Statement

The lemma of Itô , or formula of Itô of the principal results of the theory of the stochastic Calcul is one. This lemma offers a means of handling it Brownian Movement or solutions of stochastic differential equations (EDS).

If $X$ is the solution of the EDS

$X\; (T)\; =X\; (0)\; +\; \backslash \; int\_0^t\; \backslash \; sigma\; (X\; (S),\; S)\; \backslash ,\; dB\; (S)$

+ \ int_0^t m (X (S), S) \, ds,

or of $dX\; (T)\; =\; \backslash \; sigma\; (X\; (S),\; S)\; \backslash ,\; dB\; (S)\; +m\; (X\; (S),\; S)\; \backslash ,\; ds$,

where $B$ is a Brownian Movement, and if $f\; (X,\; T)$ is a function of class $\backslash \; mathcal\; \{C\}\; ^\; \{1,2\}\; (\backslash \; mathbb\; \{R\}\; \_+,\; \backslash \; mathbb\; \{R\})$), then

$df\; (X\; (T),\; T)\; =\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; T\}\; +\; m\; (X\; (T),\; T)\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\}\; +\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash \; sigma\; (X\; (T),\; T)\; ^2\; \backslash \; frac\; \{\backslash \; partial^2\; F\}\; \{\backslash \; partial\; x^2\}\; \backslash \; right)\; dt\; +\; \backslash \; sigma\; (X\; (T),\; T)\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; X\}\; \backslash ,\; dB\_t.$

In the case of a Brownian Movement $\backslash \; frac\; \{\backslash \; sigma^2\}\; \{2\}$ corresponds to the coefficient of diffusion and m at mean velocity of the particle. (see Equation of Fokker-Planck). In finance $\backslash \; sigma$ is volatility and m the drift of the price of under unclaimed. (see Model Black-Scholes for example)

Applications

• the formula of Itô is one of the angular stones of the stochastic Calcul, and is used in very many fields: Mathematical applied, Physical, Finance, Biology, Mechanical quantum, treatment of the sigal, etc.

In stochastic Calculation,

• It makes it possible to establish the link between the solutions of EDS and the differential operators of the second order, and thus between the theory of the Probabilités and that of the partial derivative equations.

• It makes it possible to affirm the existence of solutions of EDS under conditions (very) weak of regularity on the coefficients.

History

The formula of Itô was shown for the first time by the Japanese mathematician Kiyoshi Itô in the years 1940.

The mathematician Wolfgang Doeblin had on its side outlined a theory similar before committing suicide with the defeat of its battalion in June 1940. Its work was envoys in a fold sealed with the Academy of Science who was open only in 2000.

An example: the Model Black-Scholes

The Brownian Movement is often used in Finance like the simplest model of evolution of stock exchange courts. It is about the stochastic solution of the differential equations:

$$

D X (T) = \ sigma X (T) \, dB (T) + \ driven X (T) \, dt, where $B$ is a Brownian Movement.

Let us note that if $\backslash \; sigma=0$, then we are vis-a-vis a differential equation ordinary whose solution is $X\; (T)\; =X\; (0)\; \backslash \; exp\; (\backslash \; driven\; T)$.

By posing $f\; (X\; (X),\; T)\; =log\; (X\; (T))$ one obtains thanks to the formula of Itô:

$d\; (log\; (X\; (T)))\; =\; \backslash \; left\; (X\; (T)\; \backslash \; driven\; \backslash \; dfrac\; \{1\}\; \{X\; (T)\}\; +\; 0\; +\; \backslash \; dfrac\; \{1\}\; \{2\}\; X\; (T)\; ^2\; \backslash \; sigma^2\; \backslash \; left\; (-\; \backslash \; dfrac\; \{1\}\; \{X\; (T)\; ^2\}\; \backslash \; right)\; \backslash \; right)\; dt\; +\; X\; (T)\; \backslash \; sigma\; \backslash \; dfrac\; \{1\}\; \{X\; (T)\}\; dB\_t$

$d\; (log\; (X\; (T)))\; =\; \backslash \; left\; (\backslash \; driven\; -\; \backslash \; dfrac\; \{\backslash \; sigma^2\}\; \{2\}\; \backslash \; right)\; dt\; +\; \backslash \; sigma\; dB\_t$

One can then integrate and:

$$

X (T) =X (0) \ exp \ left (\ sigma B (T) + \ driven T \ frac {1} {2} \ sigma^2 T \ right)

References

• C.G. Gardiner. Handbook off Stochastic Methods (3^ème ED.), Springer, 2004. ISBN 3540208828

• I. Karatzas and S. Shreve. Brownian Motion and Stochastic Calculus , Graduate Texts in Mathematics (2^ème ED.), Springer, 2004. ISBN 0387976558.

• B. Øksendal. Stochastic Differential Equations: Year Introduction With Applications (6^ème ED.), Springer, 2005. ISBN 3540047581

• ( popularizing work ) G. Pages and C. Bouzitat. While passing by chance… the probabilities of under the days, Vuibert, 1999. ISBN 2711752585

• D. Revuz and Mr. Yor. Continuous Martingales and Brownian Motion , (3^ème ED.), Springer, 2004.ISBN 3540643257

• L.C.G. Rogers and D. Williams. Diffusions, Markov processes and martingales (2^ème ED.), Cambridge Mathematical Library, Cambridge University Near, 2000. ISBN 0521775930

• Karlin S, Taylor H M: With first race in stochastic processes. Academic Close, (1975)

• Karlin S, Taylor H M: With second race in stochastic processes. Academic Close, (1981)

• Schuss Z: Theory and applications off stochastic differntial equations. Wiley Series in Probability and Statistics, (1980)

… and any treating work of the Brownian movement and of stochastic calculation.

See too

• Integral of stochastic Itô
• Calculation

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