Stochastic Differential Equation.

[Ben K]

Hi Folks, I' m an old timer trying to get back on the math horse so to speak. I studied this 20 years ago. My interests are Ito Calculus and financial mathematics, although this question may be simple to some of you and it is fundamentally a differential equations problem. Thanks for any guidance. I will try to text write this correctly.

Ssubt=Ssub0e^(u minus sigma^2/t e^sigmaBt satisfies the equation dS=muSdt+sigmaSdBsubt

This is geometric Brownian motion

You use Ito's lemma .

my question is the muSdt term....In other words the dt term

When doing this out in Ito's Lemma you get for the dt term, (I will use small "a" for partial derivative symbol)

(aS/at + 1/2a^2S/aB^2subt)dt.........how does that end up equaling ((mu-sigma^2/2)S+sigma^2/2S)....which then equals muS in the GBM equation above

I apologize for any confusion. I don't know of there is a forum that deals with fundamental Ito Calculus.

Thank You

 

[stapel]

Hi Folks, I'm an old-timer trying to get back on the math horse so to speak. I studied this 20 years ago. My interests are Ito Calculus...

I've heard of differential calculus, integral calculus, and multi-variate calculus, but not "Ito" calculus. Are you referring to this?

it is fundamentally a differential equations problem. Thanks for any guidance. I will try to text write this correctly.

Ssubt=Ssub0e^(u minus sigma^2/t e^sigmaBt satisfies the equation dS=muSdt+sigmaSdBsubt

Is the equation above meant to be as follows?

. . . . .\(\displaystyle \Large{ S_t\, =\, S_0\, e^{(\,u\, -\, ^{\sigma^2}/_t\,)},\, \mbox{ where }\, e^{\,\sigma Bt}\, \mbox{ satisfies }\, dS\, =\, \mu\, S\, dt\, +\, \sigma\, S\, dB_t }\)

If not, please reply with corrections.

This is geometric Brownian motion. You use Ito's lemma.

Which is this...?

Thank you! (And please wait for somebody who has at least some familiarity with this topic to reply. I'm just trying to clarify terms, but this is way out of my league!)