Ito

One of the areas of my research is stochastic differential equations (SDE). I posted about it several times before. One of the things students and collaborators keep asking me about SDEs is the weird stochastic Itô Calculus. Itô Calculus is different from what you learn in 101 calculus. In particular, the chain rule is not longer valid. Let me explain it with an example. Suppose you have the following equation ...

Stochastic differential equations (SDEs) are basically inhomogenous ordinary differential equations that depend on an external stochastic process. Typically, that stochastic process is white noise, which is the mathematical idealization of the noise found in nature. This idealization is handy, because it simplifies the mathematical description. However, this idealization comes at some cost: traditional calculus is no longer valid and you have to use the so-call Itô calculus. This introduces some non intuitive changes. ...

November was a rather sad month in the world of stochastic differential equations. In the 26th we were suppose to be celebrating the birth of one of the best mathematicians in history, Norbert Wiener, who gives name to the Wiener process, usually denoted W(t). However, in the 10th, Kiyoshi Itô, the father of stochastic differential equations, passed away. Interestingly both are present in a simple stochastic differential equation like this ...

Kiyoshi Itô (90), professor emeritus at kyoto University, has become the first winner of the Gauss Prize. This prize is to honor scientist whose mathematical research has had an impact outside mathematics. Ito’s work, mainly in establishing a well defined calculus (named Ito’s calculus) to treat high irregular noise functions has got widespread application in describing several stochastic processes across fields like economics, biology, chemistry, physics, etc. Ito’s calculus is behind the pricing of options introduced by Black, Scholes and Merton (which got them a Nobel price). ...

Authors: Esteban Moro and Henri Schurz Journal: SIAM Journal of Scientific Computing, Volume 29 Issue 4, Pages 1525-1549 (2007). LINK | arXiv Abstract: Construction of splitting-step methods and properties of related non-negativity andboundary preserving numerical algorithms for solving stochastic differential equations (SDEs) of Ito-type are discussed. We present convergence proofs for a newly designed splitting-step algorithm and simulation studies for numerous numerical examples ranging from stochastic dynamics occurring in asset pricing theory in mathematical finance (SDEs of CIR and CEV models) to measure-valued diffusion and superBrownian motion (SPDEs) as met in biology and physics. ...