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?



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 [edit]. 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.

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

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


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.


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