Ein stochastischer Prozess
(
X
t
)
t
≥
0
{\displaystyle (X_{t})_{t\geq 0}}
Itō-Prozess , falls
X
t
=
X
0
+
∫
0
t
a
s
d
s
+
∫
0
t
b
s
d
W
s
{\displaystyle X_{t}=X_{0}+\int _{0}^{t}a_{s}\,{\rm {d}}s+\int _{0}^{t}b_{s}\,{\rm {d}}W_{s}}
für zwei stochastische Prozesse
a
s
{\displaystyle a_{s}}
b
s
{\displaystyle b_{s}}
stochastische Integration ). In Differentialschreibweise:
d
X
t
=
a
t
d
t
+
b
t
d
W
t
.
{\displaystyle {\rm {d}}X_{t}=a_{t}\,{\rm {d}}t+b_{t}\,{\rm {d}}W_{t}\,.}
Ist
h
:
R
+
×
R
→
R
{\displaystyle h\colon \mathbb {R} _{+}\times \mathbb {R} \to \mathbb {R} }
Y
t
:=
h
(
t
,
X
t
)
{\displaystyle Y_{t}:=h(t,X_{t})}
[ 2]
d
Y
t
=
∂
h
∂
t
(
t
,
X
t
)
d
t
+
∂
h
∂
x
(
t
,
X
t
)
d
X
t
+
1
2
∂
2
h
∂
x
2
(
t
,
X
t
)
(
d
X
t
)
2
=
(
∂
h
∂
x
(
t
,
X
t
)
a
t
+
∂
h
∂
t
(
t
,
X
t
)
+
1
2
∂
2
h
∂
x
2
(
t
,
X
t
)
b
t
2
)
d
t
+
∂
h
∂
x
(
t
,
X
t
)
b
t
d
W
t
.
{\displaystyle {\begin{aligned}{\rm {d}}Y_{t}&={\frac {\partial h}{\partial t}}(t,X_{t})\,{\rm {d}}t+{\frac {\partial h}{\partial x}}(t,X_{t})\,{\rm {d}}X_{t}+{\frac {1}{2}}{\frac {\partial ^{2}h}{\partial x^{2}}}(t,X_{t})({\rm {d}}X_{t})^{2}\\&=\left({\frac {\partial h}{\partial x}}(t,X_{t})\,a_{t}+{\frac {\partial h}{\partial t}}(t,X_{t})+{\frac {1}{2}}{\frac {\partial ^{2}h}{\partial x^{2}}}(t,X_{t})\,b_{t}^{2}\right){\rm {d}}t+{\frac {\partial h}{\partial x}}(t,X_{t})\,b_{t}\,{\rm {d}}W_{t}\,.\end{aligned}}}
Hierbei bezeichnen
∂
h
∂
t
{\displaystyle {\tfrac {\partial h}{\partial t}}}
∂
h
∂
x
{\displaystyle {\tfrac {\partial h}{\partial x}}}
partiellen Ableitungen der Funktion
h
{\displaystyle h}
(
d
X
t
)
2
=
b
t
2
d
t
{\displaystyle ({\rm {d}}X_{t})^{2}=b_{t}^{2}\,{\rm {d}}t}
d
t
{\displaystyle {\rm {d}}t}
d
W
t
{\displaystyle {\rm {d}}W_{t}}
Die Formel lässt sich auf
n
{\displaystyle n}
X
=
(
X
1
,
…
,
X
n
)
{\displaystyle X=(X_{1},\dots ,X_{n})}
h
:
[
0
,
∞
)
×
R
n
{\displaystyle h:[0,\infty )\times \mathbb {R} ^{n}}
C
1
{\displaystyle C^{1}}
C
2
{\displaystyle C^{2}}
Y
(
t
)
:=
h
(
t
,
X
(
t
)
)
{\displaystyle Y(t):=h(t,X(t))}
d
Y
(
t
)
=
∂
h
∂
t
(
t
,
X
(
t
)
)
d
t
+
∑
i
=
1
n
∂
h
∂
i
(
t
,
X
(
t
)
)
d
X
i
(
t
)
+
1
2
∑
i
,
j
=
1
n
∂
2
h
∂
i
∂
j
(
t
,
X
(
t
)
)
d
[
X
i
,
X
j
]
(
t
)
.
{\displaystyle {\rm {d}}Y(t)={\frac {\partial h}{\partial t}}(t,X(t))\,{\rm {d}}t+\sum \limits _{i=1}^{n}{\frac {\partial h}{\partial i}}(t,X(t))\,{\rm {d}}X_{i}(t)+{\frac {1}{2}}\sum \limits _{i,j=1}^{n}{\frac {\partial ^{2}h}{\partial i\partial j}}(t,X(t)){\rm {d}}[X_{i},X_{j}](t).}
Sei
(
X
t
)
t
≥
0
=
(
X
t
1
,
…
,
X
t
d
)
t
≥
0
{\displaystyle (X_{t})_{t\geq 0}=(X_{t}^{1},\dotsc ,X_{t}^{d})_{t\geq 0}}
R
d
{\displaystyle \mathbb {R} ^{d}}
Semimartingal und sei
F
∈
C
2
(
R
d
,
R
)
{\displaystyle F\in C^{2}(\mathbb {R} ^{d},\mathbb {R} )}
(
F
(
X
t
)
)
t
≥
0
{\displaystyle (F(X_{t}))_{t\geq 0}}
F
(
X
t
)
−
F
(
X
0
)
=
∑
j
=
1
d
∫
0
+
t
∂
F
∂
x
j
(
X
s
−
)
d
X
s
j
+
1
2
∑
j
,
k
=
1
d
∫
0
+
t
∂
2
F
∂
x
j
∂
x
k
(
X
s
−
)
d
[
X
j
,
X
k
]
s
c
+
∑
0
<
s
≤
t
(
F
(
X
s
)
−
F
(
X
s
−
)
−
∑
j
=
1
d
∂
F
∂
x
j
(
X
s
−
)
Δ
X
s
j
)
.
{\displaystyle {\begin{aligned}F(X_{t})-F(X_{0})=&\sum _{j=1}^{d}\int _{0+}^{t}{\frac {\partial F}{\partial x^{j}}}(X_{s-}){\rm {d}}X_{s}^{j}+{\frac {1}{2}}\sum _{j,k=1}^{d}\int _{0+}^{t}{\frac {\partial ^{2}F}{\partial x^{j}\partial x^{k}}}(X_{s-}){\rm {d}}[X^{j},X^{k}]_{s}^{c}\\&{}+\sum _{0<s\leq t}\left(F(X_{s})-F(X_{s-})-\sum _{j=1}^{d}{\frac {\partial F}{\partial x^{j}}}(X_{s-})\Delta X_{s}^{j}\right).\end{aligned}}}
Hierbei ist
X
s
−
=
lim
u
↑
s
X
u
{\displaystyle \textstyle X_{s-}=\lim _{u\uparrow s}X_{u}}
Δ
X
s
j
=
X
s
j
−
X
s
−
j
{\displaystyle \Delta X_{s}^{j}=X_{s}^{j}-X_{s-}^{j}}
Sprungprozess . Mit
[
X
j
,
X
k
]
c
{\displaystyle [X^{j},X^{k}]^{c}}
quadratische Kovariation der stetigen Anteile der Komponenten
X
j
{\displaystyle X^{j}}
X
k
{\displaystyle X^{k}}
X
{\displaystyle X}
[
X
j
,
X
k
]
c
=
[
X
j
,
X
k
]
{\displaystyle [X^{j},X^{k}]^{c}=[X^{j},X^{k}]}
Schreibt man den Ausdruck
[
X
j
,
X
k
]
t
c
:=
[
X
j
,
X
k
]
t
−
∑
s
≤
t
Δ
X
s
j
Δ
X
s
k
{\displaystyle [X^{j},X^{k}]_{t}^{c}:=[X^{j},X^{k}]_{t}-\sum \limits _{s\leq t}\Delta X_{s}^{j}\Delta X_{s}^{k}}
f
∈
C
2
(
R
d
,
R
)
{\displaystyle f\in C^{2}(\mathbb {R} ^{d},\mathbb {R} )}
f
(
X
t
)
−
f
(
X
0
)
=
∑
j
=
1
d
∫
0
+
t
∂
f
∂
x
j
(
X
s
−
)
d
X
s
j
+
1
2
∑
j
,
k
=
1
d
∫
0
+
t
∂
2
f
∂
x
j
∂
x
k
(
X
s
−
)
d
[
X
j
,
X
k
]
s
+
∑
0
<
s
≤
t
(
Δ
f
(
X
s
)
−
∑
j
=
1
d
∂
f
∂
x
j
(
X
s
−
)
Δ
X
s
j
−
1
2
∑
k
,
j
=
1
d
∂
2
f
∂
x
j
∂
x
k
(
X
s
−
)
Δ
X
s
j
Δ
X
s
k
)
.
{\displaystyle {\begin{aligned}f(X_{t})-f(X_{0})=&\sum _{j=1}^{d}\int _{0+}^{t}{\frac {\partial f}{\partial x^{j}}}(X_{s-}){\rm {d}}X_{s}^{j}+{\frac {1}{2}}\sum _{j,k=1}^{d}\int _{0+}^{t}{\frac {\partial ^{2}f}{\partial x^{j}\partial x^{k}}}(X_{s-}){\rm {d}}[X^{j},X^{k}]_{s}\\&{}+\sum _{0<s\leq t}\left(\Delta f(X_{s})-\sum _{j=1}^{d}{\frac {\partial f}{\partial x^{j}}}(X_{s-})\Delta X_{s}^{j}-{\frac {1}{2}}\sum _{k,j=1}^{d}{\frac {\partial ^{2}f}{\partial x^{j}\partial x^{k}}}(X_{s-})\Delta X_{s}^{j}\Delta X_{s}^{k}\right).\end{aligned}}}
wobei
Δ
f
(
X
s
)
:=
f
(
X
s
)
−
f
(
X
s
−
)
{\displaystyle \Delta f(X_{s}):=f(X_{s})-f(X_{s-})}
Das Integrationsgebiet
1
[
0
+
,
t
]
{\displaystyle 1_{[0+,t]}}
1
(
0
,
t
]
{\displaystyle 1_{(0,t]}}
Sei
X
=
(
X
1
,
…
,
X
n
)
{\displaystyle X=(X^{1},\dots ,X^{n})}
R
n
{\displaystyle \mathbb {R} ^{n}}
f
∈
C
2
(
R
n
,
R
)
{\displaystyle f\in C^{2}(\mathbb {R} ^{n},\mathbb {R} )}
f
(
X
)
{\displaystyle f(X)}
[ 3]
f
(
X
t
)
−
f
(
X
0
)
=
∑
i
=
1
n
∫
0
+
t
∂
f
∂
x
i
(
X
s
−
)
∘
d
X
s
i
+
∑
0
<
s
≤
t
(
f
(
X
s
)
−
f
(
X
s
−
)
−
∑
i
=
1
n
∂
f
∂
x
i
(
X
s
−
)
Δ
X
s
i
)
.
{\displaystyle f(X_{t})-f(X_{0})=\sum \limits _{i=1}^{n}\int _{0+}^{t}{\frac {\partial f}{\partial x_{i}}}(X_{s-})\circ dX_{s}^{i}+\sum \limits _{0<s\leq t}\left(f(X_{s})-f(X_{s-})-\sum \limits _{i=1}^{n}{\frac {\partial f}{\partial x_{i}}}(X_{s-})\Delta X_{s}^{i}\right).}
Bearbeiten
Hans Föllmer erweiterte die Formel von Itō auf (deterministische) Funktionen mit beschränkter quadratischer Variation.[ 4]
Sei
f
∈
C
2
{\displaystyle f\in C^{2}}
x
:
[
0
,
∞
[
→
R
{\displaystyle x:{[0,\infty [}\to \mathbb {R} }
Càdlàg-Funktion mit endlicher quadratischer Variation. Dann gilt
f
(
x
t
)
=
f
(
x
0
)
+
∫
0
t
f
′
(
x
s
−
)
d
x
s
+
1
2
∫
]
0
,
t
]
f
″
(
x
s
−
)
d
[
x
,
x
]
s
+
∑
0
≤
s
≤
t
(
f
(
x
s
)
−
f
(
x
s
−
)
−
f
′
(
x
s
−
)
Δ
x
s
−
1
2
f
″
(
x
s
−
)
(
Δ
x
s
)
2
)
)
.
{\displaystyle {\begin{aligned}f(x_{t})&=f(x_{0})+\int _{0}^{t}f'(x_{s-})\mathrm {d} x_{s}+{\frac {1}{2}}\int _{]0,t]}f''(x_{s-})d[x,x]_{s}\\&+\sum _{0\leq s\leq t}\left(f(x_{s})-f(x_{s-})-f'(x_{s-})\Delta x_{s}-{\frac {1}{2}}f''(x_{s-})(\Delta x_{s})^{2})\right).\end{aligned}}}
Für
Y
t
=
sin
(
W
t
)
{\displaystyle Y_{t}=\sin(W_{t})}
d
Y
t
=
cos
(
W
t
)
d
W
t
−
1
2
sin
(
W
t
)
d
t
{\displaystyle {\rm {d}}Y_{t}=\cos(W_{t})\,{\rm {d}}W_{t}-{\tfrac {1}{2}}\sin(W_{t})\,{\rm {d}}t}
S
t
=
S
0
e
r
t
−
1
2
σ
2
t
+
σ
W
t
{\displaystyle S_{t}=S_{0}e^{rt-{\frac {1}{2}}\sigma ^{2}t+\sigma W_{t}}}
eine Lösung der stochastischen Differentialgleichung von Black und Scholes
d
S
t
=
r
S
t
d
t
+
σ
S
t
d
W
t
{\displaystyle {\rm {d}}S_{t}=rS_{t}\,{\rm {d}}t+\sigma S_{t}\,{\rm {d}}W_{t}}
ist. Hierzu wählt man
X
t
=
W
t
{\displaystyle X_{t}=W_{t}}
a
t
=
0
,
b
t
=
1
{\displaystyle a_{t}=0,\;b_{t}=1}
Dann ergibt die Formel mit
h
(
t
,
x
)
=
S
0
e
r
t
−
1
2
σ
2
t
+
σ
x
{\displaystyle h(t,x)=S_{0}e^{rt-{\frac {1}{2}}\sigma ^{2}t+\sigma x}}
d
S
t
=
[
(
r
−
σ
2
2
+
σ
2
2
)
S
0
e
r
t
−
1
2
σ
2
t
+
σ
W
t
]
d
t
+
[
σ
S
0
e
r
t
−
1
2
σ
2
t
+
σ
W
t
]
d
W
t
=
r
S
t
d
t
+
σ
S
t
d
W
t
.
{\displaystyle {\rm {d}}S_{t}=\left[\left(r-{\frac {\sigma ^{2}}{2}}+{\frac {\sigma ^{2}}{2}}\right)S_{0}e^{rt-{\frac {1}{2}}\sigma ^{2}t+\sigma W_{t}}\right]{\rm {d}}t+\left[\sigma S_{0}e^{rt-{\frac {1}{2}}\sigma ^{2}t+\sigma W_{t}}\right]{\rm {d}}W_{t}=rS_{t}\,{\rm {d}}t+\sigma S_{t}\,{\rm {d}}W_{t}\,.}
Ist
(
W
t
)
t
≥
0
{\displaystyle (\mathbf {W} _{t})_{t\geq 0}}
d
{\displaystyle d}
F
:
R
d
→
R
{\displaystyle F\colon \mathbb {R} ^{d}\to \mathbb {R} }
Y
t
=
F
(
W
t
)
{\displaystyle Y_{t}=F(\mathbf {W} _{t})}
d
Y
t
=
∇
F
(
W
t
)
T
⋅
d
W
t
+
1
2
Δ
F
(
W
t
)
d
t
{\displaystyle \mathrm {d} Y_{t}=\nabla F(\mathbf {W} _{t})^{\mathsf {T}}\cdot \mathrm {d} \mathbf {W} _{t}+{\frac {1}{2}}\Delta F(\mathbf {W} _{t})\,\mathrm {d} t}
wobei
∇
F
{\displaystyle \nabla F}
Gradienten und
Δ
F
{\displaystyle \Delta F}
Laplace-Operator von
F
{\displaystyle F}
Es gibt verschiedene Varianten von Itō-Formeln für unendlich-dimensionale Räume (z. B. Pardoux[ 5] [ 6] [ 7]
Philip E. Protter: Stochastic Integration and Differential Equations (2nd edition), Springer, 2004, ISBN 3-540-00313-4 .
↑ Kiyoshi Itô: On a formula concerning stochastic differentials . In: Nagoya Math. J. Band 3 , 1951, S. 55–65 (projecteuclid.org ).
↑ Hui-Hsiung Kuo: Introduction to Stochastic Integration. Springer, 2006, ISBN 978-0387-28720-1 , S. 103 (eingeschränkte Vorschau in der Google-Buchsuche).
↑ Philip E. Protter: Stochastic Integration and Differential Equations . Hrsg.: Springer. 2004, ISBN 3-540-00313-4 , S. 277–278 .
↑ Hans Föllmer: Calcul d'Ito sans probabilités . In: Séminaire de probabilités de Strasbourg . Band 15 , 1981, S. 143–144 (numdam.org ).
↑ E. Pardoux, E: Équations aux dérivées partielles stochastiques de type monotone . In: Séminaire Jean Leray . Nr. 3 , 1974 (numdam.org ).
↑ I. Gyöngy und N. V. Krylov: Ito formula in banach spaces . In: Springer, Berlin, Heidelberg (Hrsg.): Arató, M., Vermes, D., Balakrishnan, A.V. (eds) Stochastic Differential Systems . Band 36 , 1981, doi :10.1007/BFb0006409
↑ Z. Brzezniak, J. M. A. M. van Neerven, M. C. Veraar und L. Weis: Ito's formula in UMD Banach spaces and regularity of solutions of the Zakai equation . 2008.
.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fd827aa1e55b160692bf22b26cfc993bee3a6ff)
![{\displaystyle {\begin{aligned}F(X_{t})-F(X_{0})=&\sum _{j=1}^{d}\int _{0+}^{t}{\frac {\partial F}{\partial x^{j}}}(X_{s-}){\rm {d}}X_{s}^{j}+{\frac {1}{2}}\sum _{j,k=1}^{d}\int _{0+}^{t}{\frac {\partial ^{2}F}{\partial x^{j}\partial x^{k}}}(X_{s-}){\rm {d}}[X^{j},X^{k}]_{s}^{c}\\&{}+\sum _{0<s\leq t}\left(F(X_{s})-F(X_{s-})-\sum _{j=1}^{d}{\frac {\partial F}{\partial x^{j}}}(X_{s-})\Delta X_{s}^{j}\right).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23758421b1b1cf32a6c54a4ecdb59230a212c6ec)
![{\displaystyle [X^{j},X^{k}]^{c}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfb3280b1de91dcf4fb6073f0b969d122c09a4a5)
![{\displaystyle [X^{j},X^{k}]^{c}=[X^{j},X^{k}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2a43b9f05a2932d1e7c431e8037ff81d6088d25)
![{\displaystyle [X^{j},X^{k}]_{t}^{c}:=[X^{j},X^{k}]_{t}-\sum \limits _{s\leq t}\Delta X_{s}^{j}\Delta X_{s}^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b7e256d7be97ec943487fec9fe4822ef1f52ea8)
![{\displaystyle {\begin{aligned}f(X_{t})-f(X_{0})=&\sum _{j=1}^{d}\int _{0+}^{t}{\frac {\partial f}{\partial x^{j}}}(X_{s-}){\rm {d}}X_{s}^{j}+{\frac {1}{2}}\sum _{j,k=1}^{d}\int _{0+}^{t}{\frac {\partial ^{2}f}{\partial x^{j}\partial x^{k}}}(X_{s-}){\rm {d}}[X^{j},X^{k}]_{s}\\&{}+\sum _{0<s\leq t}\left(\Delta f(X_{s})-\sum _{j=1}^{d}{\frac {\partial f}{\partial x^{j}}}(X_{s-})\Delta X_{s}^{j}-{\frac {1}{2}}\sum _{k,j=1}^{d}{\frac {\partial ^{2}f}{\partial x^{j}\partial x^{k}}}(X_{s-})\Delta X_{s}^{j}\Delta X_{s}^{k}\right).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f1acdfc2a743bfc0881687eac0e0c7741e4af7a)
![{\displaystyle 1_{[0+,t]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa37f15af3f93df5391db786346bc8d7485dc6ef)
![{\displaystyle 1_{(0,t]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00eeb8b9a89f59cb17dbbaf9d29bffd678ef65cc)
![{\displaystyle {\begin{aligned}f(x_{t})&=f(x_{0})+\int _{0}^{t}f'(x_{s-})\mathrm {d} x_{s}+{\frac {1}{2}}\int _{]0,t]}f''(x_{s-})d[x,x]_{s}\\&+\sum _{0\leq s\leq t}\left(f(x_{s})-f(x_{s-})-f'(x_{s-})\Delta x_{s}-{\frac {1}{2}}f''(x_{s-})(\Delta x_{s})^{2})\right).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2c66519e0ead3a3187aacf652d264607741b8e9)
![{\displaystyle {\rm {d}}S_{t}=\left[\left(r-{\frac {\sigma ^{2}}{2}}+{\frac {\sigma ^{2}}{2}}\right)S_{0}e^{rt-{\frac {1}{2}}\sigma ^{2}t+\sigma W_{t}}\right]{\rm {d}}t+\left[\sigma S_{0}e^{rt-{\frac {1}{2}}\sigma ^{2}t+\sigma W_{t}}\right]{\rm {d}}W_{t}=rS_{t}\,{\rm {d}}t+\sigma S_{t}\,{\rm {d}}W_{t}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4361106c796478db21b77b0f13fbab02bf7939a)
