You can calculate a derivative of densities for stochastic processes in some generalised sense which I do not at present understand, and do the normal calculus thing you do with a derivative. Stochastic differential equations arise, presumably ones in some sense involving this generalised derivative, can then solve some kind of problems for you. Or something.
Clearly this is a placeholder for a topic I do not have time for right now. 🏗
1 References
Bell. 2006. The Malliavin Calculus.
Bichteler, Gravereaux, and Jacod. 1987. Malliavin Calculus for Processes with Jumps. Stochastics Monographs, v. 2.
Di Nunno, Øksendal, and Proske. 2009. Malliavin Calculus for Lévy Processes with Applications to Finance. Universitext.
Friz. 2005. “An Introduction to Malliavin Calculus.”
Levajkovic, and Selesi. 2011. “Chaos Expansion Methods for Stochastic Differential Equations Involving the Malliavin Derivative, Part I.” Publications de l’Institut Mathematique.
Nualart. 2006. The Malliavin Calculus and Related Topics. Probability and Its Applications.
Øksendal. 1997. An Introduction to Malliavin Calculus with Applications to Economics.
Osswald. 2012. Malliavin Calculus for Lévy Processes and Infinite-Dimensional Brownian Motion: An Introduction. Cambridge Tracts in Mathematics 191.
Sanz-Solé. 2004. A Course on Malliavin Calculus.
Schiller. 2009. “Malliavin Calculus for Monte Carlo Simulation with Financial Applications.”
Zhang. 2004. “The Malliavin Calculus.”