Let \(n \in \mathbb {N}\). Then

which concludes the proof.

By Lemma 7.48, the process \(A\) is predictable if \(A_0\) is \(\mathcal{F}_0\)-measurable and for all integer \(n\), \(A_{n+1}\) is \(\mathcal{F}_n\)-measurable. As \(A_0 = 0\) from Lemma 12.2, it is \(\mathcal{F}_0\)-measurable. Lemma 12.4 allows to conclude the proof.

Let \(X\) be a submartingale and let \(A\) be its predictable part. Then for all \(n \geq 0\), from Lemma 7.9 we have that almost surely

The first equality comes from Lemma 12.3. As \(\mathbb {N}\) is countable, we deduce that almost surely, for all \(n \in \mathbb {N}\), \(A_{n+1} \ge A_n\). Thus, \((A_n)_{n \in \mathbb {N}}\) is almost surely nondecreasing.

Let \(B\) be the bound of \((\phi (f_n))_{n\in \mathbb {N}}\). Then for all \(n\in \mathbb {N}\) and \(g\in convex(f_n,f_{n+1},\cdots )\) we have \(\phi (g)\le B\) by convexity of \(\phi \). Let \(r_n = \inf (\phi (g) \mid g\in convex(f_n, f_{n+1},\ldots ))\). By construction \((r_n)_{n\in \mathbb {N}}\) is nondecreasing. Let \(A = \sup _{n \ge 1} r_n\), which is finite (as \(A \le B\)) and for each \(n\) we may pick some \(g_n\in convex(f_n, f_{n+1},\ldots )\) such that \(\phi (g_n) \le A+1/n\) by \(\inf \) and \(\sup \) definitions.

Let \(\varepsilon \in (0, \delta /4)\). By properties of \(\sup \) there exists \(\bar{n}\) such that \(r_{\bar{n}} \ge A-\varepsilon \) and such that \(\frac{1}{\bar{n}} \le \varepsilon \). Let \(m \ge k \ge \bar{n}\). We have \((g_k+g_m)/2 \in convex(f_k,f_{k+1},\ldots )\) and it follows since \((r_n)_{n\in \mathbb {N}}\) is nondecreasing that \(\phi ((g_k+g_m)/2) \ge A - \varepsilon \). Hence due to the ordering of \(m,k,\bar{n}\),

Thus, for \(n, m \ge \bar{n}\), \((g_n, g_m) \notin S_\delta \).

Consider \(\phi : H \to \mathbb {R}_+\) defined by \(\phi (f) = \| f\| _2^2\), which is convex. Then Lemma 12.9 applied to \((f_n)_{n\in \mathbb {N}}\) and \(\phi \) gives us functions \(g_n\in convex(f_n,f_{n+1},\cdots )\) such that for every \(\delta {\gt}0\) there exists \(N\) such that for \(n,m\geq N\), \((g_n,g_m)\notin S_\delta \). Thus for every \(\delta {\gt}0\) there exists \(N\) such that for \(n,m\geq N\),

But the left-hand side is equal to \(\| g_n - g_m\| _2^2/4\) by the parallelogram identity, hence \((g_n)_{n\in \mathbb {N}}\) is a Cauchy sequence in \(H\) and thus converges in \(H\) by completeness.

The second point is a direct consequence of the first one, so we only prove the first one. Let \(\varepsilon {\gt}0\). By convergence of \(x_n\), there exists \(\bar{n}\) such that for all \(n \ge \bar{n}\), \(\Vert x_n-x \Vert \le \varepsilon \). Let \(a_{n, m}\) be convex weights such that \(y_n = \sum _{m = n}^{N_n} a_{n, m} x_m\). By triangular inequality it follows that for \(n \ge \bar{n}\),

First by lemma 12.10 applied to \((x_n^{(1)})_{n\in \mathbb {N}}\) in the Hilbert space \(E\), there exist \(g_n^1 \in convex(x_n^{(1)}, x_{n+1}^{(1)}, \ldots )\) (call its weights \(\lambda ^{1,n}_n,\cdots ,\lambda ^{1,n}_{N^1_n}\)) such that \(g_n^1\) converges to some \(g^1\).

Secondly define \(\tilde{g}_n^2\), convex combination of \(x_n^{(2)}, x_{n+1}^{(2)}, \ldots \) with weights \(\lambda ^{1,n}_n,\cdots ,\lambda ^{1,n}_{N^1_n}\). Applying lemma 12.10 to \((\tilde{g}_n^2)_{n\in \mathbb {N}}\) gives us \(g_n^2 \in convex(\tilde{g}_n^2, \tilde{g}_{n+1}^2, \ldots )\) (call its weights \(\lambda ^{1,n}_n,\cdots ,\lambda ^{2,n}_{N^2_n}\)) such that \(g_n^2\) converges to some \(g^2\). \(g_n^2\) is a convex combination of \(x_n^{(2)}, x_{n+1}^{(2)}, \ldots \) with weights \((\lambda ^{2,n}_\cdot ) * (\lambda ^{1,\cdot }_\cdot )\).

We continue iterating this process inductively. At iteration \(k\) we have weights \((\lambda ^{k,n}_\cdot * \ldots * \lambda ^{1,\cdot }_\cdot )\). We define \(\tilde{g}_n^{k+1}\) as the convex combination of \(x_n^{(k+1)}, x_{n+1}^{(k+1)}, \ldots \) with those weights. We apply Lemma 12.10 to \((\tilde{g}_n^{k+1})_{n\in \mathbb {N}}\) to get \(g_n^{k+1} \in convex(\tilde{g}_n^{k+1}, \tilde{g}_{n+1}^{k+1}, \ldots )\) such that \(g_n^{k+1}\) converges to some \(g^{k+1}\). We denote its weights by \(\lambda ^{k+1,n}_n,\cdots ,\lambda ^{k+1,n}_{N^{k+1}_n}\).

We have thus defined, for all \(k, n \in \mathbb {N}\), convex weights \((\lambda ^{k,n}_m)\) (that are zero for \(m {\lt} n\)) such that \(\sum _{m \ge n}((\lambda ^{k,n}_\cdot * \ldots * \lambda ^{1, \cdot }_\cdot ))_m x_m^{(k)}\) converges to \(g^k\).

Let \(i \in \mathbb {N}\). By Lemma 12.11, there is uniform convergence over all convex combinations of the sequence \(\left(\sum _{m \ge n} \left((\lambda ^{i,n}_\cdot ) * \ldots * (\lambda ^{1,\cdot }_\cdot )\right)_m x_m^{(i)}\right)_{n \in \mathbb {N}}\) to \(g^i\). All sums \(\sum _{m \ge n} \left((\lambda ^{k,n}_\cdot ) * \ldots * (\lambda ^{1,\cdot }_\cdot )\right)_m x_m^{(i)}\) for \(k \ge i\) are convex combinations of \(\sum _{m \ge n} \left((\lambda ^{i,n}_\cdot ) * \ldots * (\lambda ^{1,\cdot }_\cdot )\right)_m x_m^{(i)}\), hence they converge to \(g^i\) uniformly in \(k\).

Let \((\lambda ^{k,n}_\cdot )_{k, n \in \mathbb {N}}\) be convex weights satisfying the conclusion of Lemma 12.12, and let \((g^i)_{i\in \mathbb {N}}\) be the sequence of limits of the sums. Let \(\eta ^n_m = (\lambda ^{n,n}_\cdot * \ldots * \lambda ^{1,\cdot }_\cdot )_m\). We show that for all \(i \in \mathbb {N}\), the sequence \(\left(\sum _{m \ge n} \eta ^n_m x_m^{(i)}\right)_{n \in \mathbb {N}}\) converges to \(g^i\).

Let \(i \in \mathbb {N}\). By Lemma 12.13, for all \(\varepsilon {\gt} 0\), there exists \(\bar{n}\) such that for all \(n \ge \bar{n}\), for all \(k \ge i\), \(\left\Vert \sum _{m \ge n} \left((\lambda ^{k,n}_\cdot ) * \ldots * (\lambda ^{1,\cdot }_\cdot )\right)_m x_m^{(i)} - g^i\right\Vert \le \varepsilon \). Hence for \(n \ge \max (\bar{n}, i)\),

Use Lemma 12.14 in the Hilbert space \(L^2(E)\) with the sequence of sequences \((f_n^{(i)})\), which are bounded in \(L^2(E)\) for each \(i \in \mathbb {N}\).

For \(i,n\in \mathbb {N}\) set \(f_{n}^{(i)}:=f_n \mathbb {1}_{(|f_n|\leq i)}\) such that \(f_{n}^{(i)}\in L^2\). Using 12.15 there exist for every \(n\) convex weights \(\lambda _n^{n}, \ldots , \lambda _{N_n}^{n}\) such that the functions \( \lambda _n^{n} f_n^{(i)} + \ldots +\lambda _{N_n}^{n} f_{N_n}^{(i)}\) converge in \(L^2\) for every \(i\in \mathbb {N}\). By uniform integrability, \(\lim _{i\to \infty }\| f^{(i)}_n- f_n\| _1=0\), uniformly with respect to \(n\). Hence, once again, uniformly with respect to \(n\),

Thus \((\lambda _n^{n} f_n + \ldots +\lambda _{N_n}^{n} f_{N_n})_{n\geq 1}\) is a Cauchy sequence in \(L^1\).

Let \(\phi : (\Omega \to [0, \infty ]) \to [0, \infty ]\) be defined by \(\phi (X) = \mathbb {E}[e^{-X}]\). Then \(\phi \) is convex and \(\phi (f_n) \le 1\) for all \(n\). By Lemma 12.9, there exist \(g_n \in convex( f_n, f_{n+1}, \cdots )\) such that for all \(\delta {\gt}0\), for \(N\) large enough and \(n, m \ge N\),

For \(\varepsilon {\gt} 0\), let \(B_\varepsilon = \{ (x, y) \in [0, \infty ]^2 \mid \vert x - y \vert \ge \varepsilon \text{ and } \min \{ x, y\} \le 1/\varepsilon \} \). Then for all \(x, y\),

Hence for any pair of random variables \((X, Y)\) with values in \([0, \infty ]\),

On the other hand, for \((x, y) \in B_\varepsilon \), there exists \(\delta _\varepsilon {\gt} 0\) such that

Thus,

For \(n, m \ge N\) large enough so that we can apply the first inequality of this proof with \(\delta = \varepsilon \delta _\varepsilon \), we deduce that

As \(\varepsilon \) is arbitrary, we deduce that \((e^{-g_n})_{n\in \mathbb {N}}\) is a Cauchy sequence in \(L^1\) and thus converges in \(L^1\) to some random variable \(h\). Therefore, it has a subsequence \((e^{-g_{n_k}})_{k\in \mathbb {N}}\) converging almost surely to \(h\). Finally, the subsequence of \(g_n\) converges almost surely to \(g = -\log (h)\).

Trivial

Trivial

\(S\) is a submartingale:

Since \(A^n_{t}\) is predictable, \(A^n_{t + 2^{-n}T}\) is adapted. The hitting time of an adapted process is a stopping time (we use the discrete time version of that result here, not the full Début theorem).

WLOG \(S_T=0\) and \(S_t\leq 0\) (else consider \(S_t-\mathbb {E}\left[S_T\vert \mathcal{F}_{t}\right]\)).

We have that \(0=S_T=M^n_T+A^n_T\). Thus

Since \(M^n\) is a martingale it follows by optional sampling that for any \((\mathcal{F}_t)_{t\in \mathcal{D}_n}\) stopping time \(\tau \)

Let \(c{\gt}0\). By Lemma 12.25, \(\tau _n(c)\) (Definition 12.24) is a stopping time. By construction \(A^n_{\tau _n(c)}\leq c\). It follows that

Since \((A^n_T{\gt}c)=(\tau _n(c){\lt}T)\) we have

Now we notice that \((\tau _n(c){\lt}T)\subseteq (\tau _n(c/2){\lt}T)\), thus

It follows

We may notice that

which goes to \(0\) uniformly in \(n\) as \(c\) goes to infinity. This implies that \(\int _{(A^n_T{\gt}c)}A^n_TdP\) is uniformly bounded in \(n\) due to the fact that \(S\) is of class \(D\). And so also the \(L^1\) norm is uniformly bounded.

\(M^n_T=S_T-A^n_T\), also \(S\) is of class \(D\) and \(A^n_T\) is uniformly integrable.

Let \(t\in [0,T]\) and \(s\in \mathcal{D}^T\) such that \(t{\lt}s\). We have

Since the above is true uniformly in \(s\) in particular since \(f\) is right-continuous

By lemma 12.30 it is enough to show that \(\liminf _n f_n(t)\geq f(t)\). Let \(s\in \mathcal{D}^T\) such that \(t{\gt}s\). We have

Since the above is true uniformly in \(s\) in particular since \(f\) is continuous in \(t\)

By theorem 11.11

By lemma 12.29 \((M^n_T)_{n\in \mathbb {N}}\) is uniformly bounded in \(L^1\), thus by lemma 12.16 there are convex weights \(\lambda ^n_n,\cdots ,\lambda ^n_{N_n}\) such that \(\mathcal{M}^n_T\stackrel{L^1}{\rightarrow }M\), where \(\mathcal{M}^n:=\lambda ^n_nM^n+\cdots +\lambda ^n_{N_n}M^{N_n}.\)

By construction and 12.32

By construction \(M_t\) is a martingale and thus by theorem 11.11 admits a cadlag martingale modification (\(M_t\) is a version of \(\mathbb {E}[M\vert \mathcal{F}_t]\) and thus passing to modification does not pose any problem).

We may notice that by Jensen’s inequality, the tower lemma and lemma 12.33

By Lemma 12.36

\(\mathcal{D}^T\) is countable we can arrange the elements as \((t_n)_{n\in \mathbb {N}}\). Given \(t_0\in \mathcal{D}^T\) there exists a subsequence \(k^{0}_n\) for which \(\mathcal{A}^{k^{0}_n}_{t_0}\) converges to \(A_{t_0}\) over the set \(\Omega \setminus E_{0}\) where \(P(E_{0})=0\). Suppose we have a sequence \(k^m_n\) for which \(\mathcal{A}^{k^j_n}_{t_j}\) converges to \(A_{t_j}\) over the set \(\Omega \setminus E_{m}\) where \(P(E_{m})=0\) for each \(j=0,\cdots ,m\). From this subsequence we may extract a new subsequence \(k^{m+1}_n\) for which \(\mathcal{A}^{k^{m+1}_n}_{t_{m+1}}\) converges to \(A_{t_{m+1}}\) over the set \(\Omega \setminus E_{m+1}\) where \(P(E_{m+1})=0\). By construction over this subsequence the convergence for \(t_0,\cdots ,t_m\) still applies. With a diagonal argument we obtain the final result with \(E=\bigcup _n E_n\).

Since \(\mathcal{A}^n_t\) is increasing on \(\mathcal{D}^T\) by lemma 12.37 also \(A\) is almost surely increasing on \(\mathcal{D}^T\). Since \(S,M\) are cadlag also \(A\) is cadlag (thus right-continuous). It follows that \(A\) must be increasing on \([0,T]\).

Let \(\sigma _n:=\inf \left(t\in \mathcal{D}^T_n\vert t{\gt}\tau \right)\). By construction of \(A^n\) we have \(A^n_\tau =A^n_{\sigma _n}\). Also \(\sigma _n\searrow \tau \). Since \(S\) is of class \(D\) and cadlag we have

Firstly we notice that \(\liminf _n \mathbb {E}[A_\tau ^n] \leq \limsup _n \mathbb {E} [\mathcal{A}_\tau ^n ] \leq \mathbb {E}[\limsup _n \mathcal{A}_\tau ^n ] \leq \mathbb {E}[ A_\tau ]\), where the first inequality is justified by the definition of limsup and liminf and the fact that

the third inequality by 12.30. Let’s prove the second inequality: observe that

thus it follows that \(\mathcal{A}^n_\tau - (\mathcal{A}^n_\tau -A_1)_+\leq A_1\); since \(A_1\) is an integrable guardian the inverse Fatou Lemma may be applied to show together with limsup properties that

where the first equality is justified by the fact that \(\mathcal{A}^n_\tau \leq \mathcal{A}^n_1\rightarrow A_1\) almost surely. Due to lemma 12.39 and 12.30 the first sequence of inequalities is a sequence of equalities, thus we know that \(A_\tau - \limsup _n \mathcal{A}_\tau ^n \) is an a.s. nonnegative function with null expected value, and thus it must be almost everywhere null.

By construction \(M\) is a cadlag martingale and \(A_0=0\) and by lemma 12.38 \(A\) is increasing. It suffices to show that \(A\) is predictable. \(A^n,\mathcal{A}^n\) are left continuous and adapted, and thus they are predictable (measurable wrt the predictable sigma algebra (the one generated by left-cont adapted processes)). It is enough to show that \(\omega -a.e.\), \(\forall t\in [0,T]\), \(\limsup _n\mathcal{A}^n_t(\omega )=A_t(\omega )\).

By lemma 12.31 that is true for any continuity point of \(A\). Since \(A\) is increasing it can only have a finite amount of jumps larger than \(1/k\) for any \(k\in \mathbb {N}\). Consider now \(\tau _{q,k}\) the family of stopping times equal to the \(q\)-th time that the process \(A_t\) has a jump higher than \(1/k\). This is a countable family. Given a time \(t\) and a trajectory \(\omega \) there are only two possibilities: either \(A\) is continuous or not at time \(t\) along \(\omega \). If \(A\) is continuous at time \(t\) we have \(\limsup _n\mathcal{A}^n_t(\omega )=A_t(\omega )\), if it jumps there exists \(q(\omega ),k(\omega )\) such that \(t=\tau _{q(\omega ),k(\omega )}(\omega )\). Due to lemma 12.40 we know that \(\limsup _n A^n_{\tau _{q,k}} = A_{\tau _{q,k}}\) for each \(q,k\) almost surely. Thus, since it is an intersection of a countable amount of almost sure events \(\forall \omega \in \Omega '\) with \(P(\Omega ')=1\), for each \(q,k\) \(\limsup _n A^n_{\tau _{q,k}}(\omega ) = A_{\tau _{q,k}}(\omega )\) (\(\omega \) does not depend upon \(q,k\)). Consequently, \(\forall \omega \in \Omega '\) we have \(\limsup _n\mathcal{A}^n_t(\omega )=\limsup _n\mathcal{A}^n_{\tau _{q(\omega ),k(\omega )}}(\omega )=A_{\tau _{q(\omega ),k(\omega )}}(\omega )=A_t(\omega )\)