[Daniel_Van_Zant]

Is stochastic calculus something that requires a computer to stimulate many possible unfolding of events, or is there a more elegant mathematical way to solve for some of the important final outputs and probability distributions if you know the distribution of dW? This is an awesome article. I've seen stochastic calculus before but this is the first time I really felt like I started to grok it.

[sfpotter]

In case the other responses to your question are a little difficult to parse, and to answer your question a little more directly:

- Usually, you will only get analytic answers for simple questions about simple distributions.

- For more complicated problems (either because the question is complicated, or the distribution is complicated, or both), you will need to use numerical methods.

- This doesn't necessarily mean you'll need to do many simulations, as in a Monte Carlo method, although that can be a very reasonable (albeit expensive) approach.

More direct questions about certain probabilities can be answered without using a Monte Carlo method. The Fokker-Planck equation is a partial differential equation which can be solved using a variety of non-Monte Carlo approaches. The quasipotential and committor functions are interesting objects which come up in the simulation of rare events that can also be computed "directly" (i.e., without using a Monte Carlo approach). The crux of the problem is that applying standard numerical methods to the computation of these objects faces the curse of dimensionality. Finding good ways to compute these things in the high-dimensional case (or even the infinite-dimensional case) is a very hot area of research in applied mathematics. Personally, I think unless you have a very clear physical application where the mathematics map cleanly onto what you're doing, all this stuff is probably a bit of a waste of time...

[Daniel_Van_Zant]

Thanks for the explanation this was very helpful. You've given me a whole new list of stuff to Google. The quasipotential/comittor functions especially seem quite interesting although I'm having a bit of trouble finding good resources on them.

[kkylin]

It depends a bit on exactly what you want to calculate, but in general things like the probability density function of the solution of a stochastic differential equation (SDE) at time t satisfies a partial differential equation (PDE) that is first order in time and second order in space [0]. (This PDE is known to physicists as the Fokker-Planck equation and to mathematicians as the Kolmogorov forward equation.) Except in special examples, the PDE will not have exact analytical solutions, and a numerical solution is needed. Such a numerical solution will be very expensive in high dimensions, however, so in high-dimensional problems it is cheaper to solve the SDE and do Monte Carlo sampling, rather than try to solve the PDE.

Edit: sometimes people are interested in other types of questions, for example the solution when certain random events occur. Analogous comments apply. Also, while stochastic calculus is very useful for working with SDEs, if your interest is other types of Markov (or even non-Markov) processes you may need other tools.

Edit again: as another commenter mentioned, in special cases the SDE itself may also have exact solutions, but in general not.

[0] This statement is specific to stochastic differential equations, i.e., a differential equation with (gaussian) white noise forcing. For other types of stochastic processes, e.g., Markov jump processes, the evolution equation for distributions have a different form (but some general principles apply to both, e.g., forms of the Chapman-Kolmogorov equation, etc).

[FabHK]

Certain simple stochastic differential equations can be solved explicitly analytically (like some integrals and simple ordinary differential equations can be solved explicitly), for example the classic Black Scholes equation. More complicated ones typically can't be solved in that way.

What one often wishes to have is the expectation of a function of a stochastic process at some point, and what can be shown is that this expectation obeys a certain (deterministic) partial differential equation. This then can be solved using numerical PDE solvers.

In higher dimensions, though, or if the process is highly path-dependent (not Markovian), one resorts to Monte Carlo simulation, which does indeed simulate "many possible unfolding of events".

[LeonardoTolstoy]

It has been a while since I studied along these lines (stochastic chemical reaction simulations in my case) but I think the answer is often yes, but not always (I don't think). A random walk for example will be a normal distribution (and you know the mean, and you know the variance is going to infinity), so I do think in that case you end up with an elegant analytical solution if I'm understanding correctly as the inputs can determine the function the variance follows through time.

But often no, you need to run a stochastic algorithm (e.g. Gillespie's algorithm in the case of simple stochastic chemical kinetics) as there will be no analytical solution.

Again it has been a while though.

[yoyoma1234]

For normal distributions I think do - black scholes is an analytical solution to option pricing. Been a while since I studied stochastic calculus

I question why this is the second highest article on hacker news currently, can’t imagine many people reading this website are REALLY in this field or a related one, or if it’s just signaling like saying you have a copy of Knuths books or that famous lisp one

[anvuong]

Depends on what you want to know. If you want to get some trajectories then simulation of the stochastic differential equation is required. But if you just want to know the statistics of the paths, then in many cases you can write and try to solve the Fokker-Planck equation, which is a partial differential equation, to get the path density.