Malliavin Calculus - Invariance Principle

Invariance Principle

The usual invariance principle for Lebesgue integration over the whole real line is that, for any real number ε and integrable function f, the following holds

This can be used to derive the integration by parts formula since, setting f = gh and differentiating with respect to ε on both sides, it implies

\int_{-\infty}^\infty f' \,d \lambda = \int_{-\infty}^\infty (gh)' \,d \lambda = \int_{-\infty}^\infty g h'\, d \lambda + \int_{-\infty}^\infty g' h\, d \lambda.

A similar idea can be applied in stochastic analysis for the differentiation along a Cameron-Martin-Girsanov direction. Indeed, let be a square-integrable predictable process and set

If is a Wiener process, the Girsanov theorem then yields the following analogue of the invariance principle:

E(F(X + \varepsilon\varphi))= E \left [F(X) \exp \left ( \varepsilon\int_0^1 h_s\, d X_s - \frac{1}{2}\varepsilon^2 \int_0^1 h_s^2\, ds \right ) \right ].

Differentiating with respect to ε on both sides and evaluating at ε=0, one obtains the following integration by parts formula:

E(\langle DF(X), \varphi\rangle) = E\Bigl.

Here, the left-hand side is the Malliavin derivative of the random variable in the direction and the integral appearing on the right hand side should be interpreted as an Itô integral. This expression also remains true (by definition) if is not adapted, provided that the right hand side is interpreted as a Skorokhod integral.

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