Stochastic differential equations lecture notes

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# Stochastic differential equations lecture notes

Definition and properties of a stochastic process, classical and modern classifications of stochastic processes. Also, a review on basic operations in Python. Discrete random variables, continuous random variables, expectation of a random variable, jointly distributed random variables, moment generating functions, conditional probabilities.

Python code for this lecture are available here. Forward and backward Kolmogorov differential equations, Poisson processes, birth and death processes, birth and death processes with immigration. Step plot of the Poisson process. Python code: poissonprocess. The mistakes have been corrected, the lecture note 4b and Python code have been updated.

Standard Brownian motion, geometric Brownian motion, and Brownian motion with drift. Disclaimer This is not an official course offered by Boston University. The sole aim of this page is to share the knowledge of how to implement Python in numerical stochastic modeling to anyone, for free.

Description Most complex phenomena in nature follow probabilistic rules. To study natural phenomena more realistically, we use stochastic models that take into account the possibility of randomness. In this course, introductory stochastic models are used to analyze the inherent variation in natural processes. For this purpose, numerical models of stochastic processes are studied using Python. Lecture Notes Lecture Brief Review on Stochastic Processes Definition and properties of a stochastic process, classical and modern classifications of stochastic processes.

Brief Review on Probability Theory Discrete random variables, continuous random variables, expectation of a random variable, jointly distributed random variables, moment generating functions, conditional probabilities. Continuous-Time Markov Chain Models Forward and backward Kolmogorov differential equations, Poisson processes, birth and death processes, birth and death processes with immigration.

Lecture 5 :.Download the video from iTunes U or the Internet Archive. Description: This lecture covers the topic of stochastic differential equations, linking probablity theory with ordinary and partial differential equations.

The following content is provided under a Creative Commons license. But now, we're given this equation. And we have to go backwards, want to find the stochastic process that satisfies this equation.

### An introduction to stochastic partial differential equations

So goal is to find a stochastic process, xt, satisfying this equation. In other words, we want x of t to be the integral of mu ds plus sigma [? We want that.

And so these type of equations are called differential equations, I hope you already know that, also for PDE, partial differential equations. So even when it's not stochastic, these are not easy problems. At least not easy in the sense that, if you're given an equation, typically you don't expect to have a closed form solution.

So even if you find this x, most of the time, it's not in a very good form.

## Stochastic Modeling using Python

Still, a very important result that you should first note before trying to solve any of the differential equation is that, as long as mu and sigma are reasonable functions, there does exist a solution. And it's unique. So we have the same correspondence with this PDE. You're given a PDE, or given a differential equation, not a stochastic differential equation, you know that, if you're given a reasonable differential equation, then a solution exists. So the same principle holds in stochastic world.

Now, let me state it formally. This stochastic equation, star, has a solution that is unique, of course with [? And given the initial points-- so if you're given the initial point of a stochastic process, then a solution is unique.

Just check, yes-- as long as mu and sigma are reasonable. One way it can be reasonable, if it satisfies this conditions. These are very technical conditions. But at least let me parse to you what they are. They say, if you fix a time coordinate and you change x, you look at the difference between the values of mu when you change the second variable and the sigma. Then the change in your function is bounded by the distance between the two points by some constant k.

So mu and sigma cannot change too much when you change your space variable a little bit. It can only change up to the distance of how much you change the coordinate. So that's the first condition. Second condition says, when you go to your x, very similar condition. Essentially it says it cannot blow up too fast, the whole thing. This one is something about the difference between two values.

This one is about how it expands as your space variable grows. These are technical conditions. And many cases, they will hold. So don't worry too much about the technical conditions. Important thing here is that, given a differential equation, you don't expect to have a good closed form. But you do expect to have a solution of some form. OK, let's work out some examples. Here is one of the few stochastic differential equations that can be solved.Overall, I would absolutely recommend Nordic Visitor to others since we had an experience above and beyond what we were expecting.

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SC_V2_0 What is a Stochastic Differential Equation?

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