Change of variable formulas for non-anticipative functionals on path space ✩

By Rama Cont, David-Antoine Fournié. 2010

222 citations recorded by the source corpus.

We derive a change of variable formula for non-anticipative functionals defined on the space of Rd-valued right-continuous paths with left limits. The functionals are only required to possess certain directional derivatives, which may be computed pathwise. Our results lead to functional extensions of the Ito formula for a large class of stochastic processes, including semimartingales and Dirichlet processes. In particular, we show the stability of the class of semimartingales under certain functional transformations.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/84939fe6346018f0d0a8154660b4b00e84e9295f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Finance

1. Jianfeng Zhang
2. Shige Peng
3. Frederi G. Viens
4. Marcel Nutz
5. Falei Wang
6. David-Antoine Fourni'e
7. Nicolas Perkowski
8. Rama Cont

In Mathematical Physics

1. Jianfeng Zhang
2. Frederi G. Viens
3. Shige Peng
4. David-Antoine Fourni'e
5. Nicolas Perkowski
6. Rama Cont
7. Li-Xin Zhang
8. Massimiliano Gubinelli

Citations in the corpus

• A martingale approach for fractional Brownian motions and related path dependent PDEs (2017, methodology): “The main idea of the proof follows that of Dupire [16] and Cont & Fournie [12].”
• BSDE, path-dependent PDE and nonlinear Feynman-Kac formula (2011, background): “With a very different point of view, Dupire [8, 2009] introduced a new type of a functional Itô’s formula which non-trivially generalized the classical one (See Cont and Fournié [4, 5, 6] for a more general and more systematic research).”
• Pathwise integration and change of variable formulas for continuous paths with arbitrary regularity (2018, methodology): “y use of probabilistic notions of convergence. 1.3 Extension to path-dependent functionals An important generalization of F¨ollmer’s pathwise Itˆo formula is to the case of path-dependent functionals [9] of paths S∈ V2(π) using Dupire’s notion of functional derivative [13]; see [8] for an overview. ”
• Path-Dependent Optimal Stochastic Control and Viscosity Solution of Associated Bellman Equations (2012, unknown): “Recently Dupire [14] introduced horizontal and vertical derivatives in the path space of a path functional, non-trivially generalized the classical Itô formula to a functional Itô formula (see Cont & Fournie [7, 8] for a more general and systematic research), and provided a functional extension of t”
• Functional Ito calculus and stochastic integral representation of martingales (2010, methodology): “In the companion paper [5] we give a non-probabilistic proof of Theorem 4.”