Requirements for square integrable in the Doob-Meyer-Decomposition

Question

Hey i have given a non negative supermartingale $(J_{t})_{t\in[0,T]}$ of Class D. So there exists a Doob meyer decomposition $J_{t}=M_{t}-A_{t}$ where $M_{t}$ is uniformly integrable since $(J_{t})$ is of class D. What requirements are needed to make $M_{t}$ square integrable? Since i want to apply martingale representation theorem. In my literature it is only said that: since $\{M_{t}: 0\leq t\leq T\}$ bounded integrable Martingale, it is square integrable. But i dont get this.

Answer 1

Ok in your case if got it right you have :

$Z_t=KS_t$ is dominating $X_t=e^{-rt}.\Phi(S_t)$, so as the snell envelop $J_t$ is the "smallest supermaringale" dominating $X_t$. We have then (note $Z_t$ is a martingale on $[0,T]$ so it is also a supremartingale):

$$0\leq J_t\leq Z_t$$

Now we have using Itô's isometry that $\forall~ t\in [0,T]$ :

$$E[M^2_t]=E[\langle M\rangle_t]=E[\langle J\rangle_t]\leq E[\langle Z\rangle_t]<+\infty$$

Important Note

In your case $J_t$ is continuous so there is no jump part in the quadratic variation of this process. That's because under your assumptions, $X_t$ is regular in the sense of Karatzas and Shreve "Brownian Motion and Stochastic Calculus" definition 4.12 and by theorem 4.16 page 28 therein. In any case, I think that with a little bit of work you can figure out the same result in the jump case.