Continuous Local Martingale Question

#1
Continuous Local Martingale Question (involving expected values)

I have attempted the following local martingale question. I'm not sure if the proof is correct.

If M is in the space of continuous local martingales, and [tex]"p"[/tex] is a stopping time of the following filtration [tex]{F_t}[/tex],then [tex](E(M^2_p)) \leq E<M>_p[/tex] where [tex](M^2)_\infty = \lim_{t\rightarrow\infty} M^2_t[/tex]

[tex] (E(M^2_p)) = E(\lim_{t\rightarrow\infty} M^2_\min{t,p}) [/tex] (just to clarify the notation reads min of t and the stopping time p)

[tex]\leq \lim_{t\rightarrow\infty} E(M^2_\min{t,p})[/tex] (Fatou's lemma)

[tex]= \lim_{t\rightarrow\infty} E(<M>_\min{t,p})[/tex] (seen many textbooks apply this step without explaining why, perhaps they have used a condition not used in this question)

[tex]= E(<M>_p)[/tex] (monotone convergence)

Source of Q: Karaztas and Shreve, Brownian Motion and Calculus, Problem 5.19

http://books.google.com.au/books?id...artingale of class dl is a martingale&f=false
 
Last edited:

BGM

TS Contributor
#2
Sorry cannot read the whole question. Please check the attached image.

Besides the resolution of the solution image is a bit low to me.

For both parts you may consider using math tags here to type the information with LaTeX syntax.