Brownian - Weiner - understand logic

[rcaus]

Dear Hubbers,

From student reviews, Brownian / Weiner is quite confusing but important for pricing of option ( Black SM) . I decided to get a kick start last month and spent lots of time on internet research , BPP manual and stochastic calculus book . Despite that I have previously the subject of " options , money market" yet only in actuarial exam you need to understand formulas and their proof etc.

I can understand the concept and work some of the question but it appears that I am parrot learning. Pls clarify some of the logics for me.

Random walk > fine to me with the e.g of coin, pollen, drunk man.
Qu 1) Xn(t) >> does the X for one pollen and then n stands for a huge amount of pollen( or infinity)
Qu2 ) If time t tend to s . Does it also tned to infinity.
Qu3 > Does both t ( time) and n ( pollen) tend to infinity.
Qu4 > Per the book I cannot understand which one comes first one pollen in relation to time and then add n pollen or vice-versa

Brownian / Weiner

We often see these terms

dX(t) = αdt + σdZ(t)

Qu 1) Is it the same pollen e.g but why we also have a dZ(t)

Qu 2) Is this also same as dS(t) = αdt + σdZ(t) except that its adapted to share price S(t)

Qu2) If Z(t) is a brownian it is also a martingale.

Qu3) To confuse me more I often see B(t) and also W(t)


Pls help me to understand

Regards
RCaus

 

[jensen]

Qu 1) Is it the same pollen e.g but why we also have a dZ(t)

Qu 2) Is this also same as dS(t) = αdt + σdZ(t) except that its adapted to share price S(t)

Qu2) If Z(t) is a brownian it is also a martingale.

Qu3) To confuse me more I often see B(t) and also W(t)

A1) not sure but i think dZt is the random/diffusion term.

A2) Think it's suppose to be dS(t) = S[dt + σdZ(t)]. Maybe someone can verify.

A3) I guess, Yes.
A4) B(t) = standard brownian motion, with drift = 0, diffusion = 1

W(t) on the other hand is general brownian motion. You may think of these as, standard normal distribution and the general normal distribution where the standard one has mean = 0, std dev = 1, whereas the general one is mean = mu and std dev = sigma.