See 8.2.1 of Pascucci.

See 8.2.1 of Pascucci.

See 8.2.3 of Pascucci.

See 8.2.3 of Pascucci.

Let \(r_n = \inf (\| g\| _2:g\in convex(f_n, f_{n+1},\ldots ))\). Let \(A=\sup _{n\geq 1} r_n\). \(A\) is finite by boundedness of \((f_n)_{n\in \mathbb {N}}\) and for each \(n\) we may pick some \(g_n\in convex(f_n, f_{n+1},\ldots )\) such that \( \| g_n\| _2\leq A+1/n\) by \(\inf \) and \(\sup \) definitions. Let \(\epsilon {\gt}0\). By construction \((r_n)_{n\in \mathbb {N}}\) is increasing. By properties of \(\sup \) there exists \(\bar{n}\) such that \(r_{\bar{n}}\geq A-\epsilon \) and such that \(\frac{1}{\bar{n}}\leq \epsilon \). Let \(m\geq k\geq \bar{n}\). \((g_k+g_m)/2 \in convex(f_k,f_{k+1},\ldots )\). It follows since \((r_n)_{n\in \mathbb {N}}\) is increasing that \(\| (g_k+g_m)/2\| _2\geq A-\epsilon \). Hence due to the ordering of \(m,k,\bar{n}\)

By completeness, \((g_n)_{n\geq 1}\) converges in \(\| .\| _2\).

Let \(\epsilon {\gt}0\). By convergence of \(x_n\) we have \(\exists \bar{n}\) such that \(\forall n\geq \bar{n}\) \(|x_n-x|\leq \epsilon \). By triangular inequality it follows that

Firstly by lemma 8.7 over \((f_n^{(1)})_{n\in \mathbb {N}}\) there exist convex weights \(\prescript {1}{}{\lambda }^n_n,\cdots ,\prescript {1}{}{\lambda }^n_{N^1_n}\) such that \(g^1_n=\sum _{m=n}^{N^1_n}\prescript {1}{}{\lambda }^n_mf_m^{(1)}\) converges to some \(g^1\). Secondly apply the lemma to \((\tilde{g}^2_n=\sum _{m=n}^{N^1_n}\prescript {1}{}{\lambda }^n_mf^{(2)}_m)_{n\in \mathbb {N}}\), there exists convex weights \(\tilde{\lambda }^n_n,\cdots ,\tilde{\lambda }^n_{\tilde{N}_n}\) such that \(g^2_n=\sum _{m=n}^{\tilde{N}_n}\tilde{\lambda }^n_m\tilde{g}_m^{(2)}=\sum _{m=n}^{N^2_n}\prescript {2}{}{\lambda }^n_mf_m^{(2)}\) converges to some \(g^2\). Notice that \(\sum _{m=n}^{N^2_n}\prescript {2}{}{\lambda }^n_mf_m^{(1)}=\sum _{m=n}^{\tilde{N}_n}\tilde{\lambda }^n_m\tilde{g}_m^{(1)}\) and thus this sequence by lemma 8.8 converges still to \(g^1\). By iteration we may define \(\prescript {i}{}{\lambda }^n_n,\cdots ,\prescript {i}{}{\lambda }^n_{N^i_n}\) convex weights such that if used on \((f^j_n)_{n\in \mathbb {N}}\) they make the sequence convergent if \(1\leq j\leq i\). At this point consider \(\lambda ^n_m=\prescript {n}{}{\lambda }^n_m\). Since \(\forall m\geq i\) \(\sum _{j=n}^{N^m_n}\prescript {m}{}{\lambda }^n_j f^{(i)}_j\rightarrow g^i\) and even better \(\forall \epsilon {\gt}0\) \(\exists \bar{n}\), \(\forall n\geq \bar{n}\), \(\forall m\geq i\) \(|\sum _{j=n}^{N^m_n}\prescript {m}{}{\lambda }^n_j f^{(i)}_j - g^i|\leq \epsilon \) (this works by lemma 8.8 uniformity of convergence w.r.t. triangular array) this concludes.

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 8.9 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 \(L^1\).

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 8.18, \(\tau _n(c)\) (Definition 8.17) 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 8.23 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 8.4

By lemma 8.22 \((M^n_T)_{n\in \mathbb {N}}\) is uniformly bounded in \(L^1\), thus by lemma 8.10 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 8.25

By construction \(M_t\) is a martingale and thus by theorem 8.4 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 8.26

By Lemma 8.29

\(\mathcal{D}^T\) 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 8.30 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 8.23. 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 8.32 and 8.23 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 8.31 \(A\) 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 8.24 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 8.33 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 \(\limsup _n\mathcal{A}^n_t(\omega )=\limsup _n\mathcal{A}^n_{\tau _{q(\omega ),k(\omega )}}(\omega )=A_{\tau _{q(\omega ),k(\omega )}}(\omega )=A_t(\omega )\)

By Lemma 7.22, it suffices to show that if \(X\) is a submartingale with \(X_0=0\) then it is locally of class D.

TODO