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I really need help, suppose we have $$E[x(t)^2]\leq C\exp(Dt),$$ where $C$ and $D$ are positive constants, Is x^2 a martingale? Can we apply the martingale convergence theorem?

fidel
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1 Answers1

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The answer would typically be no. Take $x(t) := \exp(t/2)$ to be deterministic. Then clearly $x(t)^2=\exp(t)$ is a submartingale with respect to any filtration, but not a martingale, and it does not converge in $\mathbb{R}$.

nullUser
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  • i'm asking about x^2 – fidel Jun 03 '13 at 16:37
  • thanks a lot, but i heard from some one that the expectation goes to infinity doesn't mean forcely that the process do so, thus, mybe the process converges – fidel Jun 04 '13 at 01:17
  • @fidel The process $x(t):= \exp(t/2)$ does not converge in $\mathbb{R}$, this is obvious. The martingale convergence theorem assumes you already know your process is a supermartingale (bounded in $L^1$, say) with right continuous paths. One cannot use the martingale convergence theorem to show a process is a martingale. Of course, it is possible for a process $x(t)$ to converge without being a martingale. – nullUser Jun 04 '13 at 15:37
  • you are alright, thanks a lot teacher, it seems you are so used with stochastics and martingales , i'm biginner, thanks alot for second time, could i have your email, i have a lot of questions – fidel Jun 04 '13 at 16:18
  • nullUser, i cited an important question and no one answered me the title is :stochastic integrals and inequalities (boundedness) – fidel Jun 04 '13 at 16:28
  • @fidel If you have a lot of questions, you are more than welcome to ask them all here (I know I did when I was starting out), though I am not volunteering to be your personal teacher. Just make sure that you really try to understand the answers. Both of your questions I answered with elementary measure theory alone, so I would suggest brushing up on measure theory and measure theoretic probability before diving further into stochastic analysis. – nullUser Jun 04 '13 at 17:43