Now, the model that we obtained is a model with a non-linear diffusion. The equation that we obtained for our QG model is a particular case of a general nonlinear stochastic differential equation or as SD for short, known as the Langevin equation. The Langevin Equation is probably one of the most famous as the ancient signs. So, in this video I wanted to talk a bit about the Langevin equation. First, let's start with a bit of history. The story of modern finance start with a PhD thesis of Louis Bachelier in 1900. Bachelier was developing his own model of Brownian motion as a free diffusion few years ahead of Eeinstein. He then applied it to model stock prices and came up with what is today known as the arithmetic Brownian motion or ABM model. His work has stayed completely unnoticed to the mathematical community until it was discovered or rediscovered by Andrey Kolmogorov, a Russian mathematician. Kolmogorov appreciated Bachelier's work and told about it to Paul Levy. Then Leonard Savage translated Bachelier's work to English and this is how Paul Samuelson learnt about this work. The contribution of Paul Samuelson from 1965 was the invention of the geometric Brownian motion or GBM model, as a way to improve a problem with the ABM model. The problem was that in the ABM model a stock price can go negative, and in contrast in a GBM model a stock price not only cannot go negative, it can't even become zero. In other words, the boundary x equal zero is unaccessible in a GBM model. We will talk more about it later, but for now I want to continue a bit with this short historical know. So, about five years after Bachelier's work, Einstein published his work on the theory of the Brownian motion, and this work sparked lots of new research among physicists. The French physicist Paul Langevin studied the simplified model of Brownian motion for particles in an external field that produce force action on the Brownian particles. For example, this can be a potential of intermolecular forces in a liquid. In 1908 he published a paper on this topic where he suggested what later became known as the Langevin equation. His work was extended later in 1920s by Fokker and Planck who studied related phenomena in physics. The Langevin equation for a Brownian particle in an external potential is shown here in equation six. In this equation x stands for a particle position, delta x stands for it's time derivative and x with two dots means the second derivative in time. Parameter gamma describes diffusion in a system and u of x is an external force potential for example, potential formed by some other fast particles in a liquid. Finally, dot v is an icon sites stand for a Gaussian white noise. One example of a system that gives rise to the Langevin equation is one heavy particles with mass capital M are subject to a force potential of light particles of mass small m. They are light particles are at equilibrium at temperature T and have a Maxwell distribution of their velocities v, given by equation eight here. It turns out that the analysis of such system yields directly the Langevin equation. You can find more details about it in a book by Schuss cited on this slide. On this graph you can see an example of a potential UX that can arise in a Langevin equation. This potential has two metastable states, and we will talk quite a bit about such potential in this lecture and the next one. Now, the Langevin equation can also be written in an equivalent phase-space form. To this end, we write it as two equations. The first equation says that V is a time derivative of X, and the second equation gives the time evolution of V. This means that the solution of the Langevin equation is actually a pair of two variables, the particle position x t and it's velocity V t. In the many cases it's useful to consider a limit of the Langevin equation that is obtained if we take the dissipation coefficient gamma to be very large. This limit is called overdamped Brownian motion or the Smoluchowski limit. It turns out that in this limit the original Langevin equation becomes the overdamped Langevin equation shown here in equation 11. There is a way to actually derive this result. It uses the phase space from a form that I showed you above, plus, it is carrying of time by a factor of gamma. You can find the derivation in the book of Schuss that I already mentioned. Now this equation is only of first order in time G which is first unlike the original Langevin equation that was of the second order in time. In fact, there overdamped version of the Langevin equation is probably used more often than its' full version at least outside of physics. For example, remember our discussion of stochastic gradient descent method for training machine learning algorithms. In fact minimization of the lost function, there can exactly be described as a Langevin equation. Just think of x as a model parameter and of your facts as a loss function that you want to minimize. The role of Brownian noise in the Langevin equation will be played by the observational noise in a mini batch procedure of the stochastic gradient descent method. Now, if we take the Smoluchowski limit for free Brownian particle without any potential we obtain equation 12 on this slide from the Langevin equation. We can rewrite it as shown in equation 13 where the constant D is Einstein's diffusion coefficient. You can easily recognize an equation 13, the ABM Model without a drift. We can also compare the Langevin equation with the either diffusion, with-drift. The Langevin equation is shown here in equation 14, and I wrote the noise term [inaudible] a general noise term that can be either white noise or not white noise, but for now we can assume that we deal only with the white noise. On the other hand, we can write data diffusion law for the GBM model and it's shown here in equation 15. So if we compare these two equations we can conclude that the GBM model corresponds to a potential u of x, shown here in equation 16. It's very interesting that this is a potential of an inverted harmonic oscillator. I say inverted because of the negative sign in this expression. This means a negative mass of a particle. On this graph you can see this potential. If you put a particle exactly at point x equals zero, this will be a point of instability. A particle will roll down, and what it means we will see in the next videos
