt ) = {f (Xt )b(Xt ) + f (Xt )σ(Xt )2 }dt + f 0 (Xt )σ(Xt )dWt
2
0
Why
1 00
” f (Xt )σ(Xt )2 ”
2
comes in ?
• simplest case (b ≡ 0, σ(x) = 1)
(1)
1 00
df (Wt ) = f (Wt )dt + f 0 (Wt )dWt
2
4 Brownian motion Wt
• Wt is nowhere differentiable
4 Brownian motion Wt
• Wt is nowhere differentiable
• Wt is not of finite variation:
sup
N
X
|Wti+1 − Wti | = ∞,
a.s.
∆(T ) i=0
∆(T ) := {0 = t0 < t1 < t2 , . . . , < tN = T, N = 1, 2, . . .}
4 Brownian motion Wt
• Wt is nowhere differentiable
• Wt is not of finite variation:
sup
N
X
|Wti+1 − Wti | = ∞,
a.s.
∆(T ) i=0
∆(T ) := {0 = t0 < t1 < t2 , . . . , < tN = T, N = 1, 2, . . .}
• dWt cannot be defined in an ordinary sense
4 Brownian motion Wt
• Wt is nowhere differentiable
• Wt is not of finite variation:
sup
N
X
|Wti+1 − Wti | = ∞,
a.s.
∆(T ) i=0
∆(T ) := {0 = t0 < t1 < t2 , . . . , < tN = T, N = 1, 2, . . .}
• dWt cannot be defined in an ordinary sense
• Then, how define dWt ?
4 Brownian motion Wt
• Wt is nowhere differentiable
• Wt is not of finite variation:
sup
N
X
|Wti+1 − Wti | = ∞,
a.s.
∆(T ) i=0
∆(T ) := {0 = t0 < t1 < t2 , . . . , < tN = T, N = 1, 2, . . .}
• dWt cannot be defined in an ordinary sense
• Then, how define dWt ?
• Hint! look at the quadratic variation
X
lim
|Wti+1 − Wti |2 = T
|∆(T )|→0
5 Brownian motion Wt (cont’d)
Note that for 0 = t0 < t1 · · · < tN +1 = T
WT2
−
=
W02
PN
=
PN
2
2
(W
−
W
ti+1
ti )
i=0
i=0 (Wti+1
− Wti ) +
2
PN
i=0
2Wti (Wti+1 − Wti )
5 Brownian motion Wt (cont’d)
Note that for 0 = t0 < t1 · · · < tN +1 = T
−
WT2
=
W02
=
PN
2
2
(W
−
W
ti+1
ti )
i=0
PN
i=0 (Wti+1
− Wti ) +
2
PN
i=0
2Wti (Wti+1 − Wti )
As |∆(T )| → 0 we have
Z
WT2 − W02 = T +
(5.1)
T
2Wt dWt
0
if we define
Z
T
Wt dWt :=
0
lim
|∆|(T )→0
X
Wti (Wti+1 − Wti )
Z
WT2 − W02 = T +
(5.1)
T
2Wt dWt
0
(5.1) means
Z
(5.2)
f (WT ) − f (W0 ) =
0
T
1 00
f (Wt )dt +
2
for f (x) = x2 since f 0 (x) = 2x, f 00 (x) = 2.
Z
0
T
f 0 (Wt )dWt
Z
WT2 − W02 = T +
(5.1)
T
2Wt dWt
0
(5.1) means
Z
(5.2)
f (WT ) − f (W0 ) =
0
T
1 00
f (Wt )dt +
2
Z
T
f 0 (Wt )dWt
0
for f (x) = x2 since f 0 (x) = 2x, f 00 (x) = 2.
For general f (5.2) can be seen by using Taylor expansion
f (Wti+1 ) − f (Wti ) = f 0 (Wti )(Wti+1 − Wti )
+ 12 f 00 (Wti )(Wti+1 − Wti )2 + o(|Wti+1 − Wti |2 )
6 Brownian motion Wt (cont’d)
P
i
Wti+1 (Wti+1 − Wti )
=
P
− Wti )(Wti+1 − Wti )
P
+ i Wti (Wti+1 − Wti )
i (Wti+1
implies
lim
|∆(T )|→0
X
i
Z
T
Wti+1 (Wti+1 − Wti ) = T +
Wt dWt
0
6 Brownian motion Wt (cont’d)
P
i
Wti+1 (Wti+1 − Wti )
=
P
− Wti )(Wti+1 − Wti )
P
+ i Wti (Wti+1 − Wti )
i (Wti+1
implies
lim
X
|∆(T )|→0
By defining
Z T
0
Z
T
Wti+1 (Wti+1 − Wti ) = T +
Wt dWt
0
i
X Wti+1 + Wti
(Wti+1 − Wti )
Wt ◦ dWt := lim
2
|∆(T )|→0
i
we see that
lim
|∆(T )|→0
X
i
Z
Wti+1 (Wti+1 − Wti ) =
0
T
1
Wt ◦ dWt + T
2
7 Definition of stochastic integral
Itô integral
For predictable ϕ(t, ω)
Z
T
ϕ(t, ω)dWt :=
0
lim
|∆(T )|→0
X
i
ϕ(ti , ω)(Wti+1 − Wti )
7 Definition of stochastic integral
Itô integral
For predictable ϕ(t, ω)
Z
T
ϕ(t, ω)dWt :=
0
lim
X
|∆(T )|→0
ϕ(ti , ω)(Wti+1 − Wti )
i
Stratonovich integral
For semimartingale Y (t, ω)
RT
0
Y (t, ω) ◦ dWt
:=
=
=
lim|∆(T )|→0
P
Y (ti+1 )+Y (ti )
(Wti+1
i
2
P
− Wti )
lim|∆(T )|→0 i 2Y (ti )+Y (t2i+1 )−Y (ti ) (Wti+1 − Wti )
RT
1
Y
dW
+
t
t
2 hY, W it
0
8 Itô’s differential rule (II)
• Semi-martingale:
Z
Z
t
Xt = X 0 +
t
σ(Xs )dWs ≡ X0 + At + Mt
b(Xs )ds +
0
0
8 Itô’s differential rule (II)
• Semi-martingale:
Z
Z
t
Xt = X 0 +
t
σ(Xs )dWs ≡ X0 + At + Mt
b(Xs )ds +
0
0
•Itô’s differential rule for a semi-martingale:
Rt 0
f (Xt ) − f (X0 ) = 0 {f (Xs )b(Xs ) + 21 f 00 (Xs )σ(Xs )2 }ds
Rt 0
+ 0 f (Xs )σ(Xs )dWs
=
=
Z
hM, M it =
Rt
0
Rt
0
0
f (Xs )dXs +
Rt
0
f 00 (Xs )dhM, M is
f 0 (Xs ) ◦ dXs
t
σ(Xs )2 ds,
0
1
2
hf 0 (X. ), X. it =
Z
t
0
f 00 (Xs )σ(Xs )2 ds
9 Stochastic exponential
Let
Rt
σ(Xs )dWs
0
Yt = e
≡ eMt ,
f (x) = ex
9 Stochastic exponential
Let
Rt
σ(Xs )dWs
0
Yt = e
≡ eMt ,
f (x) = ex
Then, by Itô’s formula we have
Yt
df (Mt ) = dYt = Yt σ(Xt )dWt + σ(Xt )2 dt,
2
namely,
dYt
1
= σ(Xt )dWt + σ(Xt )2 dt
Yt
2
9 Stochastic exponential
Let
Rt
σ(Xs )dWs
0
Yt = e
≡ eMt ,
f (x) = ex
Then, by Itô’s formula we have
1
df (Mt ) = dYt = Yt σ(Xt )dWt + Yt σ(Xt )2 dt,
2
namely,
Thus we see that
satisfies
dYt
1
= σ(Xt )dWt + σ(Xt )2 dt
Yt
2
Rt
Rt
1
σ(Xt )dWt − 2
σ(Xt )2 dt
0
Zt = e 0
dZt
= σ(Xt )dWt
Zt
• Differential equations determining a Markoff processes, Journal of
Pan-Japan mathematical colloquium No.1077(1942) (original Japanese:
Zenkoku Sizyo Sugaku Danwakai-si)
• Stochastic differential equations in a differentiable manifold, Nagoya
Mathematical Journal vol.1, 35-47(1950)
• On stochastic differential equations, Memoirs of American
Mathematical Society vol. 4,1-51 (1951)
• On a formula concerning stochastic differentials, Nagoya math.
Journal vol.3, 55-65 (1951)
10 Kolmogorov equation
• SDE
(10.1)
dXt = b(Xt )dt + σ(Xt )dWt ,
Xtx : a solution to (10.1)
X0 = x
10 Kolmogorov equation
• SDE
(10.1)
dXt = b(Xt )dt + σ(Xt )dWt ,
X0 = x
Xtx : a solution to (10.1)
• Itô’s differential rule for u = u(t, x)
du(t, Xtx )
=
x
(t,
X
)
{ ∂u
t
∂t
+
∂u
x
x
(t,
X
)b(X
)
t
t
∂x
+
1 ∂2u
x
x 2
(t,
X
)σ(X
) }dt
2
t
t
2 ∂x
x
x
+ ∂u
(t,
X
)σ(X
t
t )dWt
∂x
x
= { ∂u
(t,
X
t ) + Lu(t, Xt )}dt +
∂t
where
∂u
x
x
(t,
X
)σ(X
t
t )dWt
∂x
2
∂u
1
∂
u
2
Lu(t, x) := b(x) (t, x) + σ(x)
(t, x)
2
∂x
2
∂x
11 Kolmogorov equation (cont’d)
Set u(s, x) := E[g(XTx −s )] . If it is smooth, then it satisfies

∂u
0≤s<T

∂t (s, x) + Lu(s, x) = 0,

u(T, x) = g(x)
11 Kolmogorov equation (cont’d)
Set u(s, x) := E[g(XTx −s )] . If it is smooth, then it satisfies

∂u
0≤s<T

∂t (s, x) + Lu(s, x) = 0,

u(T, x) = g(x)
Because Itô’s formula applied to u(t, x) gives us
R t−s ∂u
x
u(t, Xt−s ) − u(s, x) = 0 {( ∂t + Lu)(s + r, Xrx )}dr
+
R t−s
0
∂u
x
x
(t,
X
)σ(X
t
t )dWt
∂x
and Markov property implies
x
]]
u(s, x) = E[g(XTx −s )] = E[E[g(XTx −s )|Xt−s
¯
y
x
x
]] = E[u(t, Xt−s
)]
= E[ E[g(XT −t )¯y=X x |Xt−s
t−s
12 Optimal portfolio selection
Market model
Price of a riskless asset:
dSt0 = St0 r(Xt )dt,
S00 = 1
Price of a risky asset:
dSt1 = St1 {α(Xt )dt +
2
X
σj (Xt )dWtj },
S01 = s0
j=1
Economic factor:
dXt = β(Xt )dt +
2
X
J=1
λj (Xt )dWtj ,
X0 = x
Wealth process
Vt = Nt0 St0 + Nt1 St1
Nti , i = 0, 1: numbers of share invested to i-th security Sti
Wealth process
Vt := Nt0 St0 + Nt1 St1
Nti , i = 0, 1: numbers of share invested to i-th security Sti
Portfolio proportion
i i
N
t St
i
ht :=
, i = 0, 1,
Vt
h0t + h1t = 1
Denote h1t by ht in what follows since h0t = 1 − h1t .
Wealth process
Vt := Nt0 St0 + Nt1 St1
Nti , i = 0, 1: numbers of share invested to i-th security Sti
Portfolio proportion
i i
N
t St
i
ht :=
, i = 0, 1,
Vt
h0t + h1t = 1
Denote h1t by ht in what follows since h0t = 1 − h1t .
Dynamics of the wealth process
dVt = Vt {r(Xt ) + ht (α(Xt ) − r(Xt ))}dt + Vt ht σ(Xt )dWt ,
V0 = v0
Dynamics of the wealth process
dVt = Vt {r(Xt ) + ht (α(Xt ) − r(Xt ))}dt + Vt ht σ(Xt )dWt ,
V0 = v0
Dynamics of the wealth process
dVt = Vt {r(Xt ) + ht (α(Xt ) − r(Xt ))}dt + Vt ht σ(Xt )dWt ,
V0 = v0
Since hV it =
Rt
0
Vs2 h2s σσ ∗ (Xs )ds we have by Itô’s formula
dVtγ = γVtγ−1 dVt + 12 γ(γ − 1)Vtγ−2 dhV it
= γVtγ [{r(Xt ) + ht (α(Xt ) − r(Xt ))}dt + ht σ(Xt )dWt ]
+ 21 γ(γ − 1)Vtγ h2t σσ ∗ dt
Dynamics of the wealth process
dVt = Vt {r(Xt ) + ht (α(Xt ) − r(Xt ))}dt + Vt ht σ(Xt )dWt ,
V0 = v0
Since hV it =
Rt
0
Vs2 h2s σσ ∗ (Xs )ds we have by Itô’s formula
dVtγ = γVtγ−1 dVt + 12 γ(γ − 1)Vtγ−2 dhV it
= γVtγ [{r(Xt ) + ht (α(Xt ) − r(Xt ))}dt + ht σ(Xt )dWt ]
+ 21 γ(γ − 1)Vtγ h2t σσ ∗ dt
Therefore we have
dVtγ
Vtγ
= γ{r(Xt ) + ht (α(Xt ) − r(Xt )) + 12 (γ − 1)h2t σσ ∗ (Xt )}dt
+γht σ(Xt )dWt
Once more again by Itô’s formula we have
RT
RT
RT 2 ∗
γ2
γ
η(Xs ,hs )ds+γ
hs σ(Xs )dWs − 2
hs σσ (Xs )ds
0
0
VTγ = v0γ e 0
where
1−γ 2 ∗
h σσ (x) + h(α(x) − r(x)) + r(x)
η(x, h) = −
2
Once more again by Itô’s formula we have
RT
RT
RT 2 ∗
γ2
γ
η(Xs ,hs )ds+γ
hs σ(Xs )dWs − 2
hs σσ (Xs )ds
0
0
VTγ = v0γ e 0
where
1−γ 2 ∗
h σσ (x) + h(α(x) − r(x)) + r(x)
η(x, h) = −
2
Minimizing
log E[VT (h)γ ]
by selecting h amounts to minimizing
RT 2 ∗
RT
RT
γ2
hs σσ (Xs )ds
γ
η(Xs ,hs )ds+γ
hs σ(Xs )dWs − 2
0
0
]
log E[v0γ e 0
Utility maximization and risk-sensitive portfolio optimization
Power utility maximization:
1
sup E[VT (h)γ ],
h γ
γ<0
is equivalent to risk-senstive portfolio optimization:
inf log E[VT (h)γ ],
h
γ<0
Utility maximization and risk-sensitive portfolio optimization
Power utility maximization:
1
sup E[VT (h)γ ],
h γ
γ<0
is equivalent to risk-senstive portfolio optimization:
inf log E[VT (h)γ ],
h
γ<0
and so
inf
h
γ γ
log E[v0 e
RT
0
η(Xs ,hs )ds+γ
RT
0
2
hs σ(Xs )dWs − γ2
RT
0
h2s σσ ∗ (Xs )ds
]
Through change of measure
¯
RT
2 RT 2
h¯
γ
dP ¯
γ
hs σ(Xs )dWs − 2
hs σσ ∗ (Xs )ds
0
=e 0
,
¯
dP FT
inf
h
γ
log E[v0γ e
RT
0
η(Xs ,hs )ds+γ
RT
0
turns out to be
γ
inf log E h [v0γ e
h
2
hs σ(Xs )dWs − γ2
RT
0
η(Xs ,hs )ds
]
RT
0
h2s σσ ∗ (Xs )ds
]
Through change of measure
¯
RT
2 RT 2
h¯
γ
dP ¯
γ
hs σ(Xs )dWs − 2
hs σσ ∗ (Xs )ds
0
=e 0
,
¯
dP FT
inf
h
γ
log E[v0γ e
RT
0
η(Xs ,hs )ds+γ
RT
0
turns out to be
γ
inf log E h [v0γ e
h
γ2
hs σ(Xs )dWs − 2
RT
0
η(Xs ,hs )ds
RT
0
h2s σσ ∗ (Xs )ds
]
]
whose H-J-B (Hamilton-Jacobi-Bellman) equation is
(12.1)
∂v
1
1
∗ ∂2v
∗ ∂v 2
+
λλ
+
λλ
| ∂x |
2
∂t
2
∂x
2
∂v
+ inf h {[β(x) + γλσ ∗ (x)h] ∂x
+ γη(x, h)} = 0
v(T, x) = γ log v0
Analysis of the H-J-B equation (12.1) gives us optimal portfolio strategy
ĥ(t, Xt )
under suitable conditions, where
1
∂v
∗
−1
∗
(σσ (x)) (α(x) − r(x)) + σλ (x) (t, x).
ĥ(t, x) =
1−γ
∂x
∂v
and ∂x
(t, x) is the solution of the H-J-B equation.
Namely, ĥ(t, Xt ) attains the minimum of
RT
γ
η(Xs ,hs )ds
log E h [v0γ e 0
], γ < 0
which is equivalent to utility maximization.
Down-side risk minimization
Consider a down-side risk minimization on a long term:
1
1
J(κ) := inf lim inf log P ( log VT (h) ≤ κ)
h T →∞ T
T
for a given target growth rate κ
and risk-sensitive portfolio optimization on a long term:
1
χ(γ) := inf lim inf log E[VT (h)γ ]
h T →∞ T
Down-side risk minimization
Consider a down-side risk minimization on a long term:
1
1
J(κ) := inf lim inf log P ( log VT (h) ≤ κ)
h T →∞ T
T
for a given target growth rate κ
and risk-sensitive portfolio optimization on a long term:
1
χ(γ) := inf lim inf log E[VT (h)γ ]
h T →∞ T
We can show that
J(κ) = −
inf
sup{γk − χ(γ)} = − sup{γκ − χ(γ)}
k∈(−∞,κ] γ<0
γ<0
for limγ→∞ χ0 (γ) < κ < χ0 (0−) under suitable conditions.
Moreover, for given target growth rate κ we take γ(κ) which attains
sup{γκ − χ(γ)} = γ(κ)κ − χ(γ(κ))
γ<0
γ(κ)
and choose an optimal strategy ĥt
which minimize
1
lim inf log E[VT (h)γ ].
T →∞ T
γ(κ)
Then, this ĥt
attains the minimum of
1
1
lim inf log P ( log VT (h) ≤ κ)
T →∞ T
T
Moreover, for given target growth rate κ we take γ(κ) which attains
sup{γκ − χ(γ)} = γ(κ)κ − χ(γ(κ))
γ<0
γ(κ)
and choose an optimal strategy ĥt
which minimize
1
lim inf log E[VT (h)γ ].
T →∞ T
γ(κ)
Then, this ĥt
attains the minimum of
1
1
lim inf log P ( log VT (h) ≤ κ)
T →∞ T
T
γ(κ)
we can obtain ĥt
through analysis of H-J-B equation of ergodic type
corresponding to risk-sensitive portfolio optimization:
1
χ(γ) := inf lim inf log E[VT (h)γ ].
h T →∞ T
Thank you for your attention !