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280C, Math Spring 2005 Exponential Martingales In what follows, (, F, P) is the canonical sample space of the Brownian motion (Bt )t0 with B0 = 0; other notation is that used in class. Given H L2 let M denote the associated local martingale: loc t (1) Mt := 0 Hs dBs , t 0. Now dene a strictly positive continuous adapted process Z by (2) Zt := exp Mt 1 2 M t , t 0. Clearly Z0 = 1, and it follows...
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280C, Math Spring 2005 Exponential Martingales In what follows, (, F, P) is the canonical sample space of the Brownian motion (Bt )t0 with B0 = 0; other notation is that used in class. Given H L2 let M denote the associated local martingale: loc
t
(1)
Mt :=
0
Hs dBs ,
t 0.
Now dene a strictly positive continuous adapted process Z by (2) Zt := exp Mt
1 2
M
t
,
t 0.
Clearly Z0 = 1, and it follows easily from Its formula that o
t t
(3)
Zt = 1 +
0
Zs dMs = 1 +
0
Zs Hs dBs .
In other wor...
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