# Continuous Local Martingale Question

#### chrishello

##### New Member
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 $$"p"$$ is a stopping time of the following filtration $${F_t}$$,then $$(E(M^2_p)) \leq E<M>_p$$ where $$(M^2)_\infty = \lim_{t\rightarrow\infty} M^2_t$$

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

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

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

$$= E(<M>_p)$$ (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

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#### BGM

##### TS Contributor

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.

#### chrishello

##### New Member
Hi BGM, I just transformed the question and answer into Latex syntax as you suggested.