# Binomial representation of stochastic processes

It is common knowledge that a random walk can be represented in the form of a binomial process. Is it possible to represent any generic stochastic process (including non-linear) of the form $dX=adt+bdW(t)$ (where $W(t)$ is a Wiener process) as a binomial process? Are there any papers which have dealt with this topic?

Thanks.

For a martingale $dX=a(X,t)\,dt+b(X,t) dW(t)$ where $a$ and $b$ are not constant, your tree will not recombine in general . This is the main issue. See for instance: Florescu, I. and F. G. Viens (2008, March). Stochastic volatility: Option pricing using a multinomial recombining tree. Applied Mathematical Finance 15 (2), 151-181. It deals with the case of stochastic volatility, i.e.

$dS=aS\,dt+b(Y_t)S dW(t)$

$dY=\alpha(\nu − Y_t)dt + \psi(Y_t)dZ(t)$

• Very good point to relate the issue to the question of recombination! Commented Jun 8, 2015 at 6:09
• While the trees do not recombine in general, it is possible to transform certain stochastic processes in order to obtain recombining trees. (see Simple Binomial Processes as Diffusion Approximations in Financial Models ) Commented Jun 8, 2015 at 16:34

Yes, this is trivially true once you know that every continuous local martingale is a time-changed brownian motion. Therefore, if you change your time variable $t$ in $dX=a\,dt+b\,dW(t)$ to the right $t^\prime$ you can get a standard tree representation.

Now, the correct time change may be difficult or impossible to figure out, so this theorem is of limited use.

One then becomes tempted to try making a tree adapted to the original SDE. As @lehalle notes, you are then likely to end up with a tree that does not recombine. You may be able to overcome that flaw by interpolating nodes back onto a recombined tree and applying some reasonable boundary conditions, but at that point most people just go with PDE solvers instead.