4.1 Proof of Theorem 3.2 and Remark 3.3
To establish (
3.2), we consider an
\((X,\gamma )\in\mathcal{A}_{\mathcal {X}}\) that superreplicates
\(G\) on
\(\mathfrak {P}^{\epsilon}\) for some
\(\epsilon>0\). Since
\(X\) is bounded and
\(\gamma \) is admissible, we can find suitable
\(M>0\) such that
$$\begin{aligned} X(\mathbb {S})+\int_{0}^{T}\gamma _{u} \,\mathrm {d}\mathbb {S}_{u}\ge G(\mathbb {S})-M\lambda_{\mathfrak {P}}(\mathbb {S}), \end{aligned}$$
(4.2)
where we recall that
\(\lambda_{\mathfrak {P}}(\omega)=\inf_{\upsilon\in \mathfrak {P}}\| \omega-\upsilon\|\wedge1\). Next, for each
\(N\ge1\), we pick
\(\mathbb {P}^{(N)}\in \mathcal {M}_{\mathcal {X},\mathcal {P},\mathfrak {P}}^{1/N}\) such that
$$\mathbb {E}_{\mathbb {P}^{(N)}}[G(\mathbb {S})]\ge\sup_{\mathbb {P}\in \mathcal {M}_{\mathcal {X},\mathcal {P},\mathfrak {P}}^{1/N}}\mathbb {E}_{\mathbb {P}}[G(\mathbb {S})]-\frac{1}{N}. $$
Since
\(\gamma \) is progressively measurable, the integral
\(\int_{0}^{\cdot} \gamma _{u}(\mathbb {S})\,\mathrm {d}\mathbb {S}_{u}\), defined pathwise via integration by parts, agrees a.s. with the stochastic integral under any
\(\mathbb {P}^{(N)}\). Then by (
2.1), the stochastic integral is a
\(\mathbb {P}^{(N)}\)-supermartingale and hence
\(\mathbb {E}_{\mathbb {P}^{(N)}}[\int_{0}^{T} \gamma _{u}(\mathbb {S})\,\mathrm {d}\mathbb {S}_{u}]\le0\). Therefore, from (
4.2),
$$ \mathbb {E}_{\mathbb {P}^{(N)}}[X(\mathbb {S})]\ge \mathbb {E}_{\mathbb {P}^{(N)}}[G(\mathbb {S})-M\lambda_{\mathfrak {P}}(\mathbb {S})] \ge \sup_{\mathbb {P}\in \mathcal {M}_{\mathcal {X},\mathcal {P},\mathfrak {P}}^{1/N}}\mathbb {E}_{\mathbb {P}}[G(\mathbb {S})]-\frac{1}{N}-\frac{2M}{N}. $$
(4.3)
Also note that
\(X\) takes the form
\(a_{0} + \sum_{i=1}^{m} a_{i} X_{i}\),
\(X_{i}\in \mathcal {X}\), and hence by the definition of
\(\mathcal {M}_{\mathcal {X},\mathcal {P},\mathfrak {P}}^{1/N}\),
$$\begin{aligned} |\mathcal {P}(X)-\mathbb {E}_{\mathbb {P}^{(N)}}[X(\mathbb {S})]|\longrightarrow 0 \qquad \text{as } N\to\infty. \end{aligned}$$
Together with (
4.3), this yields
\(\mathcal {P}(X)\ge \widetilde {P}_{\mathcal {X},\mathcal {P},\mathfrak {P}}(G)\) and (
3.2) follows because
\((X,\gamma )\in\mathcal{A}_{\mathcal {X}}\) was arbitrary.
To establish (
3.3), we show the converse inequality in three steps.
Step 1: Duality without constraints. This is the crucial and also the most technical part of the proof which we defer to Sect.
5. The duality in (
3.4) follows as a special case of Theorem
5.1, which is stated and proved in Sect.
5.
Step 2: Calculus of variation approach. Fix
\(G\). Note that any
\((X,\gamma )\) that superreplicates
\(G-N\lambda_{\mathfrak {P}}\) on ℐ also superreplicates
\(G-N/M\) on
\(\mathfrak {P}^{\frac{1}{M}}\). It follows that for any fixed
\(M,N\geq1\),
$$\begin{aligned} \widetilde {V}_{\mathcal {X},\mathcal {P},\mathfrak {P}}(G) &= \inf\{\mathcal {P}(X):\exists(X,\gamma )\in\mathcal{A}_{\mathcal {X}}, \epsilon>0 \text{ such that} \\ &\phantom{=::\inf\{\mathcal {P}(X):}\text{$(X,\gamma )$ superreplicates $G$ on }\mathfrak {P}^{\epsilon}\}\\ & \le\frac{N}{M} + \inf\{\mathcal {P}(X): \text{$\exists(X,\gamma )\in\mathcal{A}_{\mathcal {X}}$ which} \\ &\phantom{=:\frac{N}{M} + \inf\{\mathcal {P}(X)::}\text{superreplicates $G-N\lambda_{\mathfrak {P}}$ on $ \mathcal {I}$}\}. \end{aligned}$$
Taking the infimum over
\(M\) and then over
\(N\), we obtain
$$\begin{aligned} \widetilde {V}_{\mathcal {X},\mathcal {P},\mathfrak {P}}(G) &\le\inf_{N\ge0} \inf\{\mathcal {P}(X): \text{$\exists(X,\gamma )\in \mathcal{A}_{\mathcal {X}}$ which} \\ &\phantom{=:\inf_{N\ge0} \inf\{\mathcal {P}(X)::}\text{superreplicates $G-N\lambda_{\mathfrak {P}}$ on $ \mathcal {I}$}\} \\ &= \inf_{N\ge0}{V}_{\mathcal {X},\mathcal {P}, \mathcal {I}}(G-N\lambda_{\mathfrak {P}}) = \inf_{N\ge0}\widetilde{V}_{\mathcal {X},\mathcal {P}, \mathcal {I}}(G-N\lambda_{\mathfrak {P}}). \end{aligned}$$
(4.4)
On the other hand, given any
\((X,\gamma )\in\mathcal{A}_{\mathcal {X}}\) and
\(\epsilon>0\) such that
\((X,\gamma )\) superreplicates
\(G\) on
\(\mathfrak {P}^{\epsilon}\), by the admissibility of
\((X,\gamma )\) and boundedness of
\(X\) and
\(G\), if
\(N>0\) is sufficiently large, then
$$\begin{aligned} X(S)+ \int_{0}^{T}\gamma _{u}(S)\,\mathrm {d}S_{u}\ge G(S)-N\lambda_{\mathfrak {P}},\qquad S\in \mathcal {I}, \end{aligned}$$
that is,
\((X,\gamma )\) superreplicates
\(G-N\lambda_{\mathfrak {P}}\) on ℐ. It follows that we have equality in (
4.4). We also have
$$\begin{aligned} \widetilde{V}_{\mathcal {X},\mathcal {P}, \mathcal {I}}(G-N\lambda_{\mathfrak {P}}) & = \inf_{X\in \mathrm {Lin}(\mathcal {X})}\big(\mathcal {P}(X) +\inf\{x\in \mathbb {R}:\exists \gamma \in\mathcal{A} \text{ such that} \\ &\phantom{=::\inf_{X\in \mathrm {Lin}(\mathcal {X})}\big(\mathcal {P}(X) +\inf\{x\in \mathbb {R}:} \text{$(x,\gamma )$ superreplicates} \\ &\phantom{=::\inf_{X\in \mathrm {Lin}(\mathcal {X})}\big(\mathcal {P}(X) +\inf\{x\in \mathbb {R}:} \text{$G-N\lambda_{\mathfrak {P}}-X$ on $ \mathcal {I}\}\big)$} \\ &= \inf_{X\in \mathrm {Lin}(\mathcal {X})}\big(\mathcal {P}(X)+ \mathbf {V}_{\mathcal {I}}(G - N\lambda_{\mathfrak {P}} - X)\big)\\ &= \inf_{X\in \mathrm {Lin}(\mathcal {X})}\big(\mathcal {P}(X)+\mathbf {P}_{\mathcal {I}}(G-N\lambda_{\mathfrak {P}}-X)\big), \end{aligned}$$
where the last equality is justified by Theorem
5.1 as
\(\lambda_{\mathfrak {P}}\) and
\(X\) are bounded and uniformly continuous. Combining the above with (
4.4), we conclude that (
3.5) holds.
Step 3: Application of the minimax theorem. We rewrite (
3.5) and apply a minimax argument to get
$$\begin{aligned} \widetilde {V}_{\mathcal {X},\mathcal {P},\mathfrak {P}}(G)&= \inf_{X\in \mathrm {Lin}(\mathcal {X}),\,N\ge0}\big(\mathbf {P}_{\mathcal {I}}(G-X-N\lambda _{\mathfrak {P}})+\mathcal {P}(X)\big) \\ &= \lim_{N\to\infty} \inf_{X\in \mathrm {Lin}_{N}(\mathcal {X})}\bigg(\sup_{\mathbb {P}\in \mathcal {M}_{ \mathcal {I}}}\mathbb {E}_{\mathbb {P}} [G-X-N\lambda_{\mathfrak {P}}]+\mathcal {P}(X)\bigg) \\ &= \lim_{N\to\infty} \sup_{\mathbb {P}\in \mathcal {M}_{ \mathcal {I}}}\inf_{X\in \mathrm {Lin}_{N}(\mathcal {X})}\big(\mathbb {E}_{\mathbb {P}} [G-X-N\lambda_{\mathfrak {P}}]+\mathcal {P}(X)\big) \end{aligned}$$
(4.5)
$$\begin{aligned} &\le \lim_{N\to\infty}\sup_{\mathbb {P}\in \mathcal {M}_{\mathcal {X},\mathcal {P},\mathfrak {P}}^{\eta_{N}}}\mathbb {E}_{\mathbb {P}}[G] = \widetilde {P}_{\mathcal {X},\mathcal {P},\mathfrak {P}}(G), \end{aligned}$$
(4.6)
for
\(\eta_{N} = 2 \kappa/\sqrt{N}\) with
\(\kappa=1+ \|G\|_{\infty}\), where
\(\|G\|_{\infty}=\sup_{S\in\varOmega}|G(S)|\). The crucial third equality follows by a minimax theorem (see e.g. Terkelsen [
50, Corollary 2]) by observing that the mapping
$$ \mathrm {Lin}_{N}(\mathcal {X})\times \mathcal {M}_{ \mathcal {I}} \ni(X,\mathbb {P}) \mapsto \mathbb {E}_{\mathbb {P}}[G(\mathbb {S})-X(\mathbb {S})-N\lambda_{\mathfrak {P}}(\mathbb {S})]+\mathcal {P}(X)\in \mathbb {R}$$
is bilinear and
\(\mathrm {Lin}_{N}(\mathcal {X})\) is convex and compact. To justify the inequality between (
4.5) and (
4.6), consider
\(\mathbb {P}\in \mathcal {M}_{ \mathcal {I}}\setminus \mathcal {M}_{\mathcal {X},\mathcal {P},\mathfrak {P}}^{\eta_{N}}\). Then in particular, either there exists
\(X^{*}\in \mathcal {X}\) such that
\(|\mathbb {E}_{\mathbb {P}}[X^{*}] -\mathcal {P}(X^{*})|> \eta_{N}\frac{1}{\sqrt{N}}\) or
\(\mathbb {P}[\mathbb {S}\notin \mathfrak {P}^{\eta_{N}}]\ge\eta_{N}\). In the former case, since
\(\pm NX^{*}\in \mathrm {Lin}_{N}(\mathcal {X})\), we obtain
$$\begin{aligned} \mathbb {E}_{\mathbb {P}}[G - NX^{*} -N\lambda_{\mathfrak {P}}]+ \mathcal {P}(NX^{*}) &\le \mathbb {E}_{\mathbb {P}}[G] - N\big(\mathbb {E}_{\mathbb {P}}[X^{*}] -\mathcal {P}(X^{*})\big)\\ & < \kappa- 2\kappa\sqrt{N}\leq -\kappa, \end{aligned}$$
where, without loss of generality, we assume
\(\mathbb {E}_{\mathbb {P}}[X^{*}] < \mathcal {P}(X^{*})\). In the latter case, we have
\(\mathbb {E}_{\mathbb {P}}[N\lambda_{\mathfrak {P}}] \geq N\frac {2\kappa}{\sqrt{N}}\frac{2\kappa}{\sqrt{N}}=4\kappa^{2}\geq4\kappa\), while
\(|\mathbb {E}_{\mathbb {P}}[X] -\mathcal {P}(X)|\leq N\frac{2\kappa}{N}=2\kappa\) for any
\(X\in \mathrm {Lin}_{N}(\mathcal {X})\). It follows that
$$\begin{aligned} \mathbb {E}_{\mathbb {P}}[G - X - N\lambda_{\mathfrak {P}}]+ \mathcal {P}(X) \leq\kappa- 4\kappa+2\kappa =-\kappa. \end{aligned}$$
On the other hand, since (
3.2) implies
\(\widetilde {V}_{\mathcal {X},\mathcal {P},\mathfrak {P}}(0)= 0\), we have
$$\widetilde {V}_{\mathcal {X},\mathcal {P},\mathfrak {P}}(G)=\widetilde {V}_{\mathcal {X},\mathcal {P},\mathfrak {P}}(G+\|G\|_{\infty})-\|G\|_{\infty} \geq \widetilde {V}_{\mathcal {X},\mathcal {P},\mathfrak {P}}(0) -\| G\|_{\infty} = -\kappa+1, $$
and hence we may restrict to measures in
\(\mathcal {M}_{\mathcal {X},\mathcal {P},\mathfrak {P}}^{\eta_{N}}\) in (
4.5). Dropping nonpositive terms, we obtain (
4.6) which completes the proof of Theorem
3.2. □
For Remark
3.3, it remains to argue that Theorem
3.2 remains true when we restrict to Brownian martingales. Specifically, given
\(T\) and a probability space
\((\varOmega ^{W},\mathbb {F}^{W},P^{W})\) with a
\(\tilde{d}\)-dimensional Brownian motion
\(W\) on
\([0,T]\), where
\(\mathbb {F}^{W}= (\mathcal {F}_{t}^{W})_{0 \leq t\leq T}\) is the
\(P^{W}\)-completion of the natural filtration of
\(W\), consider
$$ \mathbb {P}:= P^{W}\circ(Z^{\alpha})^{-1},\qquad \text{where } Z^{\alpha}:= \int _{0}^{\cdot}\alpha_{u} \,\mathrm {d}W_{u} $$
(4.7)
for some
\(\mathbb {F}^{W}\)-progressively measurable process
\(\alpha\) with values in the
\((d+K)\times\tilde{d}\) matrices such that the above vector integral is well defined. Let
\(\underline{\mathcal {M}}_{ \mathcal {I}}\) be the family of all
\(\mathbb {P}\in \mathcal {M}_{ \mathcal {I}}\) which admit such a representation. From (
3.5), as argued above, and Remark
5.2 below, we have
$$\begin{aligned} \widetilde {V}_{\mathcal {X},\mathcal {P},\mathfrak {P}}(G) &= \inf_{X\in \mathrm {Lin}(\mathcal {X}),\, N\ge0,}\bigg(\sup_{\mathbb {P}\in \underline{\mathcal {M}}_{ \mathcal {I}}}\mathbb {E}_{\mathbb {P}}[G-X-N\lambda_{\mathfrak {P}}] + \mathcal {P}(X)\bigg). \end{aligned}$$
Then by following the same argument as in Step 3 above, we can show that we have
\(\mathcal {M}_{\mathcal {X},\mathcal {P},\mathfrak {P}}^{\eta_{N}}\cap\underline{\mathcal {M}}_{ \mathcal {I}}\neq \emptyset\) when
\(N\) is sufficiently large and
$$\begin{aligned} \widetilde {V}_{\mathcal {X},\mathcal {P},\mathfrak {P}}(G) = \lim_{N\to\infty}\sup_{\mathbb {P}\in \mathcal {M}_{\mathcal {X},\mathcal {P},\mathfrak {P}}^{\eta_{N}}\cap \underline{\mathcal {M}}_{ \mathcal {I}} }\mathbb {E}_{\mathbb {P}}[G]. \end{aligned}$$
4.4 Proof of Theorem 3.16
We first make two simple observations.
We now proceed with the proof of Theorem
3.16. Recall that the inequalities
\(\widetilde {V}_{\mathcal {X},\mathcal {P},\mathfrak {P}}(G)\ge V_{\mathcal {X},\mathcal {P},\mathfrak {P}}(G)\ge P_{\pmb {\mu}, \mathfrak {P}}(G)\) hold in general. In addition, according to Theorem
3.12,
\(\widetilde {V}_{\mathcal {X},\mathcal {P},\mathfrak {P}}(G) = \widetilde{P}_{\pmb {\mu}, \mathfrak {P}}(G)\). Therefore, we only need to show that
\(\widetilde{P}_{\pmb {\mu}, \mathfrak {P}}(G) = P_{\pmb {\mu}, \mathfrak {P}}(G)\). Our proof of this equality is divided into six steps. First, using Proposition
4.5, we argue that it suffices to consider measures with “good control” on the expectation of
\(m^{(D)}(\mathbb {S})\). Next, we perform three time changes within each trading period
\([T_{i}, T_{i+1}]\). The resulting time change of
\(\mathbb {S}\), denoted by
\(\ddot{\mathbb {S}}\), allows a “good control” over its quadratic variation process. At the same time, we keep
\(G(\mathbb {S})\) and
\(G(\ddot{\mathbb {S}})\) “close”, and given a measure
with “good control” on
\(\mathbb {E}_{\mathbb {P}}[m^{(D)}(\mathbb {S})]\), since
\(\mathfrak {P}^{\eta}\) is time-invariant, the law of the time-changed price process
\(\ddot{\mathbb {S}}\) remains an element of
. Then in Step 5, given a sequence of models with improved calibration precisions, we show tightness of the quadratic variation process of the time-changed price process
\(\ddot {\mathbb {S}}\) under these measures. This then leads to tightness of the image measures via
\(\ddot{\mathbb {S}}\). In Step 6, we deduce the duality
\(\widetilde {P}_{\pmb {\mu}, \mathfrak {P}}(G) = P_{\pmb {\mu}, \mathfrak {P}}(G)\) from tightness and conclude.
Recall that
\(\mathcal {X}\) is given by (
3.7). Let
$$\begin{aligned} \mathcal {X}_{n}=\{(\kappa-\mathbb {S}^{(i)}_{T_{n}})^{+}: i=1,\ldots,d,\,\,\kappa\in \mathbb {R}_{+}\} \end{aligned}$$
and write
\(P_{\mu_{n},\mathfrak {P}}:=P_{\mathcal {X}_{n},\mathcal {P},\mathfrak {P}}\) for the associated primal problem, where the martingale measures have fixed marginals
\(\mu _{n}^{(i)}\), given by (
3.8), of the distribution of
\(\mathbb {S}_{T_{n}}\). Note that by definition,
\(P_{\mu_{n}, \mathcal {I}}=\widetilde{P}_{\mu _{n}, \mathcal {I}}\) and that since the
\(\mu_{n}^{(i)}\) have finite
\(p\)th moment, we have
\(P_{\mu_{n}, \mathcal {I}}\left(\|\mathbb {S}\|\right)<\infty\).
Step 1: Reducing to measures ℙ
with good control on
\(\mathbb {E}_{\mathbb {P}}[\sqrt{m^{(D)}(\mathbb {S})}]\). Let
\(G\) satisfy Assumption
3.14. Choose
\(\kappa\geq1\) such that
\(\|G\|\le\kappa\) and let
\(f_{e}:\mathbb {R}^{d+K}_{+}\to \mathbb {R}_{+}\) be a modulus of continuity of
\(G\), i.e.,
$$|G(\omega)-G(\upsilon)|\le f_{e}(|\omega-\upsilon|) \qquad \text{for any }\omega, \upsilon\in\varOmega $$
with
\(\lim_{x\to0} f_{e}(x) = 0\). Fix
\(D\in \mathbb {N}\). Consider
\(X_{D}:\varOmega\to \mathbb {R}\) given by
$$\begin{aligned} X_{D}(S) &= \sqrt{\sum_{j=1}^{m^{(D)}(S)\wedge2^{6D}\kappa^{2}}\, \sum _{i=1}^{d+K}\big|S^{(i)}_{\tau^{(D)}_{j}(S)}- S^{(i)}_{\tau ^{(D)}_{j-1}(S)}\big|^{2}} \\ &\ge2^{-D}\sqrt{m^{(D)}(S)\wedge(2^{6D}\kappa^{2})-1}\\ &\ge\Big(2^{-D}\big(\sqrt{m^{(D)}(S)\wedge2^{6D}\kappa^{2}} -1\big)\Big)= \kappa2^{2D}\wedge\frac{\sqrt{m^{(D)}(S)}}{2^{D}} -2^{-D}, \end{aligned}$$
where the
\(\tau^{(D)}_{i}\) and
\(m^{(D)}\) are defined in Definition
4.1. It follows from the proof of Lemma 5.4 in Dolinsky and Soner [
26] that there exists a
\(\gamma\in\mathcal{A}\) such that
$$ \int_{0}^{\tau_{m^{(D)}(S)\wedge2^{6D}\kappa^{2}}}\gamma_{u} \,\mathrm {d}S_{u} + 3(d+K)\max_{0\le j\le(m^{(D)}(S)\wedge2^{6D}\kappa^{2})}|S_{\tau ^{(D)}_{j}}|\ge X_{D}(S),\qquad S\in \mathcal {I}. $$
Hence
\(\widetilde {V}_{\mathcal {X},\mathcal {P},\mathfrak {P}}(X_{D})\le3(d+K)\widetilde {V}_{\mathcal {X},\mathcal {P},\mathfrak {P}}(\|\mathbb {S}\|\wedge(\kappa^{2}2^{5D} + 1))\). Reducing
\(\mathcal {X}\) to options with maturity
\(T_{n}\) and considering ℐ instead of
\(\mathfrak {P}\) only increases the superhedging price, and therefore
$$ 0\le \widetilde {V}_{\mathcal {X},\mathcal {P},\mathfrak {P}}(X_{D}) \le3(d+K)V_{\mathcal {X}_{n},\mathcal {P}, \mathcal {I}}\big(\|\mathbb {S}\|\wedge (\kappa^{2}2^{5D} + 1)\big)\leq3(d+K)P_{\mu_{n}, \mathcal {I}}\left(\|\mathbb {S}\|\right), $$
which is finite, and where the last inequality follows from Theorem
3.12 applied to the case of a single maturity. It now follows from sublinearity of
\(\widetilde{V}\) that
where
\(c_{2}\) is a constant independent of
\(D\) and the last inequality follows from Proposition
4.5.
Next we denote by
\(\widehat {\mathcal {M}}_{\mathcal {I}}^{\kappa}\) the set of
\(\mathbb {P}\in \mathcal {M}_{ \mathcal {I}}\) such that
$$\begin{aligned} \mathbb {E}_{\mathbb {P}}\bigg[\kappa2^{D}\wedge\frac{\sqrt{m^{(D-8)}(\mathbb {S})}}{2^{2D}}\bigg]\le2\kappa+ 2. \end{aligned}$$
(4.12)
We notice that if
\(\mathbb {P}\notin \widehat {\mathcal {M}}_{\mathcal {I}}^{\kappa}\), then
$$\begin{aligned} \mathbb {E}_{\mathbb {P}}\bigg[G(\mathbb {S})-\kappa2^{D}\wedge\frac{\sqrt{m^{(D-8)}(\mathbb {S})}}{2^{2D}}\bigg] < \kappa- 2\kappa-2 =-\kappa-2, \end{aligned}$$
while by the inequalities in (
4.11) above, for
\(D\) sufficiently large,
It follows that in (
4.11), it suffices to consider
, which in particular is nonempty.
Step 2: First time change: “squeezing paths and adding constant paths”. The first time change squeezes the evolution on
\([T_{i-1},T_{i}]\) to
\([T_{i-1},T_{i}-1/D]\) and adds a constant piece to the path on
\([T_{i}-1/D,T_{i}]\). To achieve this, define an increasing function
\(f: [0,T_{n}]\to[0,T_{n}]\) by
and then a process
\((\tilde{\mathbb {S}}_{t})_{t\in[0,T_{n}]}\) by a time change of
\(\mathbb {S}\) via
\(f\), i.e.,
\(\tilde{\mathbb {S}}_{t} = \mathbb {S}_{f(t)}\). Note that
\(f(T_{i}-1/D)=T_{i}\), as required. We argue below that (
3.9) implies that we have
\(|G(\mathbb {S}) - G(\tilde{\mathbb {S}})|\to 0\) as
\(D\to\infty\).
Now for every
\(N\in \mathbb {N}\), take
such that
Since
\(\mathbb {S}_{T_{i}} =\tilde {\mathbb {S}}_{T_{i}}\), we have in particular
\(\mathcal {L}_{\mathbb {P}^{(N)}}(\mathbb {S}_{T_{i}}) = \mathcal {L}_{\mathbb {P}^{(N)}}(\tilde {\mathbb {S}}_{T_{i}})\) for all
\(i\le n\). Also, being a time change of
\(\mathbb {S}\), the process
\((\tilde{\mathbb {S}}_{t})_{t\in[0,T_{n}]}\) is a martingale (in the time-changed filtration). It follows that its distribution
\(\mathbb {P}^{(N)} \circ(\tilde {\mathbb {S}}_{t})^{-1}\) is an element of
as
\(\mathfrak {P}^{1/N}\) is time-invariant, by Lemma
4.7.
Step 3: Second time change: introducing a lower bound on the time step. The second time change ensures that we can bound from below the difference between any two consecutive stopping times in the Lebesgue discretisation in Definition
4.1. We want to do this by adding a constancy interval of length
\(\delta\) to each step of the discretisation. As we have squeezed the paths above, we have length
\(1/D\) to use up while still keeping the time changes to within the intervals
\([T_{i-1},T_{i}]\). Taking suitably small
\(\delta\), this allows us, with high probability, to alter all the steps in the Lebesgue discretisation.
For ease of notation, it is helpful to rename the elements of the set
$$\{\tau^{(D)}_{j} : j\le m^{(D)}\}\cup\{T_{i} : i = 1,\ldots, n\} $$
as follows. We define a sequence of stopping times
\(\tau _{i,j}^{(D)}: \varOmega\to[T_{i-1},T_{i}]\) and
\(m^{(D)}_{i}: \varOmega\to \mathbb {N}_{+}\) in a recursive manner. Set
\(T_{0}=m^{(D)}_{0}(\mathbb {S}) = \tau^{(D)}_{0,-1}= 0\) and for
\(i=1,\ldots,n\), set
\(\tau^{(D)}_{i,0}(\mathbb {S})=T_{i-1}\) and let
$$\begin{aligned} \tau^{(D)}_{i,1}(\mathbb {S})&=\inf\bigg\{ t\ge T_{i-1}:\big|\mathbb {S}_{t}-\mathbb {S}_{\tau ^{(D)}_{i-1,m^{(D)}_{i-1}(\mathbb {S})-1}(\mathbb {S})}\big|=\frac{1}{2^{D}}\bigg\} \wedge T_{i},\\ \tau^{(D)}_{i,k}(\mathbb {S})&=\inf\bigg\{ t\ge\tau_{i,k-1}(\mathbb {S}):\big|\mathbb {S}_{t}-\mathbb {S}_{\tau ^{(D)}_{i,k-1}(\mathbb {S})}\big|=\frac{1}{2^{D}}\bigg\} \wedge T_{i},\\ m^{(D)}_{i}(\mathbb {S})&=m^{(D)}_{i-1}(\mathbb {S})+\min\{k\in \mathbb {N}: \tau^{(D)}_{i,k}(\mathbb {S})=T_{i}\}. \end{aligned}$$
It follows that for any
\(S\in \mathcal {I}\),
$$\begin{aligned} m^{(D)}(S)\le m^{(D)}_{n}(S) \le m^{(D)}(S) + n-1. \end{aligned}$$
(4.13)
Set
\(\varTheta= 2\lceil\kappa^{2} 2^{6D} \rceil+n\) and
\(\delta= 1/(4D\varTheta^{2})\). We now define a sequence of stopping times
\(\sigma _{i,j}:\varOmega\to[0,T_{n}]\) by
\(\sigma_{i,0}(S) := T_{i-1}\),
\(\sigma _{i,\varTheta+1}(S) := T_{i}\), and for
\(j\leq\varTheta\), we put
$$\sigma_{i,j}(S) := \big(\tau^{(D-8)}_{i, j}(S) + \delta j\big)\wedge \big(T_{i}-1/(2D)\big)\qquad \text{if }j< m^{(D-8)}_{i}(S), $$
while
\(\sigma_{i,j}(S) := T_{i}-1/(2D)\) otherwise, where
\(i=1,\ldots, n\). Then it follows from the definition that
$$T_{i-1}=\sigma_{i,0}(S)\le\sigma_{i,1}(S)\le\cdots\le\sigma _{i,\varTheta}(S)< \sigma_{i,\varTheta+1}(S)= T_{i} $$
for all
\(S\in\varOmega\). Further, since the process
\(\tilde {\mathbb {S}}\) is always constant on
\([T_{i}-1/D, T_{i}]\), we have
\(\tau^{(D-8)}_{i, j}(\tilde {\mathbb {S}})\le T_{i}-1/D\) and hence for
\(j\le\varTheta\wedge (m^{(D-8)}_{i}(\tilde {\mathbb {S}})-1)\) that
$$\sigma_{i,j}(\tilde {\mathbb {S}}) \le\tau^{(D-8)}_{i, m^{(D-8)}_{i}-1}(\tilde {\mathbb {S}}) + \delta \varTheta\le T_{i}-\frac{1}{D} + \frac{1}{4D\varTheta}< T_{i} - \frac{1}{2D}. $$
Also, for all
\(j =1,\ldots, (\varTheta\wedge (m^{(D-8)}_{i}(\tilde {\mathbb {S}})-1))\),
$$ \sigma_{i,j}(\tilde {\mathbb {S}}) - \sigma_{i,j-1}(\tilde {\mathbb {S}}) = \delta+ \big(\tau ^{(D-8)}_{i,j}(\tilde {\mathbb {S}}) - \tau^{(D-8)}_{i,j-1}(\tilde {\mathbb {S}})\big)\ge\delta.$$
We are now ready to define the time-changed process
\(\check{\mathbb {S}}\) by
Observe that
\(\check{\mathbb {S}}\) is a (continuous) time change of
\(\tilde {\mathbb {S}}\) and
\(\tilde {\mathbb {S}}_{T_{i}} = \check{\mathbb {S}}_{T_{i}} = \mathbb {S}_{T_{i}}\) for
\(i\le n\). As before, this implies that
\(\check{\mathbb {S}}\) remains a martingale and
.
We argue now that
\(|G(\mathbb {S}) - G(\check{\mathbb {S}})|\) is small for large
\(D\). To this end, we approximate a path
\(S\) with a piecewise constant function
\(\tilde{F}^{(D)}(S)\) which jump at the times
\(\tau_{i,j}^{(D)}\). A similar discretisation is used later in Sect.
5; see (
5.2). For
\(S\in\varOmega\), consider
Then the time-continuity property of
\(G\) in (
3.9) ensures that
$$\begin{aligned} |G(\mathbb {S}) - G(\tilde{\mathbb {S}})| &\le\big|G(\mathbb {S}) - G\big(\tilde{F}^{(D)}(\mathbb {S})\big)\big|+ \big|G(\tilde{\mathbb {S}}) - G\big(\tilde{F}^{(D)}(\tilde{\mathbb {S}})\big)\big| \\ & \phantom{=}{}+ \big|G\big(\tilde{F}^{(D)}(\mathbb {S})\big) - G\big(\tilde{F}^{(D)}(\tilde{\mathbb {S}})\big)\big| \\ &\le2f_{e}(2^{-D+9}) + \frac{2nL\|\mathbb {S}\|}{D}. \end{aligned}$$
(4.14)
Similarly, for any
\(S\in\varOmega\) with
\(m^{(D-8)}_{n}(\tilde {\mathbb {S}}(S))=m^{(D-8)}_{n}(S)\le\varTheta\), again by (
3.9), we have
$$\begin{aligned} \big|G\big(\tilde {\mathbb {S}}(S)\big)-G\big(\check{\mathbb {S}}(S)\big)\big| &\le\Big|G\big(\tilde {\mathbb {S}}(S)\big) - G\Big(\tilde{F}^{(D-8)}\big(\tilde {\mathbb {S}}(S)\big)\Big)\Big| \\ &\phantom{=}{} + \Big|G\big(\check{\mathbb {S}}(S)\big) - G\Big(\tilde{F}^{(D-8)}\big(\check{\mathbb {S}}(S)\big)\Big)\Big| \\ &\phantom{=}{}+ \Big|G\big(\tilde{F}^{(D-8)}(\tilde {\mathbb {S}})(S)\big) - G\Big(\tilde{F}^{(D-8)}\big(\check{\mathbb {S}}(S)\big)\Big)\Big| \\ &\le2f_{e}(2^{-D+9})+nL\|\tilde {\mathbb {S}}(S)\|\varTheta\delta \\ &\le2f_{e}(2^{-D+9}) +nL\|\mathbb {S}(S)\|/D, \end{aligned}$$
(4.15)
when
\(D\) is sufficiently large. From (
4.12), the Markov inequality gives
$$ \mathbb {P}^{(N)}[\{S\in \mathcal {I}:\, m^{(D-8)}(S)\ge\varTheta-n+2\}]\le\frac{2\kappa +2}{\kappa2^{D}}, $$
and hence by (
4.13),
$$ \mathbb {P}^{(N)}[\{S\in \mathcal {I}:\, m^{(D-8)}_{n}(S)\ge\varTheta+1\}]\le\frac{2\kappa +2}{\kappa2^{D}}. $$
(4.16)
Furthermore, by (
4.15) and (
4.16),
$$\begin{aligned} |\mathbb {E}_{\mathbb {P}^{(N)}}[G(\tilde {\mathbb {S}})] - \mathbb {E}_{\mathbb {P}^{(N)}}[G(\check{\mathbb {S}})]| &\le2\kappa \mathbb {P}^{(N)}[m^{(D-8)}_{n}(\tilde {\mathbb {S}})>\varTheta]+2f_{e}(2^{-D+9}) \\ &\phantom{=}{} +nL\mathbb {E}_{\mathbb {P}^{(N)}}[\|\mathbb {S}\|]/D \\ &\le\frac{4\kappa+4}{2^{D}} +2f_{e}(2^{-D+9}) \\ &\phantom{=}{} +nLV_{\mathcal {X}_{n}, \mathcal {P}, \mathcal {I}}(\|\mathbb {S}\|)/D. \end{aligned}$$
(4.17)
Step 4: Third time change: controlling the increments of the quadratic variation. We say that
\(\omega\in \mathcal {C}([0,T],\mathbb {R})\) admits a quadratic variation if
$$\lim_{N\to\infty}\sum_{k=0}^{m^{(N)}(\omega)-1}\Big(\omega_{\tau ^{(N)}_{k}(\omega)\land t}-\omega_{\tau^{(N)}_{k+1}(\omega)\land t}\Big)^{2} $$
exists and is a continuous function for
\(t\in[0,T]\). In this case, we denote this limit with
\(\langle\omega\rangle\) and otherwise we let
\(\langle\omega\rangle\) be zero. In addition, for
\(S\in\varOmega\), we say
\(S\) admits a quadratic variation if
\(S^{(i)}\) admits a quadratic variation for any
\(i\le d+K\).
It follows from Theorem 4.30.1 in Rogers and Williams [
46] and its proof that for any
\(\mathbb {P}\in \mathcal {M}\),
\(\langle \mathbb {S}\rangle:= (\langle \mathbb {S}^{(1)} \rangle, \ldots, \langle \mathbb {S}^{(d+K)} \rangle)\) agrees ℙ-a.s. with the classical definition of the quadratic variation of
\(\mathbb {S}\) under ℙ, i.e.,
\(\mathbb {S}^{2}-\langle \mathbb {S}\rangle\) is a ℙ-martingale. Further, Doob’s inequality gives for all
\(i\le d\) that
$$ \mathbb {E}_{\mathbb {P}^{(N)}}[\|\check{\mathbb {S}}^{(i)}\|^{p}]\le\bigg(\frac{p}{p-1}\bigg)^{p}\int_{[0,\infty)} x^{p}\mu^{(i)}_{n}(\mathrm {d}x), $$
and by the BDG inequalities, there exist constants
\(c_{p}, C_{p}\in (0,\infty)\) such that
$$ c_{p}\mathbb {E}_{\mathbb {P}^{(N)}}\big[\langle\check{\mathbb {S}}^{(i)} \rangle^{p/2}_{T_{n}}\big]\le \mathbb {E}_{\mathbb {P}^{(N)}}[\|\check{\mathbb {S}}^{(i)}\|^{p}] \le C_{p}\mathbb {E}_{\mathbb {P}^{(N)}}\big[\langle\check{\mathbb {S}}^{(i)} \rangle^{p/2}_{T_{n}}\big]. $$
It follows that
$$ \mathbb {E}_{\mathbb {P}^{(N)}}\bigg[\sum_{i=1}^{d+K}\langle\check{\mathbb {S}}^{(i)} \rangle ^{p/2}_{T_{n}}\bigg]\le K_{1}, $$
where
\(K_{1} := \frac{1}{c_{p}}((\frac{p}{p-1})^{p}\sum_{i=1}^{d}\int _{[0,\infty)} x^{p}\mu^{(i)}_{n}(\mathrm {d}x)+K\kappa^{p})\).
In the following, we want to modify
\(\check{\mathbb {S}}\) on
$$\begin{aligned} \tilde{ \mathcal {I}} &:= \{S\in \mathcal {I}: \check{\mathbb {S}}(S) \text{ admits a quadratic variation}\} \\ &\phantom{:}= \{S\in \mathcal {I}: S \text{ admits a quadratic variation}\} \end{aligned}$$
to obtain another process
\(\ddot{\mathbb {S}}\) with a better control of the quadratic variation, while its law remains in
. In fact,
\(\ddot{\mathbb {S}}\) will be obtained as a time change of
\(\check{\mathbb {S}}\) on each interval
\([\sigma_{i,j}(\tilde {\mathbb {S}}), \sigma_{i,j+1}(\tilde {\mathbb {S}}))\). Then by the continuity of
\(G\), it follows that
$$ \big|G\big(\check{\mathbb {S}}(S)\big) - G\big(\ddot{\mathbb {S}}(S)\big)\big|\le f_{e}(2^{-D+9}), \qquad \forall\, S\in\tilde{ \mathcal {I}}\cap\big\{ h\in \mathcal {I}: m^{(D-8)}_{n}\big(\tilde {\mathbb {S}}(h)\big) \le\varTheta\big\} . $$
This together with (
4.16) and the fact that
\(\mathbb {P}[\tilde{ \mathcal {I}}] = 1\) for any
\(\mathbb {P}\in \mathcal {M}_{ \mathcal {I}}\) yields
$$\begin{aligned} |\mathbb {E}_{\mathbb {P}^{(N)}}[G(\check{\mathbb {S}}) - G(\ddot{\mathbb {S}})]|&\le f_{e}(2^{-D+9}) + 2\kappa \mathbb {P}^{(N)}\big[\big\{ S\in \mathcal {I}:\, m^{(D-8)}_{n}\big(\tilde {\mathbb {S}}(S)\big)\ge \varTheta+1\big\} \big]\\ &\le f_{e}(2^{-D+9}) + \frac{4\kappa+4}{2^{D}}. \end{aligned}$$
Hence, by (
4.14) and (
4.17),
$$\begin{aligned} |\mathbb {E}_{\mathbb {P}^{(N)}}[G(\mathbb {S}) - G(\ddot{\mathbb {S}})]| &\le5f_{e}(2^{-D+9}) + \frac{2nL\|\mathbb {S}\|}{D}+ \frac{8\kappa+8}{2^{D}} \\ &\phantom{=}{}+\frac{2nLV_{\mathcal {X}_{n},\mathcal {P}, \mathcal {I}}(\|\mathbb {S}\|)}{D}. \end{aligned}$$
(4.18)
First, for every
\(i,j,k\), define
\(\rho^{(i,j,k)}:\varOmega\to [T_{i-1},T_{i}]\) by
$$\rho^{(i,j,k)}(S) = \sigma_{i,j}\big(\check{\mathbb {S}}(S)\big)+\delta(1-2^{-k+1}). $$
Then for
\(i = 1,\ldots,n\),
\(j = 0,1,\ldots\), let
\(\theta ^{(i,j,0)}_{t}=\sigma_{i,j}\) and define recursively for
\(k = 1,2,\ldots\) a change of time
\(\theta^{(i,j,k)}: \mathcal {I}\times[\rho^{i,j,k},\rho ^{i,j,k+1}] \to[T_{i-1},T_{i}]\) by
$$\begin{aligned} \theta^{(i,j,k)}_{t}(S) &= \inf\bigg\{ u\ge\theta^{(i,j,k-1)}_{\rho^{i,j,k}}(S) :\sum_{\ell =1}^{d+K} \big(\langle\check{\mathbb {S}}^{(\ell)}(S)\rangle_{u} - \langle\check{\mathbb {S}}^{(\ell)}(S) \rangle_{\theta^{(i,j,k-1)}_{\rho^{i,j,k}}}\big) \\ &\hphantom{=:\inf\bigg\{ u\ge\theta^{(i,j,k-1)}_{\rho^{i,j,k}}(S) :} > 2^{k}(t-\rho^{(i,j,k)})/\delta\bigg\} \wedge\sigma_{i,j+1}\big(\check{\mathbb {S}}(S)\big)\\ &\phantom{=:}\text{ for } t\in[\rho^{i,j,k},\rho^{i,j,k+1}], S\in \tilde{ \mathcal {I}}. \end{aligned}$$
For
\(S\in\varOmega\setminus\tilde{ \mathcal {I}}\), set
\(\theta^{(i,j,k)}_{t}(S) = t\),
\(0 \leq t\leq T_{n}\).
We consider a time change of
\(\check{\mathbb {S}}\) via the
\(\theta^{(i,j,k)}\), defined by
\(\ddot{\mathbb {S}}_{t} := \check{\mathbb {S}}_{\theta^{(i,j,k)}_{t}(\mathbb {S})}\) for
\(t\in[\rho^{(i,j,k)}(\mathbb {S}), \rho^{(i,j,k+1)}(\mathbb {S}))\) for all
\(i,j,k\) as above. Note that
\(\theta^{(i,j,k-1)}_{\rho ^{i,j,k}}=\theta^{(i,j,k)}_{\rho^{i,j,k}}\) so that the resulting process is continuous. Consider
\(S\in\tilde { \mathcal {I}}\) and
\(i,j\) such that we have
\(\sigma_{i,j+1}(\tilde {\mathbb {S}}(S)) - \sigma_{i,j}(\tilde {\mathbb {S}}(S)) > 0\), as otherwise everything collapses to one point. Then the quadratic variation of
\(\ddot{\mathbb {S}}(S)\) grows on
\([\rho^{(i,j,k)}(S), \rho^{(i,j,k+1)}(S))\) linearly at the rate
\(2^{k}/\delta\), and
\(\rho^{(i,j,k+1)}(S) - \rho ^{(i,j,k)}(S) = 2^{-k}\delta\). In particular,
\(\ddot{\mathbb {S}}\) accumulates one unit of quadratic variation over each interval
\([\rho^{(i,j,k)}(S), \rho^{(i,j,k+1)}(S))\) for
\(k\) increasing until the total quadratic variation of
\(\check{\mathbb {S}}\) on
\([\sigma_{i,j+1}(\tilde {\mathbb {S}}(S)) - \sigma_{i,j}(\tilde {\mathbb {S}}(S))]\) is exhausted. Trivially bounding the quadratic variation of
\(\check{\mathbb {S}}\) over a small interval by its quadratic variation over
\([0,T_{n}]\), we see that
$$ \sum_{\ell=1}^{d+K}\big(\langle\ddot{\mathbb {S}}^{(\ell)}(S)\rangle_{t} - \langle \ddot{\mathbb {S}}^{(\ell)}(S)\rangle_{s}\big) \le2^{k_{0}}|t-s|/\delta \quad \text{for }\sigma_{i,j}\big(\tilde {\mathbb {S}}(S)\big)\le s \le t\le\sigma_{i,j+1}\big(\tilde {\mathbb {S}}(S)\big), $$
whenever
\(S\in\tilde{ \mathcal {I}}\) is such that
\(\sum_{i=1}^{d+K}\langle \check{\mathbb {S}}^{(i)}(S) \rangle_{T_{n}} \le k_{0}\). Therefore, for such
\(S\), we have
$$ \sum_{\ell=1}^{d+K}\big(\langle\ddot{\mathbb {S}}^{(\ell)}\rangle_{t} - \langle \ddot{\mathbb {S}}^{(\ell)}\rangle_{s}\big) \le2^{k_{0}+1}|t-s|/\delta, \quad \forall s,t\in[0,T_{n}] \text{ with } |t-s|\le\delta. $$
(4.19)
We can ensure this happens with large probability since by Markov’s inequality,
$$\begin{aligned} \mathbb {P}^{(N)}\bigg[\sum_{i=1}^{d+K}\langle\ddot{\mathbb {S}}^{(i)} \rangle_{T_{n}} > k_{0}\bigg] &= \mathbb {P}^{(N)}\bigg[\sum_{i=1}^{d+K}\langle\check{\mathbb {S}}^{(i)} \rangle_{T_{n}} > k_{0}\bigg]\\ &\le\frac{\mathbb {E}_{\mathbb {P}^{(N)}}[\sum_{i=1}^{d+K}\langle\check{\mathbb {S}}^{(i)} \rangle^{p/2}_{T_{n}}]}{k_{0}^{p/2}}\le K_{1}k_{0}^{-p/2}. \end{aligned}$$
Finally, we observe that each
\(\theta^{(i,j,k)}_{t}(\mathbb {S})\) is a stopping time relative to the natural filtration of
\(\check{\mathbb {S}}\), and hence
\(\ddot{\mathbb {S}}\) is a continuous
\(\mathbb {P}^{(N)}\)-martingale.
Step 5: Tightness of the measures through tightness of the quadratic variation processes. Together with (
4.19), by the Arzelà–Ascoli theorem, the above implies that the family
\(\{\mathbb {P}^{(N)}\circ(\langle\ddot{\mathbb {S}} \rangle)^{-1}: N\in \mathbb {N}\}\) is tight in
\(\mathcal {C}([0,T_{n}], \mathbb {R}^{d+K})\). Then by Theorem VI.4.13 in Jacod and Shiryaev [
35],
\(\{\mathbb {P}^{(N)}\circ\ddot{\mathbb {S}}^{-1}\}_{N\in \mathbb {N}}\) is tight in
\(\mathbb {D}([0,T_{n}],\mathbb {R}^{d+K})\), the space of right-continuous functions with left limits. By Theorem VI.3.21 in Jacod and Shiryaev [
35], this implies that for all
\(\epsilon>0, \eta>0\), there are
\(N_{0}\in \mathbb {N}\) and
\(\theta>0\) with
$$\begin{aligned} N\ge N_{0} \Longrightarrow \mathbb {P}^{(N)}[w_{T_{n}}^{\prime}(\ddot{\mathbb {S}}, \theta )\ge\eta]\le\epsilon, \end{aligned}$$
where
\(w_{T_{n}}^{\prime}\) is defined by
$$\begin{aligned} w_{T_{n}}^{\prime}(S, \theta) &= \inf\Big\{ \max_{i\le r} \sup_{t_{i-1}\le s\le t< t_{i}}|S_{t}- S_{s}|:r\in \mathbb {N}, 0 = t_{0}< \cdots< t_{r} = T_{n},\\ &\phantom{=:\inf\Big\{ \max_{i\le r} \sup_{t_{i-1}\le s\le t< t_{i}}|S_{t}- S_{s}|:} \inf_{i< r}(t_{i}-t_{i-1})\ge\theta\Big\} . \end{aligned}$$
Clearly, for
\(S\in\varOmega\), continuity of
\(S\) implies that
$$\begin{aligned} w_{T_{n}}(S, \theta):=\sup\{|S_{t}- S_{s}|:\, 0\le s< t\le T_{n},\, t-s\le\theta \} \le2w_{T_{n}}^{\prime}(S, \theta). \end{aligned}$$
Then we have
$$\begin{aligned} N\ge N_{0} \Longrightarrow \mathbb {P}^{(N)}[w_{T_{n}}(\ddot{\mathbb {S}}, \theta)\ge2\eta ]\le\epsilon, \end{aligned}$$
which then by Theorem VI.1.5 in Jacod and Shiryaev [
35] implies that the family
\(\{\mathbb {P}^{(N)}\circ\ddot{\mathbb {S}}^{-1}: N\in \mathbb {N}\}\) is tight, now in
\(\mathcal {C}([0,T_{n}],\mathbb {R}^{d+K})\).
Step 6: Tightness gives exact duality. By tightness, there exists a converging subsequence
\(\{\mathbb {P}^{(N_{k})}\circ \ddot{\mathbb {S}}^{-1}\}\) such that
\(\mathbb {P}^{(N_{k})}\circ\ddot{\mathbb {S}}^{-1} \to \mathbb {P}\) weakly for some probability measure ℙ on
\(\varOmega\). Consequently,
$$\begin{aligned} \lim_{k\to\infty} \mathbb {E}_{\mathbb {P}^{(N_{k})}}[G(\ddot{\mathbb {S}})]=\mathbb {E}_{\mathbb {P}}[G(\mathbb {S})]. \end{aligned}$$
In addition, if ℙ is an element of
\(\mathcal {M}_{\pmb {\mu },\mathfrak {P}}\), then
where
\(e(x) := 5f_{e}(2^{-x+9}) + \frac{2nL\|\mathbb {S}\|}{x}+ \frac{c_{2}+ 8\kappa +8}{2^{x}} +\frac{2nLV_{\mathcal {X}_{n},\mathcal {P}, \mathcal {I}}(\|\mathbb {S}\|)}{x}\) and the third inequality follows from (
4.18). Recalling that
\(\widetilde {V}_{\mathcal {X},\mathcal {P},\mathfrak {P}}=\widetilde {P}_{\pmb {\mu },\mathfrak {P}}\) and letting
\(D\to\infty\), we obtain the desired equality
\(\widetilde {P}_{\pmb {\mu },\mathfrak {P}}=P_{\pmb {\mu },\mathfrak {P}}\) and conclude that
$$ \widetilde {V}_{\mathcal {X},\mathcal {P},\mathfrak {P}}(G)=V_{\mathcal {X},\mathcal {P},\mathfrak {P}}(G)=P_{\pmb {\mu}, \mathfrak {P}}(G)=\widetilde{P}_{\pmb {\mu}, \mathfrak {P}}(G). $$
It remains to argue that ℙ is an element of
\(\mathcal {M}_{\pmb {\mu}, \mathfrak {P}}\). First, it is straightforward to see that
\(\mathbb {S}\) is a ℙ-martingale and
\(\mathcal {L}_{\mathbb {P}}(S_{T_{i}}) = \mu_{i}\) for any
\(i\le n\). To show that
\(\mathbb {P}[\mathbb {S}\in \mathfrak {P}] = 1\), notice that by the Portemanteau theorem, for every
\(\epsilon>0\),
$$\mathbb {P}[\mathbb {S}\in \mkern 1.5mu\overline {\mkern -1.5mu\mathfrak {P}^{\epsilon}\mkern -1.5mu}\mkern 1.5mu] \ge\limsup_{k\to\infty} \mathbb {P}^{(N_{k})}[\mathbb {S}\in \mkern 1.5mu\overline {\mkern -1.5mu\mathfrak {P}^{\epsilon}\mkern -1.5mu}\mkern 1.5mu] \ge\limsup_{k\to\infty} \mathbb {P}^{(N_{k})}[\mathbb {S}\in \mathfrak {P}^{1/N_{k}}] =1. $$
Therefore, it follows from Remark
4.6 and monotone convergence that
$$ \mathbb {P}[\mathbb {S}\in \mathfrak {P}] = \lim_{\epsilon\searrow0}\mathbb {P}[\mathbb {S}\in \mkern 1.5mu\overline {\mkern -1.5mu\mathfrak {P}^{\epsilon}\mkern -1.5mu}\mkern 1.5mu] =1, $$
and hence
\(\mathbb {P}\in \mathcal {M}_{\pmb {\mu}, \mathfrak {P}}\). □