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Dear all, I hope you can help me with the proof of the following result:

Fact If $X$ is a continuous local martingale, then $[X]_t < \infty $ a.s. for every $t \geq 0$, where $[X]$ denote the quadratic variation of the process $X$.

I have tried in vain different ways for approaching this problem: firstly using the definition of quadratic variation and secondly using the Doob-Meyer decomposition of $X$ but in both cases I got stuck in mountains of calculations.

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2 Answers 2

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Assuming your process is one dimensional, by representation theorem there exist a Brownian motion $B$ and a (predictible) process $Y_s$ verifying $\forall t>0$, $\int_0^tY_s^2ds<+\infty$ almost surely such that :

$X_t=E[X_0]+\int_0^tY_sdB_s$ for all $t>0$

Then $[X]_t=\int_0^tY_s^2ds$ which is almost surely finite by representation theorem.

Note that your statement is only almost surely true

Regards

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  • $\begingroup$ Thanks a lot, TheBridge. I was struggling in vain with the definition of quadratic variation and mountains of calculations, but at the end the solution was easy and elegant, as you showed. Of course, my statement is true only almost surely. $\endgroup$
    – alezok
    Commented May 9, 2011 at 7:17
  • $\begingroup$ Hi, I suggest that you edit your question to follow the recommandations of Nate Eldregde Regards $\endgroup$
    – TheBridge
    Commented May 9, 2011 at 14:00
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I guess I have found another different proof of the fact I have asked. Could you check whether it is correct? Thanks

Lemma If $(M_t)_{t\geq 0}$ is a continuous local martingale, then $|M_t| < \infty$ a.s. for every $t\geq 0$.

Proof of the lemma: Suppose for sake of contradiction that $\exists s$ such that $P(|M_s|=\infty)>0$. If $(T_n)_n$ is a reducing sequence for $M$, then $\exists n$ such that $P(T_n > s)>0$. Thus $P(|M_{T_n \wedge t} |=\infty)>0$, but $M_{T_n \wedge t}$ is in $L^1$, since it is a martingale. $\Box$

Now, if we take $X_t$ continuous local martingale we have that also $X^2_t$ is a continuous local martingale and $|X^2_t| < \infty$ a.s. for every $t \geq 0$. (see comments)

The lemma applied to $X_t$ gives that $|X_t| < \infty$ a.s. for every $t \geq 0$ and thus $|X_t|^2 < \infty$ a.s. for every $t \geq 0$.

Finally, also $X^2_t-[X]_t$ is a continuous local martingale and hence we have $|X^2_t-[X]_t| < \infty$. From these facts follow that $[X]_t < \infty$ a.s. for every $t \geq 0$.

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  • $\begingroup$ if we take $X_t$ continuous local martingale we have that also $X^2_t$ is a continuous local martingale: sure about this? $\endgroup$
    – Did
    Commented May 12, 2011 at 20:27
  • $\begingroup$ Could you check whether it is correct? You could first apply Nate's recommendation. $\endgroup$
    – Did
    Commented May 12, 2011 at 20:28
  • $\begingroup$ You're right, $X^2_t$ is not necessarily a continuous local martingale. Actually the only thing I needed was $|X_t|^2 < \infty$ that simply follows the lemma applied to $X_t$. $\endgroup$
    – alezok
    Commented May 13, 2011 at 7:25

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