7 Filtrations, processes and martingales
First, recall the definitions of a filtration, an adapted process, a (sub)martingale, a stopping time and a stopped process, which are already in Mathlib.
A filtration on a measurable space \((\Omega , \mathcal{A})\) with measure \(P\) indexed by a preordered set \(T\) is a family of sigma-algebras \(\mathcal{F} = (\mathcal{F}_t)_{t \in T}\) such that for all \(i \le j\), \(\mathcal{F}_i \subseteq \mathcal{F}_j\) and for all \(t \in T\), \(\mathcal{F}_t \subseteq \mathcal{A}\).
A process \(X : T \to \Omega \to E\) is said to be adapted with respect to a filtration \(\mathcal{F}\) if for all \(t \in T\), \(X_t\) is \(\mathcal{F}_t\)-measurable.
Let \(\mathcal{F}\) be a filtration on a measurable space \(\Omega \) with measure \(P\) indexed by \(T\). A family of functions \(M : T \to \Omega \to E\) is a martingale with respect to a filtration \(\mathcal{F}\) if \(M\) is adapted with respect to \(\mathcal{F}\) and for all \(i \le j\), \(P[M_j \mid \mathcal{F}_i] = M_i\) almost surely.
Let \(\mathcal{F}\) be a filtration on a measurable space \(\Omega \) with measure \(P\) indexed by \(T\). A family of functions \(M : T \to \Omega \to E\) is a submartingale with respect to a filtration \(\mathcal{F}\) if \(M\) is adapted with respect to \(\mathcal{F}\) and for all \(i \le j\), \(P[M_j \mid \mathcal{F}_i] \ge M_i\) almost surely.
A stopping time with respect to some filtration \(\mathcal{F}\) indexed by \(T\) is a function \(\tau : \Omega \to T\) such that for all \(i\), the preimage of \(\{ j \mid j \le i\} \) along \(\tau \) is measurable with respect to \(\mathcal{F}_i\).
Let \(X : T \to \Omega \to E\) be a stochastic process and let \(\tau : \Omega \to T\). The stopped process with respect to \(\tau \) is defined by
We now give the definition of a filtered probability space satisfying the usual conditions.
For \(\mathcal{F}\) a filtration indexed by \(T\) and \(t \in T\), we define \(\mathcal{F}_{t-} = \bigsqcup _{s {\lt} t} \mathcal{F}_s\) (if that supremum is nonempty: we set \(\mathcal{F}_{\bot -} = \mathcal{F}_\bot \)) and \(\mathcal{F}_{t+} = \bigsqcap _{s {\gt} t} \mathcal{F}_s\).
Note that \(\bigsqcap \) and \(\bigsqcup \) denote the infimum and supremum in the lattice of sigma-algebras on \(\Omega \).
We say that the filtration is right-continuous if for all \(t \in T\), \(\mathcal{F}_t = \mathcal{F}_{t+}\).
We say that a filtered probability space \((\Omega , \mathcal{F}, P)\) satisfies the usual conditions if the filtration is right-continuous and if \(\mathcal{F}_0\) contains all the \(P\)-null sets.
Let \(\mathcal{F}\) be a filtration on a measurable space indexed \(\Omega \) by a linearly ordered set \(T\). Let \(S = \{ \{ \bot \} \times A \mid A \in \mathcal{F}_\bot \} \) if \(T\) has a bottom element and \(S = \emptyset \) otherwise. The predictable sigma-algebra on \(T \times \Omega \) is the sigma-algebra generated by the set of sets \(\{ (t, \infty ] \times A \mid t \in T, \: A \in \mathcal{F}_t\} \cup S\).
A process \(X : T \to \Omega \to E\) is said to be predictable with respect to a filtration \(\mathcal{F}\) if it is measurable with respect to the predictable sigma-algebra on \(T \times \Omega \).
A predictable process is progressively measurable.
Let \(X : T \times \Omega \to E\) be a predictable process, we will show that it is progressively measurable. Namely, fixing \(t \in T\), denoting
we need to show that \(\iota _t \circ X : [0, t] \times \Omega \to E\) is measurable with respect to \(\mathcal{B}([0, t]) \otimes \mathcal{F}_t\).
Denoting \(\Sigma _{\mathcal{F}}\) for the predictable \(\sigma \)-algebra generated by \(\mathcal{F}\), as \(u\) is predictable, we have that \(X^{-1}(\mathcal{B}(E)) \le \Sigma _{\mathcal{F}}\). Thus, to show that \(\iota _t \circ X\) is \(\mathcal{B}([0, t]) \otimes \mathcal{F}_t\)-measurable, it suffices to show that \(\iota _t^{-1}(\Sigma _{\mathcal{F}}) \le \mathcal{B}([0, t]) \otimes \mathcal{F}_t\). In particular, as
is suffices to show that sets of the form \(\iota _t^{-1}((s, \infty ) \times A)\) for some \(s \in T, A \in \mathcal{F}_s\) and \(\iota _t^{-1}(\{ \bot \} \times A)\) for some \(A \in \mathcal{F}_\bot \) are \(\mathcal{B}([0, t]) \otimes \mathcal{F}_t\)-measurable.
Indeed, if \(A \in \mathcal{F}_\bot \)
while for any \(s \in T\) and \(A \in \mathcal{F}_s\),
By the monotonicity of the filtration \(\mathcal{F}\), all of these cases are \(\mathcal{B}([0, t]) \otimes \mathcal{F}_t\)-measurable allowing us to conclude.
Sets of the form \((s, t] \times A\) for any \(A \in \mathcal{F}_s\) is measurable with respect to the predictable \(\sigma \)-algebra.
For \(t \le s\), the set in question is empty and thusly, trivially measurable. On the other hand, for \(s {\lt} t\), measurability follows as \((s, t] \times A = (s, \infty ) \times A \setminus (t, \infty ) \times A\).
Let \(X : \mathbb {N} \to \Omega \to E\) be a stochastic process and let \(\mathcal{F}\) be a filtration indexed by \(\mathbb {N}\). Then \(X\) is predictable if and only if \(X_0\) is \(\mathcal{F}_0\)-measurable and for all \(n \in \mathbb {N}\), \(X_{n+1}\) is \(\mathcal{F}_n\)-measurable.
Suppose first that \(X\) is predictable. Straightaway, \(X_0\) is \(\mathcal{F}_0\)-measurable as predictable implies progressively measurable which in turn implies adapted.
Fixing \(n\), we observe that for any \(S \in \mathcal{B}(E)\),
where
and
As \(X^{-1}(S) \in \Sigma _{\mathcal{F}}\) – the predictable \(\sigma \)-algebra, it suffices to show that \(\pi ^{-1}(\iota ^{-1}(\Sigma _{\mathcal{F}})) \in \mathcal{F}_n\). To this end, we again only need to show these for the generating sets of \(\Sigma _{\mathcal{F}}\):
For \(A \in \mathcal{F}_0\), measurability is clear as \(\iota ^{-1}(\{ 0\} \times A) = \varnothing \).
Similarly, for \(m {\gt} n\) and \(A \in \mathcal{F}_m\), \(\iota ^{-1}((m, \infty ) \times A) = \varnothing \).
For \(m \le n\) and \(A \in \mathcal{F}_m \le \mathcal{F}_n\) we have that \(\pi ^{-1}(\iota ^{-1}((m, \infty ) \times A)) = A\) which is \(\mathcal{F}_n\) measurable by the monotonicity of the filtration.
Now, supposing \(X_0\) is \(\mathcal{F}_0\)-measurable and \(X_{n + 1}\) is \(\mathcal{F}_n\)-measurable, we will show that \(X\) is predictable. Indeed, fixing \(S \in \mathcal{B}(E)\), we have
Thus, as \(\{ 0\} \times X_0^{-1}(S) \in \Sigma _{\mathcal{F}}\) by construction and \(\{ n + 1\} \times X_{n + 1}^{-1}(S) = (n, n + 1] \times X_{n + 1}^{-1}(S) \in \Sigma _{\mathcal{F}}\) by Lemma 7.13 and the fact that \(X_{n + 1}^{-1}(S) \in \mathcal{F}_n\), we have that \(X^{-1}(S) \in \Sigma _{\mathcal{F}}\) as required.
7.1 Local martingales
This section contains material taken mostly from [ and [ .
A localizing sequence is a sequence of stopping times \((\tau _n)_{n \in \mathbb {N}}\) such that \(\tau _n\) is non-decreasing and \(\tau _n \to \infty \) as \(n \to \infty \) (a.s.).
The constant sequence \(\tau _n = \infty \) is a localizing sequence.
Let \((\sigma _n), (\tau _n)\) be localizing sequences. Then \((\sigma _n \wedge \tau _n)\) is a localizing sequence.
Let \(P\) be a class of stochastic processes (or equivalently a predicate on stochastic processes). We say that a stochastic process \(X : \mathbb {R}_+ \to \Omega \to E\) is locally in \(P\) (or satisfies \(P\) locally) if there exists a localizing sequence \((\tau _n)_{n \in \mathbb {N}}\) such that for all \(n \in \mathbb {N}\), the process \(X^{\tau _n}\mathbb {I}_{\tau _n {\gt} 0}\) is in \(P\) (in which \(X^{\tau _n}\) denotes the stopped process). We denote the class of processes that are locally in \(P\) by \(P_{\mathrm{loc}}\).
For any class of processes \(P\), we have \(P \subseteq P_{\mathrm{loc}}\).
Take \(\tau _n = \infty \) for all \(n\).
A class of stochastic processes \(P\) is stable if whenever \(X\) is in \(P\), then for any stopping time \(\tau \), the process \(X^{\tau }\mathbb {I}_{\tau {\gt} 0}\) is also in \(P\).
If \(P, Q\) are stable classes of processes then \((P\cap Q)_{\mathrm{loc}} = P_{\mathrm{loc}}\cap Q_{\mathrm{loc}}\).
If \(P \subseteq Q\) then \(P_{\mathrm{loc}} \subseteq Q_{\mathrm{loc}}\).
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 \(X \in Q_{\mathrm{loc}}\).
For any stable class of processes \(P\), we have \((P_{\mathrm{loc}})_{\mathrm{loc}} = P_{\mathrm{loc}}\).
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 \(P\).
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 \(P(\sigma _{n,k_n} {\lt} \tau _n \wedge n) \le 2^{-n}\). Let \(\tau '_n = \tau _n \wedge \sigma _{n,k_n}\). \(\tau _n' \to \infty \) by the Borel-Cantelli lemma. Let \(\tau ''_n = \inf _{m \ge n} \tau '_m\). Then \((\tau ''_n)_{n \in \mathbb {N}}\) is a localizing sequence.
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, \(X^{\tau ''_n} = ((X^{\tau _n}\mathbb {I}_{\tau _n {\gt} 0})^{\sigma _{n,k_n}}\mathbb {I}_{\sigma _{n,k_n} {\gt} 0})^{\tau ''_n}\mathbb {I}_{\tau ''_n {\gt} 0}\). \((X^{\tau _n}\mathbb {I}_{\tau _n {\gt} 0})^{\sigma _{n,k_n}}\mathbb {I}_{\sigma _{n,k_n} {\gt} 0}\) is in \(P\) by construction and \(P\) is stable.
Let \(P, Q\) be two classes of stochastic processes such that \(P \subseteq Q_{\mathrm{loc}}\) and \(Q\) is stable. Let \(X\) be a stochastic process that satisfies \(P\) locally. Then \(X\) satisfies \(Q\) locally. In short, if \(P\) implies \(Q\) locally, then \(P\) locally implies \(Q\) locally.
A stochastic process is a local martingale if it is locally a martingale in the sense of Definition 7.18. That is, there exists a localizing sequence \((\tau _n)_{n \in \mathbb {N}}\) such that for all \(n \in \mathbb {N}\), the process \(M^{\tau _n}\mathbb {I}_{\tau _n {\gt} 0}\) is a martingale.
A stochastic process is a local submartingale if it is locally a submartingale in the sense of Definition 7.18. That is, there exists a localizing sequence \((\tau _n)_{n \in \mathbb {N}}\) such that for all \(n \in \mathbb {N}\), the process \(M^{\tau _n}\mathbb {I}_{\tau _n {\gt} 0}\) is a submartingale.
Every martingale is a local martingale.
This follows from Lemma 7.19.
The class of martingales is stable. That is, if \(M\) is a martingale and \(\tau \) is a stopping time, then the stopped process \(M^{\tau }\mathbb {I}_{\tau {\gt} 0}\) is also a martingale.
The class of submartingales is stable. That is, if \(M\) is a submartingale and \(\tau \) is a stopping time, then the stopped process \(M^{\tau }\mathbb {I}_{\tau {\gt} 0}\) is also a submartingale.
Let \(M\) be a continuous local martingale with \(M_0 = 0\). If \(M\) is also a finite variation process, then \(M_t = 0\) for all \(t\).