Take \(\tau _n = \infty \) for all \(n\).

Let \(X \in P_{\mathrm{loc}}\). Then there exists a localizing sequence \((\tau _n)_{n \in \mathbb {N}}\) such that for all \(n \in \mathbb {N}\), \(X^{\tau _n}\mathbb {I}_{\tau _n {\gt} 0} \in P\). Since \(P \subseteq Q\), for all \(n \in \mathbb {N}\), \(X^{\tau _n}\mathbb {I}_{\tau _n {\gt} 0} \in Q\). Thus \(X \in Q_{\mathrm{loc}}\).

The forward direction is trivial so we only provide proof for the reverse.

Suppose that \(X \in P_{\mathrm{loc}}\cap Q_{\mathrm{loc}}\). Then, there exists localizing sequences \((\tau _n)_{n \in \mathbb {N}}\) and \((\sigma _n)_{n \in \mathbb {N}}\) such that \(X^{\tau _n} \mathbb {I}_{\tau _n {\gt} 0}\in P\) and \(X^{\sigma _n} \mathbb {I}_{\sigma _n {\gt} 0} \in Q\). Consequently, by the stability of \(P\),

Similarly, by the stability of \(Q\), \(X^{\sigma _n \wedge \tau _n} \mathbb {I}_{\sigma _n \wedge \tau _n {\gt} 0} \in Q\). Thus, as \(\sigma _n \wedge \tau _n\) is a localizing sequence by Lemma 9.4 and \(X^{\sigma _n \wedge \tau _n} \mathbb {I}_{\sigma _n \wedge \tau _n {\gt} 0} \in P \cap Q\) for all \(n\), it follows that \(X \in (P \cap Q)_{\mathrm{loc}}\)

Using the localizing sequence defined by Lemma 9.11 suffices.

For each \(n\), since \(\sigma _{n,k} \to \infty \) a.s. as \(k \to \infty \), we may choose \(k_n \in \mathbb {N}\) such that \(P(\sigma _{n,k_n} {\lt} \tau _n \wedge n) \le 2^{-n}\). Then, defining \(\tau '_n = \tau _n \wedge \sigma _{n,k_n}\), we have \(\tau _n' \to \infty \) by the Borel-Cantelli lemma.

\((P_{\mathrm{loc}})_{\mathrm{loc}} \supseteq P_{\mathrm{loc}}\) by Lemma 9.9 so we only prove the reverse inclusion.

Let \(X\) be a process in \((P_{\mathrm{loc}})_{\mathrm{loc}}\). By definition there exists a localizing sequence \((\tau _n)_{n \in \mathbb {N}}\) such that for all \(n \in \mathbb {N}\), \(X^{\tau _n}\mathbb {I}_{\tau _n {\gt} 0}\) is in \(P_{\mathrm{loc}}\). By definition of \(P_{\mathrm{loc}}\), for each \(n\) there exists a localizing sequence \((\sigma _{n,k})_{k \in \mathbb {N}}\) such that for all \(k \in \mathbb {N}\), \((X^{\tau _n}\mathbb {I}_{\tau _n {\gt} 0})^{\sigma _{n,k}}\mathbb {I}_{\sigma _{n,k} {\gt} 0}\) is in \(P\).

By Lemma 9.12, it suffices to show that there exists a pre-localizing sequence \((\tau '_n)_{n \in \mathbb {N}}\) such that for all \(n \in \mathbb {N}\), \(X^{\tau '_n}\mathbb {I}_{\tau '_n {\gt} 0}\) is in \(P\). Thus, using the localizing sequences \(\tau '_n = \tau _n \wedge \sigma _{n, k_n}\) defined by Lemma 9.13, it remains to argue that by stability of \(P\), \(X^{\tau '_n}\mathbb {I}_{\tau '_n {\gt} 0}\) is in \(P\) for all \(n\). Indeed, this follows as \(X^{\tau '_n}\mathbb {I}_{\tau '_n {\gt} 0} = ((X^{\tau _n}\mathbb {I}_{\tau _n {\gt} 0})^{\sigma _{n,k_n}}\mathbb {I}_{\sigma _{n,k_n} {\gt} 0})^{\tau '_n}\mathbb {I}_{\tau '_n {\gt} 0}\) where \((X^{\tau _n}\mathbb {I}_{\tau _n {\gt} 0})^{\sigma _{n,k_n}}\mathbb {I}_{\sigma _{n,k_n} {\gt} 0}\) is in \(P\) by construction and \(P\) is stable.

Since \(X \in P_{\mathrm{loc}}\), then \(X \in (Q_{\mathrm{loc}})_{\mathrm{loc}}\) by assumption and Lemma 9.7. By Lemma 9.14, \((Q_{\mathrm{loc}})_{\mathrm{loc}} = Q_{\mathrm{loc}}\). Thus \(X \in Q_{\mathrm{loc}}\).

Follows directly by using the localizing sequence defined in Lemma 9.16.

If \(X\) is a.s. right continuous, then it is locally right continuous by Lemma 9.17.

On the other hand, assuming \(X\) is locally right continuous, there exists a localizing sequence \((\tau _n)_{n \in \mathbb {N}}\) such that for all \(n \in \mathbb {N}\) and \(\omega \in \Omega \), \((X^{\tau _n}\mathbb {I}_{\tau _n {\gt} 0})(\omega )\) is right continuous. Thus, for almost surely every \(\omega \) and any \(t \in T\) there exists \(N \in \mathbb {N}\) such that \(\tau _N(\omega ) {\gt} t + 1\) (not that the ordering of a.s. and for all is important). Hence, as \(X_s(\omega ) = (X^{\tau _N}\mathbb {I}_{\tau _N {\gt} 0})_s(\omega )\) on a neighborhood of \(t\), we have that \(X(\omega )\) is right continuous at \(t\). Consequently, as \(t\) was arbitrary, \(X\) is a.s. right continuous.

Same proof as in Lemma 9.18.

The forward direction follows from Lemmas 9.18 and 9.19 while the reverse direction follows from Lemma 9.17.

Trivial.

Trivial.

Follows from Lemmas 9.21 and 9.22.

This follows from Lemma 9.6.

Clearly, the stopped process \(M^{\tau }\mathbb {I}_{\tau {\gt} 0}\) is cadlag and it remains to show that it is a martingale.

Fixing \(s \le t \in T\), as \(\{ \tau {\gt} 0\} \in \mathcal{F}_0 \subseteq \mathcal{F}_s\), we have

Thus, as \(\tau \wedge t\) is a bounded stopping time, we have by the optional stopping theorem (Lemma 7.74) that \(P[M_{\tau \wedge t} \mid \mathcal{F}_{s}] = M_{(\tau \wedge t) \wedge s} = M_{\tau \wedge s}\) and so, \(P[M^{\tau }_t \mathbb {I}_{\tau {\gt} 0} \mid \mathcal{F}_s] = M^{\tau }_s \mathbb {I}_{\tau {\gt} 0}\) as required.

This follows from the definitions and Lemma 7.55.

Let \(t \in T\) and \(\tau \le t\) be a stopping time. By Lemma 7.75 and nonnegativity we get that \(0 \le X_\tau \le P[X_t \mid X_\tau ]\). Thus, we conclude by applying Lemma 7.51 and Lemma 7.47.

Assume that \(X\) is uniformly integrable. We know from Lemma 9.34 that \(X\) is of class DL. Moreover, for any finite stopping time \(\tau \), We have that \(X_\tau = \lim _{n \to +\infty } X_{\tau \land n}\). Thanks to Lemma 7.58, we deduce that \(X\) is of class D.

Conversely, if \(X\) is of class D, then applying Lemma 7.55 using constant stopping times will yield uniform integrability.

Let \(X\) be càdlàg martingale. By Lemma 7.13, \((|X_t|)_{t \in T}\) is a nonnegative càdlàg submartingale, and the result follows from Lemma 9.34 along with Lemma 7.54.

Applying Lemma 7.54, this follows from Lemma 9.34 along with Lemma 9.35.

Let \(X\) be a process with locally integrable supremum and \(\tau \) be a stopping time. Let \(t \in T\). Then \((X^\tau )^*_t = \sup _{s \le t} \| X_{\tau \land s}\| \le \sup _{s \le t} \| X_s\| = X^*_t\), and as \(X^*_t\) is integrable, so is \((X^\tau )^*_t\). Thus \((X^\tau \mathbb {I}_{\tau {\gt} 0})^*_t\) is integrable, concluding the proof.

Let \(X\) be a process of class D and \(\tau \) be a stopping time. For any finite stopping time \(\sigma \), we have that \(X_\sigma ^\tau = X_{\sigma \land \tau }\). Because \(\sigma \land \tau \) is finite and \(X\) is of class D, we deduce from Lemma 7.55 that \(\{ X_{\sigma \land \tau } \mid \sigma \text{ is a finite stopping time}\} \) is uniformly integrable, and thus that \(\{ X^\tau _\sigma \mid \sigma \text{ is a finite stopping time}\} \) is uniformly integrable. Using Lemma 7.51, we obtain that \(\{ X^\tau _\sigma \mathbb {I}_{\tau {\gt} 0} \mid \sigma \text{ is a finite stopping time}\} \) is uniformly integrable, which concludes the proof.

Let \(X\) be a process of class DL, \(\tau \) be a stopping time. Let \(t \in T\). For any stopping time \(\sigma \le t\), we have that \(X_\sigma ^\tau = X_{\sigma \land \tau }\). Because \(\sigma \land \tau \) is bounded by \(t\) and \(X\) is of class DL, we deduce from Lemma 7.55 that \(\{ X_{\sigma \land \tau } \mid \sigma \text{ is a stopping time with } \sigma \le t\} \) is uniformly integrable, and thus that \(\{ X^\tau _\sigma \mid \sigma \text{ is a stopping time with } \sigma \le t\} \) is uniformly integrable. Using Lemma 7.51, we obtain that \(\{ X^\tau _\sigma \mathbb {I}_{\tau {\gt} 0} \mid \sigma \text{ is a stopping time with } \sigma \le t\} \) is uniformly integrable, which concludes the proof.

Let \(t \in T\). For every stopping time \(\tau \) with \(\tau \le t\), we have \(\| X_\tau \| \le X^*_t\). Because by hypothesis \(X^*_t\) is integrable, we deduce from Lemma 7.52 that \(\{ X_\tau \mid \tau \text{ is a stopping time with } \tau \le t\} \) is uniformly integrable. This proves that \(X\) is of class DL.

Combine Lemma 9.41 and Lemma 9.7.

Take \(\tau _n := n\). Then

Because \(X\) is of class DL, that last set is uniformly integrable, thus

is uniformly integrable thanks to Lemma 7.55. Lemma 7.51 allows to conclude that

is uniformly integrable, thus \(X^{\tau _n} \mathbb {I}_{\tau _n {\gt} 0}\) is of class D. Obviously \(\tau _n \rightarrow +\infty \) as \(n\) goes to infinity, so \(X\) is locally of class D.

Apply Lemma 9.15 using Lemma 9.43 and Lemma 9.39.

Let \(K \subseteq T\) be a compact set and \(\omega \in \Omega \). Assume that \(X(\omega )(K)\) is not bounded. Then there exists a sequence \((t_n)\) in \(K\) such that for all \(n \in N\), \(d(X_{t_n}(\omega ), x) \ge n\), for some arbitrary \(x \in E\). Because \(K\) is compact, there is a subsequence \((t_{\phi (n)})\) that converges. Then one can extract a subsequence \((t_{\phi (\psi (n))})\) which either converges from below or from above. In both cases the sequence \((X_{t_{\phi (\psi (n))}})\) will converge, contradicting the hypotheses.

By Corollary 8.4, each \(\tau _n\) is a stopping time. Moreover, for all \(n \in \mathbb {N}\), \(X_t \ge n+1 \implies X_t \ge n\), thus \(\tau _n \le \tau _{n+1}\). Finally, for every \(\omega \in \Omega \) and \(t_0 \in T\) there exists \(N \in \mathbb {N}\) such that for all \(s \le t_0\), \(X_s \le N\) thanks to Lemma ??. Thus for all \(n \ge N\), \(\tau _n(\omega ) \ge t_0\), proving that \(\tau _n\) tends to infinity as n goes to infinity.

If \(\tau {\gt} t\), then for all \(s \le t\), \(\| X_s\| \le n\), and thus \((X^\tau )^*_t = \sup _{s \le t} \| X_{\tau \land s}\| \le n = n + \mathbb {1}_{\tau \le t} \| X_{\tau }\| \). Otherwise \((X^\tau )^*_t = \sup _{s \le \tau } \| X_s\| \). For \(s {\lt} \tau \), \(\| X_s\| \le n\), and \(\| X_\tau \| \le \| X_\tau \| \) so \(\sup _{s \le \tau } \| X_s\| \le n \lor \| X_\tau \| \le n + \| X_\tau \| = n + \mathbb {I}_{\tau \le t} \| X_\tau \| \).

Set \(\tau _n := \inf \{ t | X_t \ge n\} \). This is a localizing sequence by Lemma 9.46. For every \(t \in T\), we have by Lemma 9.47 that \((X^{\tau _n})^*_t \le n + \mathbb {I}_{\tau _n \le t} \| X_{\tau _n}\| = n + \mathbb {I}_{\tau _n \le t} \| X_{\tau _n \land t}\| \). Because X is of class DL, \(X_{\tau _n \land t}\) is integrable, so \((X^{\tau _n})^*_t\) is integrable too, so \((X^{\tau _n} \mathbb {I}_{\tau _n {\gt} 0})^*_t\) is integrable. Thus \(X\) has locally integrable supremum.

Apply Lemma 9.15 using Lemma 9.48 and Lemma 9.38.

Define the stopping times \(\sigma _n = \inf \{ t \mid \Vert X_t \Vert \ge n\} \) and set \(\tau _n = \sigma _n \wedge n\). \(\sigma _n\) is a stopping time by Lemma 8.4 and so \(\tau _n\) is also a stopping time. The times \((\tau _n)_{n \in \mathbb {N}}\) form a localizing sequence. By Lemma 9.47, we have that

It remains to show that \(X_{\tau _n}\) is integrable for each \(n\). This follows by Lemma 7.8 as \(\tau _n\) is a bounded stopping time.

By Lemma 9.50, it suffices to show that every cadlag submartingale has locally integrable supremum. This is done in Lemma 9.52.

By Lemma 9.15, it suffices to show that if \(X\) is a submartingale then it is locally of class D. This is done in Lemma 9.53.