2016/10/12 by John Armstrong, Armstrong, John, Damiano Brigo +1 · 1 citation
Economics, Econometrics and Finance · #39A50 #58A20 #58J65 #60H10 #60J60 #65D18 #Differential Geometry (math.DG) #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1610.03887
openalex publication_date 2016/10/12 · openalex created_date 2022/10/07 · openalex updated_date 2026/07/28
We define two new notions of projection of a stochastic differential equation\n(SDE) onto a submanifold: the Ito-vector and Ito-jet projections. This allows\none to systematically develop low dimensional approximations to high\ndimensional SDEs using differential geometric techniques. The approach\ngeneralizes the notion of projecting a vector field onto a submanifold in order\nto derive approximations to ordinary differential equations, and improves the\nprevious Stratonovich projection method by adding optimality analysis and\nresults. Indeed, just as in the case of ordinary projection, our definitions of\nprojection are based on optimality arguments and give in a well-defined sense\n"optimal" approximations to the original SDE in the mean-square sense. We also\nshow that the Stratonovich projection satisfies an optimality criterion that is\nmore ad hoc and less appealing than the criteria satisfied by the Ito\nprojections we introduce. As an application we consider approximating the\nsolution of the non-linear filtering problem with a Gaussian distribution and\nshow how the newly introduced Ito projections lead to optimal approximations in\nthe Gaussian family and briefly discuss the optimal approximation for more\ngeneral families of distribution. We perform a numerical comparison of our\noptimally approximated filter with the classical Extended Kalman Filter to\ndemonstrate the efficacy of the approach.\n