Stochastic Integration (Simple Version)

Transcription

1 Stochastic Integration (Simple Version) Tuesday, March 17, :03 PM Reading: Gardiner Secs , But there are boundary issues when (if ) so we can't apply the standard delta function integration result when s takes those values. But since we are integrating the result anyway, the value at one or two points won't affect the outcome of the integration, so we don't need to worry about making the value of the integrand correct at. AppSDE15 Page 1

2 Both cases can be combined: Therefore, combining our results: So we've computed the mean and autocorrelation function of the velocity. What more can we say? Because the exact solution shows that is a deterministic linear operation on the Gaussian random function, the result is also a Gaussian random function. Therefore we have a complete description. This formal calculation based on Gaussian delta-correlated force works fine, but will run into trouble in more complex calculations, so we will redo this calculation with more proper stochastic calculus that generalizes better. So now we will shift to an applied mathematical use of stochastic calculus. One point of the rigorous theory of stochastic calculus is to avoid the complications and ambiguities inherent in working with white noise (i.e., delta -correlated forces) directly since these are not well-defined objects in the conventional mathematical sense. The key way to remove this pathology is to express the dynamics in terms of changes in variables over short time intervals (i.e. increments). This will be analogous to, in our simplest model for Brownian motion, talking about increments in position (random kicks) rather than the velocity (which had infinite variance). Now we talk about increments in the current state variable, which is. Imagine integrating the momentum equation over a short time interval. AppSDE15 Page 2

3 variance). Now we talk about increments in the current state variable, which is. Imagine integrating the momentum equation over a short time interval. For small, we can approximate: because is a resolved state variable in our equation, that results from integrating a differential equation, so it is at least continuous function of time. (Mean value theorem, and by continuity, doesn't matter to leading order if we evaluate at beginning of interval). Cannot make this same argument for the thermal force integral because it is a highly irregular, nowhere to close to continuous function. So how do we discuss it? First of all, we can give it a name: An integral of force is called impulse; it represents a finite change in momentum (whereas force represents an instaneous change in momentum). So what are the properties of the random impulse: It will be Gaussian (because it is a deterministic linear operation on a Gaussian random function). The calculation proceeds in the same way as what we have just done (with exponential terms in the integrand): use bilinearity to take the integrals outside the covariance plug in the covariance of the thermal force AppSDE15 Page 3

4 plug in the covariance of the thermal force Also, the random impulse is independent on disjoint intervals. That is, choose such that To see this, we just need to show lack of correlation since the impulse is Gaussian, and that can be done by a similar calculation as above: Therefore, in summary, the random impulse (random increment in the state variable, momentum) has the following properties: Gaussian independent over nonoverlapping intervals From the March 10 lectures notes, if we compare with the properties of the increments in the simplest model for Brownian motion, we see that the random impulse has the same properties as. Therefore, we can model the random impulses using the increments in some underlying mathematical Brownian motion (Wiener process) to write: AppSDE15 Page 4

5 Notice that every quantity in this equation is well defined, for any. The continuum limit of this equation is the stochastic differential equation representation for the Langevin equation: How do we understand these stochastic differential equations? Rigorous mathematical theory: we'll discuss later. Practical (and rigorizable) interpretation is that the Euler-Marayama numerical integration method: where are iid random variables over each (nonoverlapping) time step with is a consistent approximation to the stochastic differential equation. As with ordinary differential equations, most that appear in applications cannot be solved analytically. So then what? qualitative/graphical analysis (dynamical systems); we won't go into that in this class asymptotic analysis and perturbation methods to simplify the differential equations numerical solution/simulation/integration We know there are many explict/implicit/multistep numerical methods for solving ordinary differential equations. So how do these carry over to the stochastic differential equation setting? Higham, "An Algorithmic Introduction to Numerical Solution of Stochastic Differential Equations": excellent introduction High order methods are not terribly useful for stochastic differential equations for two reasons: they are very complicated and expensive to implement (Kloeden and Platen, Numerical Solution of Stochastic Differential Equations) the solutions aren't that smooth anyway To learn about the practical use of high order methods for SDEs (second order Runge Kutta), see Talay, "Simulation of Stochastic Differential Equations" The next simplest numerical method after the Euler-Marayama method is the Milstein method (generalizes the trapezoidal rule), and it already is awkward enough to not be much used. Implicit methods are a real pain for SDE's. So in practice, almost everyone uses Euler-Marayama method with some specialized exceptions If one is concerned about conservation laws, then one may use a second order symplectic algorithm with the stochastic extension Stiff systems: Implicit methods don't work so well for SDEs. But there is modern research on two other approaches: multiscale simulation, multi-level Monte Carlo AppSDE15 Page 5

6 simulation. We'll introduce these ideas later. AppSDE15 Page 6