Stochastic Flows and Jump-Diffusions by Hiroshi Kunita

Author:Hiroshi Kunita
Language: eng
Format: epub, pdf
ISBN: 9789811338014
Publisher: Springer Singapore

Proof

Since Gδ(X) =∑lGF lW(h l) −∑lG〈DF l, h l〉, we have

Apply Proposition 5.1.2 to the above. Then the right-hand side is written as

Therefore we get (5.8).

Further, δ(X) is H-differentiable, since each term of the right-hand side of (5.7) is H-differentiable. We have indeed,

Therefore the equality of the commutation relation (5.9) holds.

Suppose . Then we have

Here we used (5.8), (5.9) and (5.8) in turn. □

Note

The derivative operator D is a basic tool in stochastic analysis on the Wiener space. It has been discussed in various contexts. Cameron [15] introduced the derivative D on the Wiener space and showed a version of the adjoint formula (5.8). It was extended and applied to potential theory on the Wiener space by Gross [34, 35]. Malliavin [77] studied the derivative operator D through the Ornstein–Uhlenbeck operator on the Wiener space. Stroock [105] discussed the finite dimensional approximation of the operator D. Ikeda–Watanabe [40, 41] and Watanabe [116] developed Malliavin’s theory by introducing norms of Sobolev-type through the Ornstein–Uhlenbeck operator.

Our definition of the derivative D is close to Cameron [15], Gross [34, 35] and Shigekawa [100]. It can be applied directly to solutions F of stochastic differential equations driven by Wiener processes and further we can compute the derivatives DF explicitly. It will be discussed in Sect. 6.​1.

The absolute continuity of the transformation T h on the Wiener space was shown by Cameron–Martin [16]. It was extended to a wider class of transformations by Maruyama [82] and Girsanov [33], and is called the Girsanov transformation (Theorem 2.2).

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