Stochastic analysis is analysis based on Ito's calculus. This calculus was developed to cope with questions arising in probability theory in which processes are modelled by motion along paths which typically are not differentiable. The development of this calculus now rests on linear analysis and measure theory.Stochastic analysis is a basic tool in much of modern probability theory and is used in many applied areas from biology to physics, especially statistical mechanics. It has become particularly well known via the Black-Scholes formula as a way of modelling financial markets and strategies.
As a branch of pure mathematics it has a rich intrinsic interest. Riemannian geometry (and degenerate versions of it) is bound up with the study of solutions of stochastic ordinary differential equations which can be considered as a model for dynamical systems with noise. These equations are also used in the study of partial differential equations, for example those arising in geometric problems.
Numerical methods are needed for computation of solutions for stochastic ordinary, and partial, differential equations.
There is currently especial interest in questions relating to stochastic calculus for motion on singular spaces (such as on the branches of a tree) and on fractals: foundational, dynamical, and geometric aspects all appear here. Stochastic analysis is also a tool for the development of analysis on infinite dimensional spaces.
Stochastic partial differential equations are partial differential equations with some noise term. The noise may be due to intrinsic randomness in the system (eg from quantum effects) or from unknown random disturbances to the dynamics being modelled. Non-random partial differential equations form a special case and the questions, techniques and applications of stocahstic pdes are at least as wide as they are for classical pdes. Questions include the existence and properties of attractors for evolution equations, travelling wave solutions, intermittency, fractal properties and the relationships with turbulence, ergodic behaviour, and control theory of stochastic pdes.