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 12.1 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.

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 12.2 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 \(\tilde{g}_n^1\), 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 12.2 to \((\tilde{g}_n^1)_{n\in \mathbb {N}}\) gives us \(g_n^2 \in convex(\tilde{g}_n^1, \tilde{g}_{n+1}^1, \ldots )\) (call its weights \(\lambda ^{2,n}_n,\cdots ,\lambda ^{2,n}_{N^2_n}\)) 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 \(\tilde{g}_n^k\) as the convex combination of \(x_n^{(k+1)}, x_{n+1}^{(k+1)}, \ldots \) with those weights:

We apply Lemma 12.2 to \((\tilde{g}_n^k)_{n\in \mathbb {N}}\) to get \(g_n^{k+1} \in convex(\tilde{g}_n^k, \tilde{g}_{n+1}^k, \ldots )\) such that \(g_n^{k+1}\) converges to some \(g^{k+1}\). We denote its weights by \(\lambda ^{k+1,n}_n,\cdots ,\lambda ^{k+1,n}_{N^{k+1}_n}\) and can then write:

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 12.3, 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 12.5, 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 12.6, 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 12.7 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 12.8 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 12.1, 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 \(g = -\log (h)\).

By definition.

By definition of the martingale part, \(M = X - A\). By Lemma 12.13, \(A_0 = 0\), thus \(M_0 = X_0 - A_0 = X_0\).

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

which concludes the proof.

Using Lemma 12.15, we have for \(n \in \mathbb {N}\),

By the martingale property, each conditional expectation in the definition of \(A\) is zero.

Since \(X\) is predictable, for all \(n \in \mathbb {N}\), \(X_{n+1}\) is \(\mathcal{F}_n\)-measurable and thus \(\mathbb {E}[X_{n+1} - X_n \mid \mathcal{F}_n] = X_{n+1} - X_n\) a.s.. We get a telescoping sum in the definition of \(A\) and thus \(A_n = X_n - X_0\) a.s. for all \(n \in \mathbb {N}\).

By definition of the martingale part, \(M = X - A\). By Lemma 12.18, \(A = X - X_0\) almost surely, thus \(M = X_0\) almost surely.

By definition of the martingale part, \(M = X - A\). By Lemma 12.17, \(A = 0\) almost surely, thus \(M = X\) almost surely.

Linearity of the conditional expectation.

By definition of the martingale part, \(M = X - A\). By Lemma 12.21, the predictable part of \(c X\) is \(c A\). Therefore, the martingale part of \(c X\) is \(c X - c A = c M\).

Linearity of the conditional expectation.

By definition of the martingale part, \(M = X - A\). By Lemma 12.23, the predictable part of \(X + Y\) is \(A + B\). Therefore, the martingale part of \(X + Y\) is \((X + Y) - (A + B) = M + N\).

By Lemma 7.50, 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 12.13, it is \(\mathcal{F}_0\)-measurable. Lemma 12.25 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 12.15. 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.

By Lemma 12.28, the predictable part \(A\) is almost surely nondecreasing. By Lemma 12.13, \(A_0 = 0\). Therefore, \(A_n \ge 0\) almost surely for all \(n \in \mathbb {N}\).

Since \(A_{n}\) is predictable, \(A_{n + 1}\) 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).

By definition and since \(X_T = 0\), \(M_T = X_T - A_T = - A_T\). Since \(M\) is a martingale it follows by optional sampling that for any stopping time \(\tau \le T\)

And then \(X_\tau = M_\tau + A_\tau = A_\tau - \mathbb {E}[A_T \mid \mathcal{F}_\tau ]\).

By Lemma 8.64, \(\tau ^T(c) {\lt} T \iff \exists s {\lt} T, A_{s+1} {\gt} c\), which is equivalent to \(\exists s \in \{ 1, \ldots , T\} , A_s {\gt} c\). By monotonicity of \(A\) (Lemma 12.28), this is equivalent to \(A_T {\gt} c\).

By Lemma 12.35,

Since that last event is \(\mathcal{F}_{\tau ^T(c)}\)-measurable, we can apply the tower property of conditional expectation to get

Now by Lemma 12.34 and Lemma 12.32,

Notice that \(\{ \tau ^T(c){\lt}T\} \subseteq \{ \tau ^T(c/2){\lt}T\} \), thus

Put together the bounds of Lemma 12.36 and Lemma 12.37.

Starting with Lemma 12.35,

WLOG \(S^n_{2^n} = S_T=0\) and \(S_t\leq 0\) (else consider \(S_t-\mathbb {E}\left[S_T\vert \mathcal{F}_{t}\right]\)).

We write \(\tau _n(c)\) for the hitting time \(\tau ^T_{A^n_{k+1}{\gt}c}\).

By Lemma 12.38,

On the other hand, by Lemma 12.39,

which goes to \(0\) uniformly in \(n\) as \(c\) goes to infinity.

The integrals in the upper bound on \(\int _{(A^n_{2^n}{\gt}c)} A^n_{2^n} dP\) are integrals of uniformly integrable random variables (by the class D assumption) over sets whose probability goes to zero uniformly. Therefore, these integrals go to zero uniformly in \(n\) as \(c\) goes to infinity.

This implies that \(\int _{(A^n_{2^n}{\gt}c)} A^n_{2^n} dP\) goes to 0 uniformly in \(n\) as \(c \to +\infty \). Hence, the \(L^1\) norm is uniformly bounded.

\(M^n_{2^n} = S_{2^n} - A^n_{2^n}\), also \(S\) is of class \(D\) hence uniformly integrable and \(A^n_{2^n}\) is uniformly integrable by Lemma 12.42.

By theorem 11.25

By lemma 12.43 \((M^n_T)_{n\in \mathbb {N}}\) is uniformly bounded in \(L^1\), thus by lemma 12.9 there are convex weights \(\lambda ^n_n,\cdots ,\lambda ^n_{N_n}\) such that \(\mathcal{M}^n_T\stackrel{L^1}{\rightarrow }M\), where \(\mathcal{M}^n_T := \lambda ^n_n \overline{M}^n_T+ \cdots + \lambda ^n_{N_n} \overline{M}^{N_n}_T\) .

By construction and 12.44

By Jensen’s inequality, the tower lemma and lemma 12.46

Since \(\overline{A}^n\) is left continuous and adapted, it is predictable (Lemma 7.51).

Either use that it is left continuous and adapted, or that it is a convex combination of predictable processes.

Since \(S\) and \(M\) are cadlag, their difference \(A = S - M\) is also cadlag.

By Lemma 12.50, for all \(t\in \mathcal{D}^T\),

By Lemma 12.61, we have convergence in \(L^1\) for every \(t\in \mathcal{D}^T\), which implies that there exists a subsequence \((k_n)\) such that \(\mathcal{A}^{k_n}_t\) converges to \(A_t\) almost surely.

Since \(\mathcal{D}^T\) is countable, to prove that \(A\) is almost surely non-decreasing on \(\mathcal{D}^T\) it is enough to show that for all \(s \le t\) in \(\mathcal{D}^T\), \(A_s \le A_t\) almost surely.

Let \(s \le t\) be two elements of \(\mathcal{D}^T\). By Lemma 12.62, there exists a subsequence \((k_n)\) such that \(\mathcal{A}^{k_n}_t\) converges to \(A_t\) almost surely. The subsequence \(\mathcal{A}_s^{k_n}\) converges to \(A_s\) in \(L^1\) and thus it has a further subsequence \((k'_n)\) that converges almost surely for \(s\) and \(t\). By Lemma 12.58, \(\mathcal{A}^{k'_n}_s \le \mathcal{A}^{k'_n}_t\) for all \(n\), thus by taking the limit we get \(A_s \le A_t\) almost surely.

\(A\) is almost surely non-decreasing on \(\mathcal{D}^T\) by Lemma 12.63. Since \(A\) is cadlag (thus right-continuous) by Lemma 12.60, it follows that \(A\) must be non-decreasing on \([0,T]\).

Since \(A\) is non-decreasing it has finitely many jumps of size higher than \(1/(k+1)\) for each \(k\). Thus, if \(A\) is discontinuous at \(t\), there exists \(k\) such that the jump of \(A\) at \(t\) is higher than \(1/(k+1)\), and thus there exists \(q\) such that \(\tau _{q, k} = t\).

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 12.66 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 \(t\)

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 12.66. 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 12.68 and 12.66 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 12.64 \(A\) 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 (Lemma 7.51). 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 12.67 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 12.69 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 \(\limsup _n\mathcal{A}^n_t(\omega )=\limsup _n\mathcal{A}^n_{\tau _{q(\omega ),k(\omega )}}(\omega )=A_{\tau _{q(\omega ),k(\omega )}}(\omega )=A_t(\omega )\)

A locally integrable increasing process is a local submartingale, thus we can apply Theorem 12.71.