When trying to solve the equationhow do we know the function to apply Ito's lemma to is ?

Hi ahw101,Let me try to answer your question:We use ln(S) because the stochastic process S is assumed to follow a geometric Brownian motion expressed as , which is a log-normal diffusion process. S is decided to follow a geometric Brownian motion to avoid that S can go negative, what is more realistic for stock prices. Moreover, ln(S) is normally distributed. For visualising this I've found this nice excel-file http://www.ssc.wisc.edu/~whs/teaching/3 ... tion.xls.I hope this assists?!Have a nice week,LiAnne

Hi Lianne,thanks very much for your response I agree with everything you are saying. My question really is though, how to choose the function just from looking at the SDE.To take another example how do you choose the function to apply Ito's lemma to for the Ornstein–Uhlenbeck processThanks.

Hi ahw101,Sorry, I'm afraid, I don't understand your problem. Could you try to explain it in another way? Best regards,LiAnne

There isn't a general way; experience and good luck will allow you to find it (sometimes) when the SDE has a closed form solution.It's the same as when you tried to solve ODEs at university or even integrals at secondary school: you did hundreds of exercises to solve equations from the simplest to the hardest, and by the end of the course you could guess (for some of them) the likely form of the integrand function. Sometime you could have an idea of a general function with some parameters that you would insert back into the equation and see for which values it matched.

Dear ahw101. There is no general recipe. Is like trying to evaluate an integral. It is kind of an art rather than anything else.

Remove the stochastic piece ( ). The resulting ODE will help you find to what function you want to apply Ito.

ahw101,I am admittedly an addict when it comes to finance and Math but I am really bad at it. So my explanations might not be 100% correct but since I work with intuition, it might help.GBM: if you see that dS/S = constant + symmetrical noise (a normal distribution. Then you can make a guess that the solution has to be something with exp() - this is the only way that the derivative can be constant. This has always been my starting point in understanding this. It also implies that only the ln() of the function is normally distributed since e^(N(u,s)) is lognormal. You can then proceed and use statistics knowledge and realise that when the expected value of a lognormal process is exp(mu+s^2/2) but you want an expected value (arbitrage free) of the stock price that is S*exp(mu), all you need to do is take the (s^2/2) away from the mu. No need for Ito's lemma!Ornstein Uhlenbeck: There are some nice write ups on how to solve this. there is no guesswork needed but some knowledge of how functions and their diffs/integrls behave. This is quite a slog to do but even I can do it! So do not give up!cheers,Alk

Am I taking you right, that you are looking for a receipt or a rule of thumg for figuring out future functions to apply Ito's Lemma to. Well, the dropping of the stochastic term may help, as foresaid. But in general there is no receipt. Luckily someone figured the ln-trick out for you. In case of Ornstein Uhlenbeck: this is a linear SDE, for which there exist an explicit solution formula (cf. Karatzas/Shreve).