14 Quasimartingales and measures derived from processes
14.1 Content of a process
Let \(E\) be a commutative monoid and \(\mathcal{A}\) be a family of sets. An additive content is a set function \(\lambda : \mathcal{A} \to E\) with value \(0\) at the empty set which is finitely additive on \(\mathcal{A}\) .
Let \(\lambda \) be an additive content on a semi-ring of sets \(C\) . Then \(\lambda \) admits a unique additive extension to the ring of sets generated by \(C\) .
Let \(X : T \to \Omega \to E\), where \(E\) is a Banach space, be an adapted process with \(X_t \in L^1\) for every \(t \in T\) . The content \(\lambda _X\) associated with \(X\) is an \(E\)-valued additive content on elementary predictable sets, given on predictable rectangles by
The content \(\lambda _X\) associated with \(X\) is uniquely defined.
An partition of a set \(A\) is a family of pairwise disjoint sets, which does not contain the empty set, and whose union is \(A\) . For a family of sets \(C\), a \(C\)-partition of \(A\) is a partition of \(A\) consisting of sets in \(C\) .
An additive content \(\lambda \) on a ring \(\mathcal{A}\) of sets, with values in a normed space \(E\), has bounded variation on a set \(A \in \mathcal{A}\) if
We can equivalently take supremum over \(C\)-partitions of \(A\) in the above definition, when \(\mathcal{A}\) is generated by the semi-ring \(C\) .
The content \(\lambda _X\) associated with a process \(X\) satisfies, for every elementary predictable set \(A\subseteq (0, t] \times \Omega \),
By Lemma 10.20, the process \(\mathbb {1}_A\) is a simple one,
where \((s_i, t_i]\) may be taken to be pairwise disjoint, thanks to the Lemma 10.17. Then
taking into account that for all \(i\) we have \(t\wedge t_i = t_i\), \(t\wedge s_i = s_i\), as \(A\subseteq (0, t] \times \Omega \) .
14.2 Quasi-martingales
Let \(E\) be a Banach space. An adapted integrable process \(X : T \to \Omega \to E\) is called a quasi-martingale if the additive content \(\lambda _X\) (on elementary predictable sets) associated with \(X\) has bounded variation on every set \((0, t] \times \Omega \) with \(t \in T\) , that is,
The content \(\lambda _X\) associated with a real-valued process \(X\) has bounded variation on \((0, t] \times \Omega \) if and only if the set \(\{ \mathbb {E}[(\mathbb {1}_A \bullet X)_t] \mid A \text{ elementary predictable}\} \) is bounded. In particular, an adapted integrable process \(X : T \to \Omega \to \mathbb {R}\) satisfies the conditions of Theorem 11.17, iff it is a quasi-martingale.
The above boundedness condition,
does not change after restricting the above supremum to sets \(A\) contained in \((0, t] \times \Omega \), since by Lemma 10.37,
Moreover, for those \(A\) the Lemma 14.8 is applicable; in effect, we can express the boundedness condition as
Let \(\mathcal{A}\) be the class of predictable rectangles, and let us use it in the definition of bounded variation of \(\lambda _X\) (see Lemma 14.7).
Fix \(t\in T\) . We aim to show that
holds if and only if
Take an elementary predictable \(A\subset B:=(0,t]\times \Omega \); choose \(A_1\), …, \(A_m\in \mathcal{A}\) to be pairwise disjoint and such that (Corollary 10.8)
Express \(A\) as a disjoint union of some \(A_{m+1}\), …, \(A_n\in \mathcal{A}\) (Lemma 10.14). Clearly, \((A_i)_{i=1}^n\) is a finite \(\mathcal{A}\)-partition of \(B=(0,t]\times \Omega \), so, by content’s additivity,
This shows the "only if" part.
For the "if" direction, take a sequence \(((A_i^k)_{i=1}^{n_k})_{k=1}^\infty \) of finite \(\mathcal{A}\)-partitions of \((0,t]\times \Omega \) such that
Let
and note that \(A_k = \bigcup _{i\in J_k} A_i^k\) is an elementary predictable subset of \((0,t]\times \Omega \) . Without loss of generality, we may assume that
which by additivity means
14.3 Measure associated with a process
TODO: reorganize and update this section, which should for now be considered a draft.
14.3.1 Monotone processes
Let \(f : T \to \mathbb {R}\) be a right-continuous monotone function on a conditionally complete linear order \(T\). We denote by \(d_+ f\) the measure on \(T\) defined by \(d_+ f((a, b]) = f(b) - f(a)\) for all \(a, b \in T\) .
Let \(X : T \to \Omega \to \mathbb {R}\) be a right-continuous adapted process which is monotone in the time variable. Then the measure \(d_+ X_\omega \) is measurable in \(\omega \) .
14.3.2 Bounded variation processes
Let \(f : T \to E\) be a function of bounded variation on a suitable order \(T\), with \(E\) a complete normed additive group. Then there exists a signed measure \(df\) on \(T\) defined by \(df((a, b]) = f(b^+) - f(a^+)\) .
Let \(X : T \to \Omega \to E\) be an adapted process which is of bounded variation in the time variable. Then the measure \(dX_\omega \) is measurable in \(\omega \) .
Denoting by \(\vert df \vert \) the total variation of the measure \(df\) , \(V_f([a, b]) = \vert df \vert ([a, b])\) .
If \(f : T \to \mathbb {R}\) is a right-continuous monotone function, then the measure \(df\) coincides with the measure \(d_+ f\) .
14.3.3 Integrable processes
Integrable -> content on the ring of sets generated by predictable rectangles. Can be extended to a measure on the predictable \(\sigma \)-algebra if the process is of class DL and ... (see Metivier, Semimartingales, 13.3).
Let \(X : T \to \Omega \to E\) be a stochastic process such that \(X_t\) is integrable for all \(t\) . For \(s {\lt} t\) and \(A \in \mathcal{F}_s\), let
Set \(\mu _X(\{ 0\} \times A) = 0\) . Then \(\mu _X\) is a content on the predictable rectangles, which can be extended to the ring of sets it generates.
TODO: lemmas stating that those 3 coincide whenever they can.