Ito's formula

Ito 's formula is a formula for changing a variable in a stochastic differential equation . The author of the formula Ito Kiyoshi is a Japanese statistical mathematician.

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Definition

Dan random process $X=(X_{t})_{t\geqslant 0}$ ${\ displaystyle X = (X_ {t}) _ {t \ geqslant 0}}$ specified on filtered probability space $\left(\Omega ,\;{\mathfrak {F}},\;({\mathfrak {F}}_{t})_{t\geqslant 0},\;P\right)$ ${\ displaystyle \ left (\ Omega, \; {\ mathfrak {F}}, \; ({\ mathfrak {F}} _ {t}) _ {t \ geqslant 0}, \; P \ right)}$ with the flow $({\mathfrak {F}}_{t})_{t\geqslant 0}$ ${\ displaystyle ({\ mathfrak {F}} _ {t}) _ {t \ geqslant 0}}$ .

Let a stochastic differential equation be given $dX_{t}=a(t,\;\omega )\,dt+b(t,\;\omega )\,dB_{t}$ ${\ displaystyle dX_ {t} = a (t, \; \ omega) \, dt + b (t, \; \ omega) \, dB_ {t}}$ or, in integral form

X_{t}=X_{0}+\int \limits _{0}^{t}a(s,\;\omega )\,ds+\int \limits _{0}^{t}b(s,\;\omega )\,dB_{s},

{\ displaystyle X_ {t} = X_ {0} + \ int \ limits _ {0} ^ {t} a (s, \; \ omega) \, ds + \ int \ limits _ {0} ^ {t} b (s, \; \ omega) \, dB_ {s},}

Where $B=\left(B_{t},\;{\mathfrak {F}}_{t}\right)_{t\geqslant 0}$ ${\ displaystyle B = \ left (B_ {t}, \; {\ mathfrak {F}} _ {t} \ right) _ {t \ geqslant 0}}$ - Brownian motion.

Now let $F(t,\;x)$ ${\ displaystyle F (t, \; x)}$ - set to $\mathbb {R} _{+}\times \mathbb {R}$ ${\ displaystyle \ mathbb {R} _ {+} \ times \ mathbb {R}}$ continuous function from class $C^{1,\;2}$ ${\ displaystyle C ^ {1, \; 2}}$ that is, having derivatives ${\frac {\partial F}{\partial t}},\ {\frac {\partial F}{\partial x}},\ {\frac {\partial ^{2}F}{\partial x^{2}}}.$ ${\ displaystyle {\ frac {\ partial F} {\ partial t}}, \ {\ frac {\ partial F} {\ partial x}}, \ {\ frac {\ partial ^ {2} F} {\ partial x ^ {2}}}.}$

Under these assumptions:

dF(t,\;X_{t})=\left[{\frac {\partial F}{\partial t}}+a(t,\;\omega ){\frac {\partial F}{\partial x}}+{\frac {1}{2}}b^{2}(t,\omega ){\frac {\partial ^{2}F}{\partial x^{2}}}\right]\,dt+{\frac {\partial F}{\partial x}}b(t,\;\omega )\,dB_{t}.

{\ displaystyle dF (t, \; X_ {t}) = \ left [{\ frac {\ partial F} {\ partial t}} + a (t, \; \ omega) {\ frac {\ partial F} {\ partial x}} + {\ frac {1} {2}} b ^ {2} (t, \ omega) {\ frac {\ partial ^ {2} F} {\ partial x ^ {2}}} \ right] \, dt + {\ frac {\ partial F} {\ partial x}} b (t, \; \ omega) \, dB_ {t}.}

Speaking more strictly, with each $t>0$ ${\ displaystyle t> 0}$ for $F(t,\;X_{t})$ ${\ displaystyle F (t, \; X_ {t})}$ The following Ito formula is valid:

F(t,\;X_{t})=F(0,\;X_{0})+\int \limits _{0}^{t}\left[{\frac {\partial F}{\partial t}}+a(s,\;\omega ){\frac {\partial F}{\partial x}}+{\frac {1}{2}}b^{2}(s,\;\omega ){\frac {\partial ^{2}F}{\partial x^{2}}}\right]\,ds+\int \limits _{0}^{t}{\frac {\partial F}{\partial x}}b(s,\;\omega )\,dB_{s}.

{\ displaystyle F (t, \; X_ {t}) = F (0, \; X_ {0}) + \ int \ limits _ {0} ^ {t} \ left [{\ frac {\ partial F} {\ partial t}} + a (s, \; \ omega) {\ frac {\ partial F} {\ partial x}} + {\ frac {1} {2}} b ^ {2} (s, \ ; \ omega) {\ frac {\ partial ^ {2} F} {\ partial x ^ {2}}} \ right] \, ds + \ int \ limits _ {0} ^ {t} {\ frac {\ partial F} {\ partial x}} b (s, \; \ omega) \, dB_ {s}.}

Multidimensional generalization

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The stochastic world - a simple introduction to stochastic differential equations

Ito's formula

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Definition

Multidimensional generalization

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