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The martingale representation theorem

So far we have concluded that if a diffusion $ X\left( t\right) $ is of the form

$\displaystyle dX\left( t\right) =\delta \left( t\right) dB\left( t\right)$   ,

then it is a martingale. The martingale representation theorem states that if $ X\left( t\right) $ is a martingale adapted to the filtration generated by $ B\left( t\right) $, then it will be of the above form.

Theorem 43 (Martingale representation)   Let $ B\left( t\right) ,t\in \left[ 0,T\right] $ be a Brownian motion on $ %% \left( \Omega ,F,P\right) $, and let $ X\left( t\right) $ be a martingale under $ P$ relative to $ \mathscr{F}\left( t\right) $. Then there exists an $ %% \mathscr{F}\left( t\right) $-measurable process $ \delta \left( t\right) $ such that
$\displaystyle X\left( t\right)$ $\displaystyle =$ $\displaystyle X\left( 0\right) +\int_{0}^{t}\delta \left( s\right) dB\left( s\right)$   , or  
$\displaystyle dX\left( t\right)$ $\displaystyle =$ $\displaystyle \delta \left( t\right) dB\left( t\right)$   .  

next up previous contents
Next: The Black-Scholes formula III Up: Modelling: Continuous state space Previous: The Black-Scholes formula II   Contents
Kyriakos 2003-03-17