Even though the mathematicians of antiquity had developed remarkably modern ideas to compute lengths, surfaces and volumes, the first rigorous construction of integration, due to Bernhard Riemann, goes back to the middle of the XIX century. His definition of ∫â–’fdx takes only a few lines, and he gives a complete characterization of the class of functions f for which the integral exists. Ever since that time, motivated by abstract considerations but also by practical questions, mathematicians have looked for generalizations of the notion of integral. Even in the framework of one variable, making sense of ∫â–’fdg for larger or classes of integrands f and integrators g remains relevant today.
To quote the most well-known direction, measure theory has made it possible to include in a general framework the case when f is measurable and g has bounded variation (i.e. is the difference of two increasing functions) ; its the Lebesgue-Stieltjes theory.
Dealing with random integrands and integrators is another development that led to stochastic calculus. In this framework, integrators are often of unbounded variation, another extension that has motivated new works even recently. The course presents an introduction to those aspects of integral calculus, illustrated at each step on the example of Brownian motion. We shall define the Itô integral, a probabilistic framework, give its elementary properties and a few applications. We shall also construct, in the deterministic framework, the Young integral, which falls short of dealing with Brownian motion, and its recent and successful generalization to integrals along rough paths.
The literature on stochastic calculus is immense. We suggest [Øks03] eand [Kuo06]. The chapters relevant for this course are at an adapted level. More advanced references include [RY05] or [KS00] for example. The literature on rough paths is much less extensive. We suggest [FH14]. The chapters relevant for this course are very readable. At a more advanced level one may consult [FV10].
Références
[FH14] Peter K. Friz and Martin Hairer, A Course on Rough Paths, Universitext, Springer, 2014.
[FV10] Peter K. Friz and Nicolas B. Victoir, Multidimensional stochastic processes as rough paths, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2010.
[KS00] Ioannis Karatsas and Steven E. Shreve, Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics, no. 113, Springer, 2000.
[Kuo06] Hui-Hsiung Kuo, Introduction to stochastic integration, 1 ed., Universitext, Springer, 2006.
[Øks03] Bernt Øksendal, Stochastic Differential Equations : An Introduction with Applications, 6 ed., Universitext, Springer, 2003.
[RY05] Daniel Revuz and Marc Yor, Continuous Martingales and Brownian Motion, 3 ed., Grundlehren der matematischen Wissenschaften, vol. 293, Springer, 2005.