# Julia package SDE.jl¶

## Layout¶

The package contains the following modules:

Diffusion
Generate Ito processes and diffusions
NonparBayes
Nonparametrically estimate the drift of a diffusion
Schauder
Provides rudimentary finite element methods and Schauder basis for NonparBayes
Lyap
Computes the solution of the continuous Lyapunov equation, useful for the generation of linear processes
Randm
Random symmetric, positive definite, stable matrix for testing purposes.
LinProc
Homogeneous vector linear processes with additive noise

## Method¶

The producure in NonparBayes is a Julia implementation of nonparametric Bayesian inference for “continuously” observed one dimensional diffusion processes with unit diffusion coefficient. The drift is modeled as linear combination of hierarchical Faber–Schauder basis functions with a Gaussian prior on the coefficients. This incorporates a Brownian motion like prior on the drift function. The posterior is then computed using Gaussian conjugacy.

This is work in progress.

## Data structures¶

I did not introduce type definitions for stochastic processes and use vectors/arrays, so it should be easy do wrap Dataframes around everything. For the meanwhile, I like the natural notation obtained by having just vectors/arrays for dW and dt

N = 100 t = linspace(0., 1., N) dt = diff(t) X = ito(2dt + 2dW1(dt))