# 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 Jan 25, 2013 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.
– SBF
Jan 25, 2013 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. Jan 27, 2013 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, 2013 at 8:58

For a GBM following SDE $$\frac{\mathrm{d}X_t}{X_t} = \mu \mathrm{d}t + \sigma \mathrm{d}W_t$$ (under some physical measure $$\mathbb{P}$$) with initial value $$X_0$$ almost surely, we have the solution $$X_t = X_0 \exp\left(\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t\right)$$ where for simplicity we have assumed $$W_t$$ to be a standard Brownian motion (Weiner process) and $$\mu$$ and $$\sigma$$ to be constants (extending to the time dependent case is not too hard).