7.7 Malliavin Calculus

We here briefly outline a hypermodel approach to the part of Malliavin calculus which we used in the computation of Vega on p. 113. We refer the reader to [Cutland and Ng, 1994] and [Cutland, 2000] for more details.

The mean idea of Malliavin calculus is to differentiate a function ? ( ?), treated as a function in ? _t , , with respect to the Brownian increment ? _t.

We first consider an extension of the It integral and will show its connection to the Malliavin derivative through an integration by parts formulas.

We let ? t:=1/ N, for some

We use the *-continuous model, i.e. we let

equipped with an internal probability measure ? given by .

Definition 7.14

Let . Then is defined by

Definition 7.15

In * L ² ( ?), a linear combination of elements of the form

is called a monomial of order K.

Given F ?* L ²( ?) we denote by F ^(K) the * L ²-projection of F onto the space of monomials of order up to K.

It can be proved that:

Theorem 7.22

There is a set dom( ?) ? L ² (W [0,1]), such that whenever u ? dom( ?), it has some SL ² lifting U( ?, t); and for all sufficiently small infinite K, U ^(K) is SL ² . We denote the...