Malliavin Derivative - Definition

Definition

Let be the Cameron-Martin space, and denote classical Wiener space:

;
;

By the Sobolev embedding theorem, . Let

denote the inclusion map.

Suppose that is Fréchet differentiable. Then the Fréchet derivative is a map

;

i.e., for paths, is an element of, the dual space to . Denote by the continuous linear map defined by

sometimes known as the H-derivative. Now define to be the adjoint of in the sense that

.

Then the Malliavin derivative is defined by

The domain of is the set of all Fréchet differentiable real-valued functions on ; the codomain is .

The Skorokhod integral is defined to be the adjoint of the Malliavin derivative:

Read more about this topic:  Malliavin Derivative

Famous quotes containing the word definition:

    “I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
    —Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)

    “The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
    —Jean Baudrillard (b. 1929)

    “... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
    —Adrienne Rich (b. 1929)