stochastic calculus question

solal · April 24th, 2012, 6:24 am

Hi,Given two correlated brownians and , correlation is rho, what is the joint law of?Thanks for your help

BrightDay · April 24th, 2012, 6:46 am

Rho is of no importance since it's not even in the formula

Also what is the "," operator? More importantly, what are you asking exactly?

emac · April 24th, 2012, 8:12 am

BrightDay has no idea what (s)he is talking about.Okay, you shoudl be able to use Ito's formula to show that the integral term has a Normal distribution with mean zero and variance (t^3)/3. W^2_t obviously has a normal distribution with mean zero, variance t. You can then use Ito's isometry to calculate the covariance of these two random variables, given that you know the covariance of W^1 and W^2. You have two normal random variables with known distribution and you know their covariance, hence you know their joint law.

BrightDay · April 24th, 2012, 8:27 am

QuoteBrightDay has no idea what (s)he is talking about.Possibly, but the formula quoted is still not well formed

list · April 24th, 2012, 10:24 am

it might be helpful to present second W_t as the stoch integral with respect to d W then write their discrete time approximations , multiply them , take expectation, and limit when the step tends to 0.

emac · April 24th, 2012, 10:31 am

No formula is quoted. The OP want to know the joint law of a pair of random variables. They have just written this pair (X,Y) in brackets.Also, list, there is no need to take discrete time approximations. The ito isometry gives you the expectation of the product of two stochastic integrals.

eh · April 24th, 2012, 10:33 am

QuoteOriginally posted by: BrightDayQuoteBrightDay has no idea what (s)he is talking about.Possibly, but the formula quoted is still not well formedYes it is.-------------------Going back to the question, everything is joint normal so it should be straightforward.First find mean and cov of W1 with its running average. Second, write W2 as , where W3 is independent of W1. You should be on the home straight by then.

BrightDay · April 24th, 2012, 2:56 pm

QuoteYes it isThanks Eh for making me see the light. I scanned the original question too quickly and misunderstood what was asked. My apologies to Solas and Emac

solal · April 26th, 2012, 12:52 am

Thx emac

list · April 26th, 2012, 4:04 pm

QuoteOriginally posted by: solalHi,Given two correlated brownians and , correlation is rho, what is the joint law of?Thanks for your helpA couple remarks. For fixed t we have 2-dimentional Gaussian vector and we do not have to use Ito calculus. If we look at the density formula we just need to find the correlation of vector's components. If we have a 2-dimensional stochastic process which is the solution od 2-d SDE then distribution of the solution is given in theoretical meaning when we write one of the Kolmogorov's equations.

foxkingdom · April 27th, 2012, 6:32 am

By applying Ito's lemma to we can show the following stochastic Integration by Parts,From this form, we know that the integral is Gaussian, and it is easy to show, the joint law is a 2-dimensional Gaussian. And its law is thus caracterized by its mean and variance matrix. Which are easy to calculate now using the transformed integral.