In stochastic analysis (a branch of advanced mathematics) we deal with continuous random processes. These processes can model for example a financial asset, such as shares or interest rates or whatever. In statistical physics we deal with statistical ensembles. These are created by diffusions. Options pricing utilizes diffusion theory heavily. Quantum mechanics can be formulated through diffusion theory, when we consider probability waves and state vectors in a Hilbert space. There is a link, and I believe the key here is the generator of the continuous Markov process: in abstract we deal with duality between partial differential equations and stochastic processes.
A special type of stochastic process is the diffusion process. These processes are characterized by continuous sample paths (no jumps). Moreover, they are Markov processes. This means that they do not possess any memory. A typical (time-homogeneous) diffusion process would look like this:
where
is a standard Wiener process or Brownian motion. Wiener process has stationary increments, and is a normal martingale, the continuous analogue of a simple random walk. It has rough sample paths, but its dual is the theory of harmonic functions, the beauties of potential theory (and of basically all complex analysis btw.)
Diffusions can be paired to elliptic second order differential operators
The function b is the drift function and sigma (diagonal matrix) represents the amplitude of the noise from independent Brownian motions. With such a diffusion we can associate a second order elliptic differential operator called the infinitesimal generator:
This is pedagogically the key object. We need to understand how this generator gives us all information we need about Markov diffusions. The generator is associated with something called the Markov semigroup. The semigroup essentially tells us that we can divide a probabilistic transition from A to C in two parts: from A to B and from B to C (Chapman-Kolmogorov equations). This corresponds to the multiplication of Markov chain matrices. The generator is just the infinitesimal version corresponding to Markov semigroups for continuous diffusions.
Options theory in short
The link between diffusions and elliptic operators gives us the beautiful duality between stochastic differential equations and partial differential equations. This means that solutions of PDEs can be seen as certain expectations of stochastic processes. This is essentially how we can solve and option price for financial markets through a Black-Scholes PDE. Duality! This can be shown via the Feynman-Kac duality formula.
Thermodynamics is about symmetry, self-adjoint infinitesimal generators and reversible diffusions
If we can define the operator adjoint of the infinitesimal generator in space L^2, we can in particular demand that the operator is self-adjoint. This gives us the class of reversible diffusions. These are of great interest to us. We can show that self-adjoint generators correspond to gradient drifts (with constant diffusion matrices). This in turn gives us ergodicity and stationary distributions for the diffusion, which allows us to calculate ensemble averages. These averages are important as they correspond to time averages for ergodic processes.
If we consider statistical mechanics, we can study a collection of particles and we perhaps wish to know what is the equilibrium distribution of particles. We may thus model the test particle through the diffusion as above. Then, as the Fokker-Planck PDE is of the form
Where L star is the operator adjoint of the infinitesimal generator of the diffusion.
The stationary solution is obviously given by:
This distribution gives us the thermodynamics of equilibrium, as the transition probability reaches a stationary state. This is due to the property of the drift and diffusions, they have to obey a property called detailed balance. The inverse problem is also interesting: given a stationary distribution, find a reversible diffusion which converges to it (fast).
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