# Approximation of SDE

I have been struggling with the following problem: If you want to find a numerical result by simulating the paths of a stochastic differential equation, in particular a geometric brownian motion I have found literature saying you should go for the exact solution (continuous time) and literature stating that you should approximate the GBM (e.g. with the Euler-Maruyama Scheme).

From what I understand, there are SDE's who can be modelled exactly, and SDE's who can't. What's true for a GBM?

From Glasserman, P. (2004). Monte Carlo Methods in Fin. Engineering:

"Chapter 3 gives examples of continuous-time stochastic processes which can be simulated exactly [...]" (p. 339). Chapter 3 then lists the GMB in 3.2 (pp. 93ff.).

From Günther & Jüngel (2010). Finanzderivate mit Matlab. :

given $$dS_t = rS_tdt + \sigma S_tdW_t$$

transform this with the Euler-Maruyama Method to

$$\Delta S_t = rS_t\Delta t + \sigma S_t \Delta W_t$$

where $$\Delta W_t = Z \cdot \sqrt{\Delta t}$$ and $$Z\sim\mathcal{N}(0,1)$$ (pp. 102f.)

I hope that helps clarifying my issue. If not I'll happily give more details. Thanks

• user1889382: Welcome to MSE! Is it possible for you to cite the references or to provide examples as that may help the MSE Community understand your question better? Regards – Amzoti Jan 25 '13 at 15:29
• The GBM has an exact closed form via the exponent of $W_t$, which you can "exactly" simulate since this is just a lognormal random variable. – Ilya Jan 25 '13 at 20:46
• A nice book covering simulation of SDEs is Iacus: Simulation and Inference for SDEs. In particular you will find the formulas for exact simulation for most popular models. – Julian Wergieluk Jan 27 '13 at 20:46
• @Ilya: Thanks, that makes sense. @ Julian: Thanks for the tip, I just looked at the book and it really helps. But now my question has changed slightly: Given that there exists a closed form solution for the GBM, I take it from chapter 2 in Iacus' book that if I'm interested in modeling the entire trajectory of the process, I still need to discretize the GBM (p. 66)? If so, why? And if not, why does he use the GBM in his approximation part? Thanks a lot!! – gu7z Jan 28 '13 at 8:58