Integral of Itô

The integral of Itō , called thus in the honor of the mathematician Kiyoshi Itō is one of the fundamental tools of the stochastic Calcul.

It is about a definite integral in a way similar to the Intégrale of Riemann like limit of a Somme of Riemann. If one gives oneself a process of Wiener (or Brownian Movement) $B: T \ times \ Omega \ to \ mathbb {R} \,$ like $X: T \ times \ Omega \ to \ mathbb {R}$ a stochastic Process adapted to the natural filtration associated with $B_t$ , then the integral of Itô

$\ int_ {has} ^ {B} X_ {T} \, \ mathrm {D} B_ {T}: \ Omega \ to \ mathbb {R}$

is defined by the limit on average quadratic of

$\ sum_ {I = 0} ^ {K - 1} X_ {t_ {I}} \ left (B_ {t_ {i+1}} - B_ {t_ {I}} \ right)$

when the step of the partition $0 = t_ {0} < t_ {1} < \ dowries < t_ {K} = T$ of $T$ tends towards 0.

These sums, considered as sums of Riemann-Stieltjes for each way of the Brownian movement given, do not converge in general; the reason is that the Brownian movement is not with limited Variations. The use of quadratic convergence is the essential point of this definition.

Properties

With the preceding notations, the stochastic process Y defined, for T real positive, by $Y_ {T} = \ int_ {0} ^ {T} {X_ {S} \ mathrm {D} B_ {S}}$ , is a Martingale. In particular, its hope is constant.

In addition, one it property known as of isometry : $E (Y_ {T} ^ {2}) = \ int_ {0} ^ {T} {E (X_ {S} ^ {2}) \ mathrm {D} S}$ . To note that this last integral is " classique" , i.e is an integral within the meaning of Riemann compared to the variable S.

See too

Process of Itô
Lemma of Itô
stochastic Calculation
Integral of Stratonovich

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