some exercises for the stochastic processes course
In this exercise, firstly, a series of random numbers that are uniformly distributed is generated. Then the Box-Muller method is used to create a normally distributed dataset from them. Then, the probability distribution function p(x) for both of these datasets is constructed. In every case, a normal graph and logarithmic graph with error bars are included. Finally, I plot xn p(x) versus x for different n values.
Wiener process is a stochastic process that describes Brownian motion. It is a continuous process described with the equation:
dW/dt = η(t) .dt
where η(t) is a random function. It can be a uniformly distributed white noise. Here I solve this equation numerically, plot a set of paths that are following this equation, and verify this relation for a Wiener process :
⟨ (W (t))2 ⟩ = t (this is an ensemble average)
The aim of this exercise is two solve a specific Langevin equation numerically. Then, calculating its correlation length and Markov length. And also computing the first, second, and fourth of Kramers-Moyal coefficients.
in this exercise, two Langevin Equations are coupled. As before, we can solve them using the Euler method. After that Kramers-Moyal coefficients are calculated and plotted, each as a surface on 3-D space.
Here I solve the equation for a Poisson Process numerically and then calculated the first 6 moments for this process. From those, we can find some characteristic parameters of our process. A Poisson jump-diffusion process is a process where as well as our normal drift and diffusion terms, like in the Langevin equation, there is another term indicating jumps. The time difference between jumps obeys a Poisson distribution.
course info : Stochastic Processes - 2017 - Sharif University of Technology - Physics Department