# Inclusion of stochastic processes classes

Why $$M^{2}_{[0,T]} \subset P^{2}_{[0,T]}$$, where:

$$M^{2}_{[0,T]} = \{f: \Omega \times [0, T] \to \mathbb{R}^{d \times m}: f - product \ measurable, (\mathcal{F_{t}})\ adapted, \mathbb{E} \int_{0}^{T} ||f(s)||^{2}ds< \infty\}$$

$$P^{2}_{[0,T]} = \{f: \Omega \times [0, T] \to \mathbb{R}^{d \times m}: f - product \ measurable, (\mathcal{F_{t}})\ adapted, \mathbb{P}( \int_{0}^{T} ||f(s)||^{2}ds< \infty ) =1\} ?$$

I can't see why probability one does not result in finite expected value.

Thank you!

This follows from the fact that if a random variable $$Y$$ is square integrable, then $$P(Y^2<\infty)=1$$ (essentially because $$nP(Y^2>n)\leqslant \mathbb E\left[Y^2\mathbf 1\{Y^2>n\}\right]\leqslant \mathbb E\left[Y^2\right]$$.
If $$X$$ is a random variable such that $$\mathbb P(X^2<\infty)=1$$ but $$\mathbb E\left[X^2\right]$$ is infinite, let $$f\colon (\omega,t)\mapsto X(\omega)$$. Then $$f$$ belongs to $$P^2_{[0,T]}$$ but not to $$M^2_{[0,T]}$$.
• But why there is no inclusion to left? What example of process is in $P^{2}$ but not in $M^{2}$? – tangerine Mar 26 '20 at 9:35