The functional Meyer–Tanaka formula

The functional Itô formula, firstly introduced by Bruno Dupire for continuous semimartingales, might be extended in two directions: different dynamics for the underlying process and/or weaker assumptions on the regularity of the functional. In this paper, we pursue the former type by proving the functional version of the Meyer–Tanaka formula. Following the idea of the proof of the classical time-dependent Meyer–Tanaka formula, we study the mollification of functionals and its convergence properties. As an example, we study the running maximum and the max-martingales of Yor and Obłój.

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The functional Itō formula under the family of continuous semimartingale measures

Stochastics and Dynamics ◽

10.1142/s0219493716500106 ◽

2016 ◽

Vol 16 (04) ◽

pp. 1650010 ◽

Cited By ~ 6

Author(s):

Harald Oberhauser

Keyword(s):

Continuous Dependence ◽

Quadratic Variation ◽

Stochastic Integrals ◽

Itô’S Formula ◽

Itô Formula ◽

Ito Formula ◽

Underlying Process ◽

Continuous Semimartingale ◽

The Family ◽

Nonlinear Functionals

Dupire [16] introduced a notion of smoothness for functionals of paths and arrived at a generalization of Itō’s formula that applies to functionals with a continuous dependence on the trajectories of the underlying process. In this paper, we study nonlinear functionals that do not have such continuity. By revisiting old work of Bichteler and Karandikar we show that one can construct pathwise versions of complex functionals like the quadratic variation, stochastic integrals or Itō processes that are still regular enough such that a functional Itō-formula applies.

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Itô stochastic integral. Itô formula. Tanaka formula

Theory of Stochastic Processes - Problem Books in Mathematics ◽

10.1007/978-0-387-87862-1_13 ◽

2009 ◽

pp. 193-213

Author(s):

Dmytro Gusak ◽

Alexander Kukush ◽

Alexey Kulik ◽

Yuliya Mishura ◽

Andrey Pilipenko

Keyword(s):

Stochastic Integral ◽

Itô Formula ◽

Ito Formula ◽

Tanaka Formula

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A generalization of the Itô formula

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171202102018 ◽

2002 ◽

Vol 31 (8) ◽

pp. 477-496

Author(s):

Said Ngobi

Keyword(s):

Sobolev Space ◽

White Noise ◽

Itô Formula ◽

Ito Formula ◽

Integrable Functions ◽

Square Integrable Functions ◽

White Noise Space

The classical Itô formula is generalized to some anticipating processes. The processes we consider are in a Sobolev space which is a subset of the space of square integrable functions over a white noise space. The proof of the result uses white noise techniques.

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