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mj
Posts: 3449
Joined: December 20th, 2001, 12:32 pm

### Exact simulation of brownian motion and of its integral

you get into tricky issues when factor reducing and then using a long step scheme since time-dependent vol and drift state-dependence render the model full factor. I discuss these issues in detail in More Mathematical Finance. See also our paper http://ssrn.com/abstract=907385

MaxwellSheffield
Posts: 70
Joined: December 17th, 2013, 11:08 pm

### Exact simulation of brownian motion and of its integral

Thank you ! I have the whole model implemented in C++, I was using a poor Euler-Scheme, and then I decided to get something more elaborated. I build my volatility vectors based on historical data(unit vectors) and market instruments( norm). When I build the unit vector, I already run a PCA to reduce the factors from 120 to let say 10 . Then I calibrate the norm to market instruments. I read in "Comparing Discretisations of the Libor market model in the spot measure" that the Predictor-corrector scheme is a superior scheme, hence my implementation issue. I tried to tackle the problem, approximating the Ito integral since my function beta is piece-wise constant, but I dont like this solution.Mj, I am going to read your paper now . Thanks.

Orbit
Posts: 492
Joined: October 14th, 2003, 5:34 pm

### Exact simulation of brownian motion and of its integral

QuoteOriginally posted by: CroackingToadHi everyone,maybe this is a very trivial question but I would like to have some suggestions on this. I would like to jointly simulate a brownian motion $W(t)$ and its integral $Y(t):=\int_0^t{W(s)ds}$.More precisely, what I would like to have is an exact updating rule that allows me to simulate $W(t_2)$ and $Y(t_2)$ given $W(t_1)$ and $Y(t_1)$. Many thanks in advanceHi all. Sorry joined the thread late. Shouldn't the density that goes with $Y(t)$ be this?...Ok, hasty effort. Y(t) is normal, expectation is zero, variance is given by E[(Y(t))^2], use Fubini / isometry, obtain var = (t^3)/3:)
Last edited by Orbit on March 3rd, 2016, 11:00 pm, edited 1 time in total.

Orbit
Posts: 492
Joined: October 14th, 2003, 5:34 pm

### Exact simulation of brownian motion and of its integral

Sorry, Alan why in OP's question do we need to simulate a joint distribution of Y(t) and W(t)? There is only one source of randomness, W(t)? Thanks

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