1 / 2017-12-07 22:58:05

Modeling explosions in financial prices.

SDE, Probability distrubition of exploding time, Bessel process,, Nonparametric estimator of exploding Diffusion function

We studied the existence and uniqueness of the one dimensional time-homogeneous stochastic differential equation with discontinuous coefficients function. Since the drift and diffusion functions (σ and µ) of SDE(1.5) depends on function g, to establish the existence and uniqueness of the strong solution of SDE we impose the standing assumption on function g. The first main result, we shown that under assumption (1) (we assume the function g is a function of bounded variation) there exists a unique and strong solution of SDE (1.5). Secondly, we consider the case the function g is discontinuous on a unique point on R, e.i. under assumption (2), we shown there a weak solution of SDE(1.5) which is unique in the sense of probability distribution and defined up until the explosion time. Then, we provide an example in the case the SDE (1.5) admits the weak solution which is unique in the sense of probability distribution and defined up until the explosion time. We derive a Generalization of Bessel type process and we compute the probability distribution of exploding time for a Generalization of Bessel process type. We also propose the nonparametric estimator for the exploding Diffusion coefficient.