13 Classes of martingales and related processes
The main reference for this part is [ HWY19 ] .
Notations for classes of processes:
\(\mathcal{V}\): finite variation processes (Definition 13.9)
\(\mathcal{V}^+\): non-decreasing finite variation processes
\(\mathcal{A}\): adapted processes with integrable variation (Definition 13.14)
\(\mathcal{A}^+\): non-decreasing adapted integrable processes
\(\mathcal{M}\): càdlàg martingales (TODO: [ HWY19 ] add uniform integrability)
\(\mathcal{M}^c\): continuous martingales
\(\mathcal{M}^2\): square-integrable martingales (Definition 13.19)
\(\mathcal{C}_{\mathrm{loc}}\) for a class \(\mathcal{C}\): processes that are locally in \(\mathcal{C}\)
Note: [ HWY19 ] use \(\mathcal{W}\) for \(\mathcal{M} \cap \mathcal{A}\).
13.1 Jumps of a process
The jumps of a process \(X : T \to \Omega \to E\) is the process \(\Delta X : T \to \Omega \to E\) defined by \((t, \omega ) \mapsto X_t(\omega ) - X_{t^-}(\omega )\) .
The jump part (or purely discontinuous part) of a process \(X : T \to \Omega \to E\) is the process \(T \to \Omega \to E\) defined by \((t, \omega ) \mapsto \sum _{0 {\lt} s \le t} \Delta X_s(\omega )\) .
The continuous part of a process \(X : T \to \Omega \to E\) is the process \(X^c : T \to \Omega \to E\) defined by \((t, \omega ) \mapsto X_t(\omega ) - \sum _{0 {\lt} s \le t} \Delta X_s(\omega )\) .
The large jump part of a process \(X : T \to \Omega \to E\) at level \(\varepsilon \) is the process \(X^{d, \varepsilon } : T \to \Omega \to E\) defined by \((t, \omega ) \mapsto \sum _{0 {\lt} s \le t} \Delta X_s(\omega ) \mathbb {1}_{\{ \Vert \Delta X_s(\omega ) \Vert {\gt} \varepsilon \} }\) .
13.2 Integrable variation
The (extended-real-valued) variation of a function \(f : T \to E\) on a set \(s\) inside a linear order is the supremum of \(\sum _i \mathrm{edist}(f(u_{i+1}, f(u_i)))\) over all finite increasing sequences \(u : N \to T\) in \(s\). We denote it by \(V_f(s)\) .
A function \(f : T \to E\) is of bounded variation on a set \(s\) if its variation \(V_f(s)\) is finite.
A function \(f : T \to E\) is of locally bounded variation on a set \(s\) if for every \(a, b\) in \(s\), the variation \(V_f(s \cap [a, b])\) is finite.
The set of points of discontinuity of a function of locally bounded variation is at most countable.
Mathlib has the result for monotone functions and has the fact that a function of locally bounded variation is the difference of two monotone functions.
A process \(X : T \to \Omega \to E\) is of finite variation if it is right-continuous and for every \(\omega \in \Omega \), the path \(t \mapsto X_t(\omega )\) is of locally bounded variation on \(T\) .
We denote by \(\mathcal{V}\) the class of finite variation processes.
A finite variation process is càdlàg.
Right-continuity is by definition. Left limits exist because a function of locally bounded variation has left limits (that should be in Mathlib’s file about locally bounded variation).
The variation of a process \(X : T \to \Omega \to E\) is the process \(V_X : T \to \Omega \to \mathbb {R}\) defined by \((t, \omega ) \mapsto V_{X(\omega )}([0,t])\) .
The variation process \(V_X\) of a process \(X\) is non-decreasing.
The variation process \(V_X\) of a right-continuous process \(X\) is right-continuous.
We denote by \(\mathcal{A}^+\) the class of non-decreasing adapted integrable processes.
We denote by \(\mathcal{V}^+\) the class of non-decreasing adapted processes.
A process with integrable variation can be written as the difference of two non-decreasing adapted integrable processes.
A process in \(\mathcal{A}_{\mathrm{loc}}\) (locally integrable variation) can be written as a sum of a local martingale and a predictable locally integrable process.
We write the process as a difference of two non-decreasing locally integrable processes, and then apply Lemma 12.73 to each of them. We take the difference of the two resulting local martingales and the difference of the two resulting predictable locally integrable processes.
Let \(M \in \mathcal{M} \cap \mathcal{A}\) be a martingale with integrable variation. Then the jump part \(A = \sum _{0 {\lt} s \le \cdot } \Delta M_s\) is an adapted process with integrable variation. Let \(A_m\) be the martingale part of the Doob decomposition of \(A\). Then \(M = M_0 + A_m\). Finally, the predictable part of the decomposition of \(A\) is continuous.
A predictable uniformly integrable martingale is continuous.
13.3 Square integrable martingales
In this section, \(E\) denotes a complete normed space.
Let \(T\) be a linear order with bottom element 0, on which we have a filtration \(\mathcal{F}\) satisfying the usual conditions. We say that a martingale \(M : T \to \Omega \to E\) is square integrable if it is càdlàg and \(\sup _{t \in T} \Vert M_t \Vert _{L^2} {\lt} \infty \) (Lean remark: use eLpNorm (M t) 2).
A square integrable martingale is uniformly integrable.
Apply Lemma 7.53.
13.3.1 The Hilbert space of square integrable martingales
If \(M\) and \(N\) are square integrable martingales and \(a \in \mathbb {R}\), then \(M + N\) and \(a M\) are square integrable martingales.
If \(M\) is a square integrable martingale, then \(\Vert M \Vert ^2\) is a submartingale.
Apply Lemma 7.15 with the convex function \(f: x \mapsto \Vert x \Vert ^2\) .
For \(M\) a square integrable martingale, the function \(t \mapsto \Vert M_t \Vert _{L^2}\) is non-decreasing.
By Lemma 13.22, \(\Vert M_t \Vert ^2\) is a submartingale. Thus, for \(s \le t\) ,
For \(M\) a square integrable martingale, we have \(M_t \to M_\infty \) almost surely and in \(L^2\) as \(t \to \infty \) .
TODO: use a martingale convergence theorem. Check whether Theorem 7.84 is what we need.
For \(M\) and \(N\) two square integrable martingales, \((M + N)_\infty \) and \(M_\infty + N_\infty \) are a.e. equal.
Lemma 7.85 with \(g := M_\infty + N_\infty \).
For \(M\) and \(N\) two square integrable martingales, \((M - N)_\infty \) and \(M_\infty - N_\infty \) are a.e. equal.
Lemma 7.85 with \(g := M_\infty - N_\infty \).
For \(M\) a square integrable martingale and \(t \in T\), we have that \(P[X_\infty | \mathcal{F}_t]\) is a.e. equal to \(X_t\).
Apply Theorem 7.90.
For \(M\) a square integrable martingale and \(\tau \) a stopping time, we have that \(P[X_\infty | \mathcal{F}_\tau ]\) is a.e. equal to \(X_\tau \).
Apply Theorem 7.91.
For \(M\) a càdlàg and uniformly integrable martingale,
By Lemma 13.27, We have that \(P[M_\infty | \mathcal{F}_t] = M t\) almost surely, thus
where the last inequality is eLpNorm_condExp_le_eLpNorm.
A càdlàg and uniformly integrable martingale \(M\) is square integrable if and only if \(M_\infty \) is in \(\mathrm L^2\).
It is a càdlàg martingale by hypothesis, and the bound follows from Lemma ??.
If \(M\) is a square integrable martingale and \(\tau \) is a stopping time, then \(M^\tau \) is a square integrable martingale.
We apply Lemma 13.30. First, \(M^\tau \) is a martingale. Moreover, \(M\) is uniformly integrable by Lemma 13.20, thus it is of class D by Lemma 9.43, thus \(M^\tau \) is uniformly integrable. Finally, we have
The first equality follows from Lemma 7.92. The first inequality is obtained as follows: if \(\tau (\omega ) = \infty \), then \(M_\tau = M_\infty \) and \(M_t \longrightarrow M_\infty \) almost surely so \(\| M_\infty (\omega )\| \le \sup _t \| M_t(\omega )\| \). Otherwise the inequality is obvious. The second inequality is Corollary 7.100.
For \(M\) a square integrable martingale,
For \(f : \Omega \to E\), we have almost surely that
If \(M\) is a square integrable random variable, then
If \(M\) is a square integrable martingale and \(\tau \) is a stopping time, then \(M_\tau \) is in \(\mathrm L^2\).
A stochastic process \(X\) is purely discontinuous if it is a square integrable martingale such that for any continuous square integrable martingale \(Y\),
We denote by \(\mathcal{M}^2(E)\) or simply \(\mathcal{M}^2\) the space of equivalence classes with respect to indistinguishability of square integrable martingales \(T \to \Omega \to E\) .
The space \(\mathcal{M}^2(E)\) is a real vector space.
We define a norm on \(\mathcal{M}^2\) by
For \(M \in \mathcal{M}^2(E)\), \(\Vert M \Vert = 0\) if and only if \(M = 0\).
By Lemma 13.32, \(\Vert M \Vert = 0\) if and only if for all \(t \in T\), \(\Vert M_t \Vert _{L^2} = 0\) . Because \(M\) is càdlàg, this implies that \(M\) is indistinguishable from \(0\).
We define an inner product on \(\mathcal{M}^2\) by
If \(T\) is separable then the space \(\mathcal{M}^2\) is a Hilbert space.
We show that the map
is a bijective isometry. It is an isometry by Definition 13.39. We define the inverse \(\psi \) as the map which to \(f\) associates a càdlàg martingale that is a modification of \(t \mapsto P[f | \mathcal{F}_t]\), given by Lemma 11.27.
Let \(M\) be a square integrable martingale. Then for any \(t\), \(M_t\) is a.e. equal to \(P[M_\infty | \mathcal{F}_t]\) by Lemma 13.27, thus \(M\) is a càdlàg modification of \(t \mapsto P[f | \mathcal{F}_t]\), thus it is indistinguishable from \(\psi (M_\infty )\).
Let \(f \in L^2(E, P)\) be a.e. strongly measurable with respect to \(\bigsqcup _t \mathcal{F}_t\). Then by Lemma 13.33 we know that almost surely \(P[f | \mathcal{F}_t]\) converges to \(f\). Because \(T\) is separable we can consider \((t_n)\) a sequence of points that tends to infinity. Because \(\psi (f)\) and \(P[f | \mathcal{F}_t]\) are modifications of each other, we have that almost surely, for all \(n \in \mathcal{N}\), \(\psi (f)_{t_n} = P[f | \mathcal{F}_{t_n}]\). But we also know that \(\psi (f)_{t_n}\) converges to \(\psi (f)_\infty \) so we conclude that \(\psi (f)_\infty \) is a.e. equal to f.
13.3.2 Elementary stochastic integrals
For \(V \in \mathcal{E}_{T, F}\) bounded by a constant \(D\), \(M \in \mathcal{M}^2(E)\) and a continuous bilinear map \(B: E \times F \to G\),
TODO: this can be improved to \(D \: \Vert B \Vert \: \Vert M_t \Vert _{L^2}\)?
Let \(C\) be a bound on \(\Vert M_t \Vert _{L^2}\) for all \(t \in T\) . Let \((s_k {\lt} t_k)_{k \in \{ 1, ..., n\} }\) and \(\eta _k\) be the intervals and random variables defining \(V\) . Let \(D\) be a bound on \(\Vert \eta _k\Vert \). Then, for all \(t\) ,
Since only at most one term of that sum is non-zero for each fixed \(t\) , we can bound the sum by the maximum of its terms. It suffices then to bound each term of that sum.
TODO: here we supposed that the intervals of the simple process are disjoint. Check with our Lean def.
For each \(k\) ,
For \(V \in \mathcal{E}_{T, F}\), \(M \in \mathcal{M}^2(E)\) and a continuous bilinear map \(B: E \times F \to G\), the elementary stochastic integral \(V \bullet _B M\) is in \(\mathcal{M}^2(G)\).
By Lemma 10.34, \(V \bullet _B M\) is càdlàg, and we know that it is a martingale by Lemma 10.44 . It remains to show that \(\sup _{t \in T} \Vert (V \bullet _B M)_t \Vert _{L^2} {\lt} \infty \) . By Lemma 13.43, this supremum is bounded by \(2 D \Vert B \Vert \sup _{t \in T} \Vert M_t \Vert _{L^2}\), which is finite since \(M \in \mathcal{M}^2(E)\) and \(V\) is bounded.
We denote by \(\mathcal{M}^{2,c}\) the submodule of \(\mathcal{M}^2\) made of continuous square integrable martingales.
Let \((M^{(n)})\) be a sequence of square integrable martingales and \(N\) be a square integrable martingale such that \(\| M^{(n)}_\infty - N_\infty \| _2 \longrightarrow _{n \to +\infty } 0\). Then there exists \(\phi : \mathbb {N} \to \mathbb {N}\) increasing such that, almost surely, \(M^{(\phi (n))}_\cdot (\omega )\) converges uniformly to \(N_\cdot (\omega )\) as \(n\) goes to infinity.
We can find \(\phi \) increasing such that \(\| M^{(\phi (n))}_\infty - N_\infty \| _2 \le 1/2^n\). Then,
Where the second inequality is Lemma 13.34. In particular, almost surely, \(\sum _n \sup _t \| M^{(\phi (n))}_t - N_t\| {\lt} \infty \) and thus \(\sup _t \| M^{(\phi (n))}_t - N_t\| \longrightarrow _{n \to +\infty } 0\), which concludes the proof.
The submodule \(\mathcal{M}^{2,c}\) is closed in the Hilbert space \(\mathcal{M}^2\).
Take \(M^{(n)}\) a sequence of continuous square integrable martingales that converges to \(N\) in \(\mathcal{M}^2\). This means that \(\| M^{(n)}_\infty - N_\infty \| \) goes to zero as \(n\) goes to infinity, which means that there exists \(\phi \) increasing such that, almost surely, \(M^{(\phi (n))}_\cdot (\omega )\) converges uniformly to \(N_\cdot (\omega )\) as \(n\) goes to infinity (Lemma 13.46). As each \(M^{(\phi (n))}(\omega )\) is continuous, \(N(\omega )\) is also continuous.
If \(M\) is a square integrable martingale, we define its continuous part as the orthogonal projection of \(M\) onto the closed subspace \(\mathcal{M}^{2,c}\). We denote it by \(M^c\).
The continuous part of a square integrable martingale is continuous.
By definition of the orthogonal projection.
The continuous part of a square integrable martingale is a square integrable martingale.
By definition of the orthogonal projection.
If \(M\) is a square integrable martingale, we define its discontinuous part by \(M^d := M - M^c\).
The discontinuous part of a square integrable martingale is a square integrable martingale.
By definition.
We denote by \(\mathcal{M}^{2,d}\) the orthogonal of \(\mathcal{M}^{2,c}\) and call its elements purely discontinuous square integrable martingales.
If \(M\) is a purely discontinuous martingale in the sense of Definition 13.36, then it belongs to \(\mathcal{M}^{2,d}\).
By definition of the orthogonal projection.
The discontinuous part of square integrable martingale is purely discontinuous.
By definition of the orthogonal projection.
If \(M\) is a square integrable martingale, then the decomposition \(M = M^c + M^d\) as the sum of a continuous and a purely discontinuous square integrable martingales is unique up to indistinguishability.
A closed subspace is the complement of its orthogonal subspace.
If \(M\) is a purely discontinuous martingale and \(\tau \) is a stopping time, then \(M^\tau \) is a purely discontinuous martingale.
By Lemma 13.31 \(M^\tau \) is a square integrable martingale. Let \(N\) be a continuous square integrable martingale. Then again by Lemma 13.31, \(N^\tau \) is a continuous square integrable martingale. We thus have
The first equality is Lemma 7.92, the second is the pull-out property of the conditional expectation, the third one is Lemma 13.28, the fourth one is the same as before and the last one is the fact that \(M\) is purely discontinuous while \(N^\tau \) is continuous.
If \(M\) is a square integrable martingale and \(\tau \) is a stopping time, then \((M^\tau )^c = (M^c)^\tau \) and \((M^\tau )^d = (M^d)^\tau \).
For \(V \in \mathcal{E}_{T, \mathbb {R}}\) and \(M, N \in \mathcal{M}^2\), we have
13.4 Locally square integrable martingales
13.4.1 Definition and basic properties
A process is locally square-integrable if it locally satisfies the square-integrable martingale property. We denote that class of processes by \(\mathcal{M}^2_{\mathrm{loc}}\) .
Every square-integrable martingale is locally square-integrable: \(\mathcal{M}^2 \subseteq \mathcal{M}^2_{\mathrm{loc}}\) .
This follows from Lemma 9.9.
If \(M \in \mathcal{M}^2_{\mathrm{loc}}\), then \(\Vert M \Vert ^2\) is a càdlàg local submartingale.
A continuous local martingale is locally square-integrable: \(\mathcal{M}^c_{\mathrm{loc}} \subseteq \mathcal{M}^2_{\mathrm{loc}}\) .
13.4.2 Predictable quadratic variation
For \(M \in \mathcal{M}^2_{\mathrm{loc}}\) with càdlàg paths, the predictable quadratic variation of \(M\) is defined as the predictable part of the Doob-Meyer decomposition of the local submartingale \(\Vert M \Vert ^2\) . We denote it by \(\langle M \rangle \) .
The predictable quadratic variation \(\langle M \rangle \) of \(M \in \mathcal{M}^2_{\mathrm{loc}}\) is a predictable process.
The predictable quadratic variation \(\langle M \rangle \) of \(M \in \mathcal{M}^2_{\mathrm{loc}}\) is càdlàg.
The predictable quadratic variation \(\langle M \rangle \) of \(M \in \mathcal{M}^2_{\mathrm{loc}}\) is locally integrable.
\(\langle M \rangle _0 = 0\) .
The predictable quadratic variation \(\langle M \rangle \) of \(M \in \mathcal{M}^2_{\mathrm{loc}}\) is non-decreasing.
For \(M \in \mathcal{M}^2_{\mathrm{loc}}\), the process \(\Vert M_t \Vert ^2 - \langle M \rangle _t\) is a local martingale.
For \(M, N \in \mathcal{M}^2_{\mathrm{loc}}\), the predictable covariation \(\langle M, N \rangle \) is a stochastic process defined by polarization of the predictable quadratic variation:
The predictable covariation \(\langle M, N \rangle \) of \(M, N \in \mathcal{M}^2_{\mathrm{loc}}\) is a predictable process.
The predictable covariation \(\langle M, N \rangle \) of \(M, N \in \mathcal{M}^2_{\mathrm{loc}}\) is càdlàg.
\(\langle M, N \rangle _0 = 0\) .
For \(M, N \in \mathcal{M}^2_{\mathrm{loc}}\), the process \(\langle M_t, N_t \rangle _E - \langle M, N \rangle _t\) is a local martingale.
The two differences are local martingales by Lemma 13.70, so their linear combination is also a local martingale.
Let \(M\) and \(N\) be square integrable martingales. Then
Let \(B\) be a standard Brownian motion. Then the quadratic variation of \(B\) is given by \(\langle B \rangle _t = t\) .
13.5 Local martingales
The large jump process of a local martingale is a process with locally integrable variation (it’s in \((\mathcal{A} \cap \mathcal{M})_{\mathrm{loc}}\)).
Let \(M\) be a local martingale. Then for any \(\varepsilon {\gt} 0\), \(M\) can be decomposed as \(M = M_0 + U + V\), where \(U\) is a locally bounded martingale (local version of both bounded and martingale) with \(\vert \Delta U \vert \le \varepsilon \) and \(U_0 = 0\) and \(V\) is a local martingale with locally integrable variation (\(V \in (\mathcal{M} \cap \mathcal{A})_{\mathrm{loc}}\)) and \(V_0 = 0\).
See [ HWY19 ] , 7.17
Remark: \(U\) is locally bounded, hence locally square integrable, so we can use the integration machinery for those to define an integral. \(V\) has locally integrable variation so we can integrate it with Stieltjes integrals.
13.6 Semimartingales
A process \(X\) is a semimartingale if it can be decomposed as \(X = M + A\), where \(M\) is a local martingale (\(M \in \mathcal{M}_{\mathrm{loc}}\)) and \(A\) is an adapted process with finite variation.
TODO: it’s adapted and cadlag.
A semimartingale \(X\) can be decomposed as \(X = M + A\), where \(M\) is a locally bounded martingale with bounded jumps and \(A\) is an adapted process with finite variation.
Decompose \(X = M + A\) as in Definition 13.80. Then decompose \(M\) as in Theorem 13.79 with \(\varepsilon = 1\). Then \(M = M_0 + U + V\) with \(U\) a locally bounded martingale with \(\vert \Delta U \vert \le 1\) and \(V\) a local martingale with locally integrable variation. Then \(X = (M_0 + U) + (A + V)\), with \(U\) a locally bounded martingale with bounded jumps and \(A + V\) an adapted process of finite variation (since \(V\) has locally integrable variation, hence finite variation).
TODO. Denoted by \(X^c\).
Let \(X\) and \(Y\) be semimartingales. Their quadratic covariation \([X, Y]\) is defined as
TODO: this is an adapted process with finite variation (it’s in \(\mathcal{V}\)).
The quadratic variation of a semimartingale \(X\) is defined as \([X] = [X, X]\) .
TODO: \([X]\) is an adapted increasing process.
A semimartingale \(X\) is a special semimartingale if it can be decomposed as \(X = M + A\), where \(M\) is a local martingale and \(A\) is an adapted process with locally integrable variation.
A special semimartingale \(X\) can be decomposed as \(X = M + A\), where \(M\) is a local martingale and \(A\) is a predictable process with finite variation and \(A_0 = 0\).