Rama Cont

Functional Ito calculus, martingale representation and path-dependent Kolmogorov equations

The Functional Ito Calculus [1, 4] is a non-anticipative calculus which extends Ito’s stochastic calculus to path-dependent functionals of stochastic processes. A key ingredient of the approach is an Ito formula for functionals [1, 2], which enables
to represent the variations of a functional on the space D([0, T], R^d) of right-continuous paths in terms of certain directional
derivatives. These directional derivatives are used related to the predictable projection of the Malliavin derivative [3] and appear in the representation of martingales as stochastic integrals. However, unlike the Malliavin derivative, their construction is pathwise and does not rely on any Gaussian properties of the underlying process.

The Functional Ito calculus allows to characterize a large class of Brownian martingales, viewed as time-dependent functionals on Wiener space, as space-time harmonic functionals, whose directional derivatives verify a functional PDE on [0, T] × C0([0, T], R^d), the path-dependent Kolmogorov equation. This PDE, which may be seen as a generalization of Kolmogorov’s backward equation to the setting of Ito processes, extends the classical relation between diffusion processes and parabolic partial differential equations to a non-Markovian setting. This class of infinite-dimensional PDEs shares many properties with parabolic PDEs in finite dimensions and leads to Feynman-Kac formulas for path-dependent functionals of a square-integrable martingale [4]. Such path-dependent Kolmogorov equations have a natural link with Backward stochastic differential equations and non-Markovian stochastic control problems.

References
[1] R Cont and D Fourni´e (2010) A functional extension of the Ito formula, Comptes Rendus de l’Acad´emie des Sciences,
Volume 348, Issues 1-2, January 2010, Pages 57-61.
[2] R Cont and D Fourni´e (2010) Change of variable formulas for non-anticipative functionals on path space, Journal of
Functional Analysis, Volume 259, No 4, Pages 1043-1072.
[3] R Cont and D Fourni´e (2013) Functional Ito calculus and stochastic integral representation of martingales, Annals of
Probability, Vol 41, No 1, 109–133.
[4] R Cont: Functional Ito calculus and functional Kolmogorov equations, in: V Bally et al: Stochastic integration by
parts and Functional Ito calculus (Lectures Notes of the Barcelona Summer School on Stochastic Analysis, July 2012),
Springer: 2016.