Tony G Deangelo 
Stochastic Differential Equations [PDF ebook] 
Basics and Applications

In this collection, the authors begin by introducing a methodology for examining continuous-time Ornstein-Uhlenbech family processes defined by stochastic differential equations (SDEs). Additionally, a study is presented introducing the mathematics of mixed effect parameters in univariate and bivariate SDEs and describing how such a model can be used to aid our understanding of growth processes using real world datasets. Results and experience from applying the concepts and techniques in an extensive individual tree and stand growth modeling program in Lithuania are described as examples. Next, the authors present a review paper on J-calculus, as well as a contributed paper which displays some new results on the topic and deepens some special properties in relation with non-differentiability of functions. Following this, this book develops the general framework to be used in our papers [2, 9, 8]. The starting point for the discussion will be the standard risk-sensitive structures, and how constructions of this kind can be given a rigorous treatment. The risk-sensitive optimal control is also investigated by using the extending part of this of problem of backward stochastic equation. In the closing article, the authors note that the square of an O-U process is the Cox-Ingersoll-Ross process used as a model for volatility in finance. The filtered form of the original hazard rate based on this new observation is also studied. If the difference between the original hazard rate and the filtered one is not significant, then the person is not affected by the new frailty.