Log-optimal and numéraire portfolios for market models stopped at a random time

Abstract

This paper focuses on numéraire and log-optimal portfolios when a market model $(S,\mathbb{F},P)$ – specified by its assets’ price $S$, its flow of information $\mathbb{F}$ and a probability measure $P$ – is stopped at a random time $\tau$. The flow of information that incorporates both $\mathbb{F}$ and $\tau$, denoted by $\mathbb{G}$, is the progressive enlargement of $\mathbb{F}$ with $\tau$. For the resulting stopped model $(S^{\tau},\mathbb{G},P)$, we study the two portfolios in different manners and describe their computations in terms of $\mathbb{F}$-observable parameters of the pair $(S, \tau )$. In particular, we single out the types of risks induced by $\tau$ that really affect the numéraire portfolio, and address the following questions: 1) What are the conditions on $\tau$ (preferably in terms of information-theoretic concepts such as entropy) that guarantee the existence of the log-optimal portfolio for $(S^{\tau},\mathbb{G},P)$ when that for $(S,\mathbb{F},P)$ already exists? 2) What are the factors that fully determine the increment in maximal expected logarithmic utility from terminal wealth for the models $(S^{\tau},\mathbb{G},P)$ and $(S,\mathbb{F},P)$, and how can one quantify them?

Appendices

Appendix A: Some $\mathbb{G}$-properties versus those in $\mathbb{F}$

Lemma A.1

Let $A$ be a nondecreasing and $\mathbb{F}$-predictable process, and suppose that $G>0$. Then the following assertions hold:

(a) For any $\mathbb{G}$-predictable process $\varphi ^{\mathbb{G}}$, there exists an $\mathbb{F}$-predictable process $\varphi ^{\mathbb{F}}$ such that $\varphi ^{\mathbb{G}}=\varphi ^{\mathbb{F}}$ on ${]\!\!]0,\tau ]\!\!]}$. Furthermore, if $\varphi ^{\mathbb{G}}>0$ (respectively $\varphi ^{\mathbb{G}}\leq 1$), then $\varphi ^{\mathbb{F}}>0$ (respectively $\varphi ^{\mathbb{F}}\leq 1$).

(b) For any $\theta \in{\mathcal{L}}(S^{\tau},\mathbb{G})$, there exists ${\varphi}\in {\mathcal{L}}(S,\mathbb{F})$ such that ${\varphi}={\theta}$ on $]\!\!]0,\tau ]\!\!]$.

(c) Let $\varphi$ be a nonnegative and $\mathbb{F}$-predictable process. Then $\varphi <\infty$ $(P\otimes A)$-a.e. on $]\!\!]0,\tau ]\!\!]$ if and only if $\varphi <\infty$ $(P\otimes A)$-a.e.

(d) Let $V$ be an $\mathbb{F}$-predictable and nondecreasing process with values in $[0,\infty ]$. Then $V^{\tau}$ is $\mathbb{G}$-locally integrable if and only if $V$ is $\mathbb{F}$-locally integrable.

(e) For any $M\in {\mathcal{M}}_{\mathrm{loc}}(\mathbb{F})$, we have .

The proof of (a) is a particular case of Aksamit et al. [1, Lemma B.1], (b) can be found in Choulli and Yansori [17, Lemma A.1], while (c) and (d) follow immediately from Aksamit et al. [1, Proposition B.2, (c)–(f)]. Assertion (e) can be found in Jeulin [34, Proposition (4.16), Chap. IV] or Dellacherie et al. [24, Théorème 76, Chap. XX].

The results in the following lemma sound new to us.

Lemma A.2

Let $A$ be a nondecreasing and $\mathbb{F}$-predictable process, and suppose that $G>0$. Then the following assertions hold:

(a) For any $\theta \in L(S^{\tau},\mathbb{G})$, there exists ${\varphi}\in L(S,\mathbb{F})$ such that ${\varphi}={\theta}$ on $]\!\!]0,\tau ]\!\!]$.

(b) Let $V$ be an $\mathbb{F}$-predictable process. Then $V I_{]\!\!]0,\tau ]\!\!]}\leq 0$ $(P\otimes A)$-a.e. if and only if $V\leq 0$ $(P\otimes A)$-a.e.

(c) Let $Y$ be an $\mathbb{F}$-optional process. Then $Y$ is $\mathbb{F}$-locally bounded if and only if $Y^{\tau}$ is $\mathbb{G}$-locally bounded.

Proof

(a) Let $\theta \in{ L}(S^{\tau},\mathbb{G})$. Then on the one hand, due to Stricker [50, Theorem 1.16, or Remark 2.2(h)], this is equivalent to the set

being bounded in probability. On the other hand, by Lemma A.1(a), we deduce that there exists an $\mathbb{F}$-predictable process $\varphi$ such that $\theta =\varphi$ on $]\!\!]0,\tau ]\!\!]$, and the $\mathbb{G}$-predictability in $\mathcal{X}^{\mathbb{G}}$ can be replaced with $\mathbb{F}$-predictability as well. Furthermore, for any $T\in (0,\infty )$, any $c>0$ and any $\mathbb{F}$-predictable $H$ bounded by one, by putting $Q:=(G_{T}/E[G_{T}])\cdot P\approx P$ and $X^{*}_{t}:=\sup _{0\leq s\leq t}\vert X_{s}\vert$ for a right-continuous with left limits process $X$, we have

This allows us to conclude, due to Stricker [50, Theorem 1.16, or Remark 2.2(h)] again, that $\varphi \in L(S^{T},\mathbb{F})$, for any $T\in (0,\infty )$. Thus (a) follows from combining this latter fact and Chou et al. [9, Theorem 4].

(b) Let $V$ be an $\mathbb{F}$-predictable process such that $V I_{]\!\!]0,\tau ]\!\!]}\leq 0$ $(P\otimes A)$-a.e. This is equivalent to

or $V^{+}=0$ $(P\otimes A)$-a.e. This is obviously equivalent to $V\leq 0$ $(P\otimes A)$-a.e.

(c) If $Y$ is $\mathbb{F}$-locally bounded, then obviously $Y^{\tau}$ is $\mathbb{G}$-locally bounded. To prove the converse, we consider a sequence of $\mathbb{G}$-stopping times that increase to infinity almost surely and such that $\vert Y^{\tau \wedge T_{n}}\vert \leq C_{n}$ for some $C_{n}\in (0,\infty )$. By applying Aksamit et al. [1, Proposition B.2(b)] and its proof to the sequence $(T_{n})$ and using $\widetilde{G}\geq G>0$ and $G_{-}>0$, we obtain a sequence $(\sigma _{n})$ of $\mathbb{F}$-stopping times that increase to infinity and $\sigma _{n}\wedge \tau =T_{n}\wedge \tau$ for all $n$. By taking the $\mathbb{F}$-optional projection on both sides of $I_{]\!\!]0,\tau ]\!\!]}\leq I_{\{\vert Y^{\sigma _{n}} \vert \leq C_{n}\}}$ and using again the positivity of $\widetilde{G}$, we obtain $\vert Y^{ \sigma _{n}}\vert \leq C_{n}$. □

The following recalls the $\mathbb{G}$-compensator of an $\mathbb{F}$-optional process stopped at $\tau$.

Lemma A.3

Let $V \in {\mathcal{A}}_{\mathrm{loc}} ({\mathbb{F}})$. Then we have

The proof of the lemma can be found in Aksamit et al. [1, Lemma 3.2].

Appendix B: Some useful martingale integrability properties

The results of this section are new and very useful.

Lemma B.1

Consider $K\in {\mathcal{M}}_{0,\mathrm{loc}}(\mathbb{H})$ with $1+\Delta K>0$, and let $H^{(0)}(K,\mathbb{H})$ be given by Definition 4.1. If $E[H^{(0)}_{T}(K,\mathbb{H})]<\infty$, then $E[\sqrt{[K,K]_{T}}]<\infty$ or, equivalently, $E[ \sup _{0\leq t\leq T}\vert K_{t}\vert ]<\infty$.

Proof

Take $K$ as above. Remark that for any $\delta \in (0,1)$ and

$$ C_{\delta}:=\min \big(-\delta -\ln (1-\delta ),\delta -\ln (1+\delta ) \big), $$

we have

$$ \Delta K-\ln (1+\Delta K)\geq {\frac{{C_{\delta} }}{{\delta}}}\vert \Delta K\vert{I}_{\{\vert \Delta K\vert >\delta \}}+{ \frac{{(\Delta K)^{2}}}{{2(1+\delta )}}} I_{\{\vert \Delta K\vert \leq \delta \}}.$$

By combining this with (4.2), on the one hand, we deduce that

$$\begin{aligned} &E\bigg[\langle K^{c}\rangle _{T}+\sum _{0< t\leq T}\vert \Delta K_{t} \vert I_{\{\vert \Delta K_{t}\vert >\delta \}}+\sum _{0< t\leq T}( \Delta K_{t})^{2} I_{\{\vert \Delta K_{t}\vert \leq \delta \}}\bigg] \\ &\leq \overline{C}_{\delta} E[H^{(0)}_{T}(K, \mathbb{H})] < \infty , \end{aligned}$$

where $\overline{C}_{\delta}:=\max (\delta /C_{\delta}, 2(1+\delta ))$. On the other hand, it is clear that

$$ [K,K]^{1/2}_{T}\leq \sqrt{\langle K^{c}\rangle _{T}}+\sum _{0< t\leq T} \vert \Delta K_{t}\vert I_{\{\vert \Delta K_{t}\vert >\delta \}}+ \sqrt{\sum _{0< t\leq T}(\Delta K_{t})^{2} I_{\{\vert \Delta K_{t} \vert \leq \delta \}}}.$$

This gives the result. □

Proposition B.2

Let $Z$ be a positive supermartingale such that $Z_{0}=1$. Then the following assertions hold:

(a) There exists a unique pair $(K,V)$ such that $K\in {\mathcal{M}}_{\mathrm{loc}}(\mathbb{H})$, $V$ is nondecreasing and ℍ-predictable, $K_{0}=V_{0}=0$, $\Delta K>-1$ and $Z=\mathcal{E}(K)\exp (-V)$. Furthermore,

$$ \ln Z= V-\ln {\mathcal{E}}(K)= V-K+H^{(0)}(K,\mathbb{H}). $$

(B.1)

(b) $-\ln Z$ is a uniformly integrable submartingale if and only if there exist a uniformly integrable martingale $N$ and a nondecreasing and predictable process $V$ such that $V_{0}=N_{0}=0$, $\Delta N>-1, Z=\mathcal{E}(N)\exp (-V)$ and

$$\begin{aligned} E [-\ln Z_{T} ]=E [V_{T}+H^{(0)}_{T}(N,\mathbb{H}) ]< \infty . \end{aligned}$$

(B.2)

(c) Suppose there exists a finite sequence $(Z^{(i)})_{i=1,\dots ,n}$ of positive supermartingales such that the product $Z:= \prod _{i=1}^{n} Z^{(i)}$ is a supermartingale. Then $-\ln Z$ is a uniformly integrable submartingale if and only if all $-\ln Z^{(i)}$, $i=1,\dots ,n$, are.

Proof

(a) Combining Jacod and Shiryaev [33, Theorem II.8.21] and $Z_{-}\geq {^{p,\mathbb{F}}(Z)}>0$ implies the existence of a unique pair $(K,U)$, where $K$ is an ℍ-local martingale and $U$ is positive nonincreasing and ℍ-predictable such that $K_{0}=0$, $U_{0}=1$, $\mathcal{E}(K)>0$ and $Z=\mathcal{E}(K)U$. Thus it is enough to put $V=-\ln U$, and the first statement of (a) is proved. By combining Definition 4.1 and Itô’s formula for $\ln {\mathcal{E}}(X)$, for any $X\in {\mathcal{M}}_{\mathrm{loc}}(\mathbb{H})$ with $\Delta X>-1$, we derive

$$\begin{aligned} -\ln{\mathcal{E}}(X)=-X +H^{(0)}(X, \mathbb{H}). \end{aligned}$$

Thus (B.1) follows immediately.

(b) Due to (a), there exists a unique pair $(N,V)$ with $N\in{\mathcal{{M}}}_{\mathrm{loc}}(\mathbb{H})$, $V$ RCLL, nondecreasing and predictable, $N_{0}=V_{0}=0$, $\Delta N>-1$, $Z=\mathcal{E}(N)\exp (-V)$ and

$$\begin{aligned} -\ln Z &=-\ln{\mathcal{E}}(N)+V=-N+H^{(0)}(N,\mathbb{H})+ V. \end{aligned}$$

(B.3)

Suppose that $-\ln Z$ is a uniformly integrable submartingale and let $(\tau _{n})$ be a sequence of stopping times that increase to infinity and such that each $N^{\tau _{n}}$ is a martingale. By stopping (B.3) with $\tau _{n}$ and taking expectations, we then get

$$ E[-\ln Z_{\tau _{n}\wedge T}]=E [V_{\tau _{n}\wedge T}+H^{(0)}_{\tau _{n} \wedge T}(N,\mathbb{H}) ].$$

As $\{ -\ln Z_{\tau _{n}\wedge T} : n\geq 1\}$ is uniformly integrable and both $V$ and $H^{(0)}(N,\mathbb{H})$ are nondecreasing, (B.2) follows by letting $n$ go to infinity. Conversely, if (B.2) holds, then $E[H^{(0)}_{ T}(N,\mathbb{H}) ]<\infty$, and by combining this with Lemma B.1 and (B.3), we deduce that $-\ln Z$ is a uniformly integrable submartingale and $N$ is a uniformly integrable martingale.

(c) A direct application of (a) to each $Z^{(i)}$ ($i=1,\dots , n$) implies the existence of $N^{(i)}\in {\mathcal{M}}_{\mathrm{loc}}(\mathbb{H})$ and nondecreasing and predictable $V^{(i)}$ such that $\Delta N^{(i)}>-1$ and $Z^{(i)}=\mathcal{E}(N^{(i)})\exp (-V^{(i)})$, $i=1,\dots ,n$. Furthermore, we get

$$\begin{aligned} -\ln Z &=-\sum _{i=1}^{n} N^{(i)} +\sum _{i=1}^{n} H^{(0)}(N^{(i)}, \mathbb{H})+ \sum _{i=1}^{n} V^{(i)}. \end{aligned}$$

Hence $-\ln Z$ is a uniformly integrable submartingale if and only if

$$\begin{aligned} E\bigg[\sum _{i=1}^{n} H^{(0)}_{T}(N^{(i)},\mathbb{H})+ \sum _{i=1}^{n} V^{(i)}_{T}\bigg]< \infty , \end{aligned}$$

or equivalently $E [H^{(0)}_{T}(N^{(i)},\mathbb{H})+V^{(i)}_{T} ]<\infty$ for all $i=1,\dots , n$, as all the processes $H^{(0)}(N^{(i)},\mathbb{H})$ and $V^{(i)}$, $i=1,\dots ,n$, are nondecreasing with null initial values. By applying (b) to each $Z^{(i)}$ for $i=1,\dots ,n$, (c) follows immediately. □

Appendix C: Martingales’ parametrisation via predictable characteristics

Consider an arbitrary general model $(X, \mathbb{H},P)$, and recall the corresponding notation given in the first paragraph of Sect. 5 up to (5.1).

For the following result, we refer to Jacod [31, Theorem 3.75] and to Jacod and Shiryaev [33, Lemma III.4.24].

Theorem C.1

Let $N\in {\mathcal{M}}_{0,\mathrm{loc}}$. Then there exist $\phi \in L^{1}_{\mathrm{loc}}(X^{c})$, $N'\in {\mathcal{M}}_{\mathrm{loc}}$ with $[N',X]=0$, $N'_{0}=0$ and functionals $f\in {\widetilde{\mathcal{P}}}$ and $g\in {\widetilde{\mathcal{O}}}$ such that the following hold:

(a) $\sqrt{(f-1)^{2}\star \mu}$ and $(\sum (\widehat{f}- a)^{2}(1-a)^{-2} I_{\{a<1\}}I_{\{\Delta X=0\}} )^{1/2}$ belong to $\mathcal{A}^{+}_{\mathrm{loc}}$.

(b) $(g^{2}\star \mu )^{1/2}\in{\mathcal{A}}^{+}_{\mathrm{loc}}$, $M^{P}_{\mu}[g\ |\ {\widetilde {\mathcal{P}}}]=0$ $(P\otimes \mu )$-a.e., $\{a=1\}\subseteq \{\widehat{f}=1\}$ and

$$\begin{aligned} N=\phi \cdot X^{c}+\left (f-1+{\frac{{\widehat{f}-a}}{{1-a}}}I_{\{a< 1 \}}\right )\star (\mu -\nu )+g\star \mu +{N'}. \end{aligned}$$

Appendix D: A result on the log-optimal portfolio: Choulli and Yansori [18]

Here, we consider the general setting and its notation as in the first paragraph of Sect. 5, where $(X,\mathbb{H},P)$ is an arbitrary general model.

Theorem D.1

Let $X$ be an ℍ-semimartingale whose predictable characteristics are $(b,c,F, A )= (b^{X},c^{X},F^{X}, A^{X} )$, and let $\mathcal{K}_{\mathrm{log}}$ be the function given by (5.3). Then the following assertions are equivalent:

(a) The set $\mathcal{D}_{\mathrm{log}}(X,\mathbb{H})$ given by (2.4) is not empty (i.e., $\mathcal{D}_{\mathrm{log}}(X,\mathbb{H})\neq\emptyset$).

(b) There exists an ℍ-predictable process $\widetilde{\psi}\in{\mathcal{L}}(X,\mathbb{H})\cap{ L}(X,\mathbb{H})$ such that for any $\varphi \in {\mathcal{L}}(X,\mathbb{H})$, we have

$$\begin{aligned} &(\varphi -\widetilde{\psi})^{\mathrm{tr}}(b-c\widetilde{\psi})+ \int \left ( { \frac{{(\varphi -\widetilde{\psi})^{\mathrm{tr}}x}}{{1+{\widetilde{\psi}}^{\mathrm{tr}}x}}}-( \varphi -\widetilde{\psi})^{\mathrm{tr}}h(x)\right )F(dx)\leq 0, \end{aligned}$$

(D.1)

(D.2)

(c) There exists a unique $\widetilde{Z}\in{\mathcal{D}}_{\mathrm{log}}(X,\mathbb{H})$ such that

$$\begin{aligned} \inf _{Z\in{\mathcal{D}}(X,\mathbb{H})}E[-\ln Z_{T}]=E[-\ln \widetilde{Z}_{T}]. \end{aligned}$$

(d) There exists a unique $\widetilde{\theta}\in \Theta (X,\mathbb{H},P)$ such that

(e) The numéraire portfolio exists and its portfolio rate $\widetilde{\Psi}$ satisfies (D.2) (with $\widetilde{\psi}$ there replaced by $\widetilde{\Psi}$).

If one of the above holds, then $(P\otimes A)$-a.e. and

Appendix E: Proof of Lemmas 3.4, 4.4 and 5.12–5.14

This section has five subsections, where we prove these lemmas respectively.

5.1 E.1 Proof of Lemma 3.4

If $Y$ is a nonnegative $\mathbb{F}$-supermartingale, the Doob–Meyer decomposition implies the existence of $M\in {\mathcal{M}}_{\mathrm{loc}}(\mathbb{F})$ and a nondecreasing and $\mathbb{F}$-predictable process $B$ such that $Y=M-B$. Then Theorem 2.3(b) implies that both and are $\mathbb{G}$-local martingales. Thus by writing

we conclude that this process is a nonnegative $\mathbb{G}$-local supermartingale and hence a $\mathbb{G}$-supermartingale. To prove the converse implication, we assume is a $\mathbb{G}$-supermartingale. As a result, for bounded $\mathbb{F}$-stopping times $\sigma _{1}$ and $\sigma _{2}$ such that $\sigma _{1}\leq \sigma _{2}$ $P$-a.s., we have

Conditioning on $\mathcal{F}_{\sigma _{1}}$ and using and

we put and derive

$$\begin{aligned} Y_{\sigma _{1}}\widetilde{\mathcal{{E}}}_{\sigma _{1}}\geq E\left [ Y_{ \sigma _{2}}\widetilde{\mathcal{{E}}}_{\sigma _{2}}+\int _{\sigma _{1}}^{ \sigma _{2}} {\frac{{Y_{s}}}{{G_{s}}}}\widetilde{\mathcal{{E}}}_{s} d D^{o, \mathbb{F}}_{s} \bigg|{\mathcal{F}}_{\sigma _{1}}\right ]. \end{aligned}$$

Combining this with ${\widetilde{G}}\widetilde{\mathcal{{E}}}=G\widetilde{\mathcal{{E}}}_{-}$, $Y\widetilde{\mathcal{{E}}}_{-}dD^{o,\mathbb{F}}=-{\widetilde{G}}Yd \widetilde{\mathcal{{E}}}$ and the integration by parts formula for $Y\widetilde{\mathcal{{E}}}$, we get

$$ E\left [ \int _{\sigma _{1}}^{\sigma _{2}}\widetilde{\mathcal{{E}}}_{s-}dY_{s} \bigg|{\mathcal{F}}_{\sigma _{1}}\right ] \leq 0.$$

Thanks to this inequality and being nonnegative, we conclude that is an $\mathbb{F}$-supermartingale. Hence the local boundedness and positivity of $\widetilde{\mathcal{{E}}}_{-}^{-1}$ imply that $Y$ is a nonnegative $\mathbb{F}$-local supermartingale and then again an $\mathbb{F}$-supermartingale. □

5.2 E.2 Proof of Lemma 4.4

The proof of the lemma is achieved in three parts.

(a) Clearly, due to the local boundedness of both $G_{-}^{-1}$ and $m$, the processes and have variations that are $\mathbb{F}$-locally integrable. Therefore, by combining Lemma A.3, $G_{-}I_{]\!\!]0,\infty [\!\![}={{}^{p, \mathbb{F}}(}I_{]\!\!]0,\tau ]\!\!]})$, $\widetilde{G} {=} G_{-}+\Delta m$ and Definition 4.1, we derive

This proves (a).

(b) and (c). On the one hand, for any $Z^{\mathbb{G}}\in{\mathcal{D}}(S^{ \tau},\mathbb{G})$, Theorem 2.3 leads to the existence of a triplet $(Z^{\mathbb{F}}, \varphi ^{(o)}, \varphi ^{ (\mathrm{pr})} )$ that belongs to

$$ {\mathcal{D}}(S, \mathbb{F})\times \mathcal{I}^{o}_{\mathrm{loc}}(N^{\mathbb{G}}, \mathbb{G})\times L^{1}_{\mathrm{loc}}\big(\mathrm{{Prog}}(\mathbb{F}),P \otimes D\big) $$

and satisfies

$$ \varphi ^{ (\mathrm{pr})}>-1,\quad -{\widetilde{G}} G^{-1}< \varphi ^{(o)}, \quad \varphi ^{(o)}({\widetilde{G}} -G)< {\widetilde{G}},\quad \quad ({P} \otimes{D})\mbox{-a.e.} $$

and

This implies that

Hence, thanks to Proposition B.2(c), we deduce that $Z^{\mathbb{G}}\in {\mathcal{D}}_{\mathrm{log}}(S^{\tau},\mathbb{G})$ if and only if is in $\mathcal{D}_{\mathrm{log}}(S^{\tau},\mathbb{G})$ and and are uniformly integrable $\mathbb{G}$-submartingales, and (4.6) follows immediately. This proves (b). On the other hand, thanks to Theorem 2.3(c), the process always belongs to $\mathcal{D}(S^{\tau},\mathbb{G})$ as soon as $Z\in {\mathcal{D}}(S,\mathbb{F})$, and hence

By combining this with (b) (or equivalently (4.6)), we conclude that (4.7) holds, and (c) is proved.

(d) Consider $Z^{\mathbb{F}}\in {\mathcal{D}}(S,\mathbb{F})$. From Proposition B.2(a), we obtain the existence of $K\in {\mathcal{M}}_{0,\mathrm{loc}}(\mathbb{F})$ and a nondecreasing and $\mathbb{F}$-predictable process $V$ such that $V_{0}=K_{0}=0$ and $Z^{\mathbb{F}}:=\mathcal{E}(K)\exp (-V)$. Therefore, we derive

(E.1)

The third equality follows from combining the fact that $V-V^{p,\mathbb{G}}\in {\mathcal{M}}_{\mathrm{loc}}(\mathbb{G})$ for any $V\in {\mathcal{A}}_{\mathrm{loc}}(\mathbb{G})$ and Lemma A.1(e). Using now (E.1), Lemma A.3 and that (so that is a uniformly integrable $\mathbb{G}$-submartingale), (4.8) follows immediately. □

5.3 E.3 Proof of Lemma 5.12

Let $W$ be a $\widetilde{\mathcal{P}}(\mathbb{F})$-measurable function such that $W\star (\mu ^{\tau}-\nu ^{\mathbb{G}})$ is a well-defined $\mathbb{G}$-local martingale and $\widehat{Wf_{m}}$ is $\mathbb{F}$-locally bounded. Then by putting

$$\widehat{W}^{\mathbb{G}}_{t}:=\int W(t,x)\nu ^{\mathbb{G}}(\{t\}, dx)=( \widehat{Wf_{m}})_{t} I_{]\!\!]0,\tau ]\!\!]}(t),\qquad a^{ \mathbb{G}}:=\widehat{1}^{\mathbb{G}} $$

and using Definition 5.1, we deduce that

$$ \sqrt{(W-\widehat{W}^{\mathbb{G}})^{2}\star \mu ^{\tau}}\in {\mathcal{A}}^{+}_{{ \mathrm{loc}}}(\mathbb{G})\qquad \mbox{and}\qquad \sqrt{\sum (\widehat{W}^{ \mathbb{G}})^{2}I_{\{\Delta S=0\}}}\in {\mathcal{A}}^{+}_{\mathrm{loc}}( \mathbb{G}).$$

(E.2)

Consider $\Phi (y):=y^{2}(1+y)^{-1}, y\geq 0$, and remark that for any $y_{1}\geq 0,y_{2}\geq 0$ and $\alpha \geq 0$, we have

$$ \Phi (y_{1}+y_{2}) \leq 2 \Phi (y_{1})+2 \Phi (y_{2}),\qquad \Phi ( \alpha y_{1})\leq \max (1,\alpha )\alpha \Phi (y_{1}).$$

(E.3)

Then by Jacod [31, Proposition III.3.68], (E.2) holds iff

$$\begin{aligned} \Phi (\vert W-\widehat{Wf_{m}}\vert )\star \mu ^{\tau}+\sum \Phi ( \vert \widehat{Wf_{m}}\vert )(1-a^{\mathbb{G}})I_{]\!\!]0,\tau ]\!\!]}\in {\mathcal{A}}^{+}_{\mathrm{loc}}(\mathbb{G}) . \end{aligned}$$

(E.4)

Thus on the one hand, by combining the fact that $E[V_{T}]=E[V^{p,\mathbb{H}}_{T}]=E[V^{o,\mathbb{H}}_{T}]$ for any nondecreasing RCLL process $V$, Lemma A.1(d), (5.28), (E.2) and that both $G_{-}$ and $1/G_{-}$ are $\mathbb{F}$-locally bounded, we conclude that (E.4) is equivalent to each of the two equivalent conditions

$$ \big(f_{m}\Phi (\vert W-\widehat{Wf_{m}}\vert )\big)\star \mu \in { \mathcal {A}}^{+}_{\mathrm{loc}}(\mathbb{F})\ \mbox{and}\ \sum \Phi (\vert \widehat{Wf_{m}}\vert )(1-\widehat{f_{m}})\in {\mathcal{A}}^{+}_{\mathrm{loc}}( \mathbb{F}) $$

(E.5)

and

$$ \big(\widetilde{G}\Phi (\vert W-\widehat{Wf_{m}}\vert )\big)\star \mu \in {\mathcal{A}}^{+}_{\mathrm{loc}}(\mathbb{F})\ \mbox{and}\ \sum \Phi ( \vert \widehat{Wf_{m}}\vert )(1-\widehat{f_{m}})\in {\mathcal{A}}^{+}_{{ \mathrm{loc}}}(\mathbb{F}).$$

(E.6)

On the other hand, combining the second condition in (E.5) with the claim that

$$\begin{aligned} Y:=(1-a)(1-\widehat{f_{m}})^{-1}I_{\{\widehat{f_{m}}< 1\}}\ \mbox{is $\mathbb{F}$-locally bounded} \end{aligned}$$

(E.7)

implies that

$$\begin{aligned} \sum \Phi (\vert \widehat{Wf_{m}}\vert )(1-a)\in {\mathcal{A}}^{+}_{\mathrm{loc}}( \mathbb{F}). \end{aligned}$$

(E.8)

Furthermore, due to (E.3) and $f_{m}=M^{P}_{\mu}[{\widetilde{G}}G_{-}^{-1}|{\widetilde{\mathcal{P}}}( \mathbb{F})]\leq G_{-}^{-1}$, we have

$$\begin{aligned} &\Phi (\vert Wf_{m}-\widehat{Wf_{m}})\vert ) \\ &\leq 2\max (1,f_{m})f_{m}\Phi (\vert W-\widehat{Wf_{m}}\vert )+2 \Phi \big(\vert \widehat{Wf_{m}}(f_{m}-1)\vert \big) \\ &\leq {\frac{{2}}{{G_{-}}}}f_{m}\Phi (\vert W-\widehat{Wf_{m}}\vert )+2 \max \big((\widehat{Wf_{m}})^{2}, \vert \widehat{Wf_{m}}\vert \big) \Phi (\vert f_{m}-1\vert ). \end{aligned}$$

Thus by combining this with the local boundedness of both $G_{-}^{-1}$ and $\widehat{Wf_{m}}$, (E.8), the first condition in (E.5) and $\sqrt{(f_{m}-1)^{2}\star \mu}\in {\mathcal{A}}^{+}_{\mathrm{loc}}(\mathbb{F})$, we get

$$\begin{aligned} \Phi (\vert Wf_{m}-\widehat{Wf_{m}})\vert )\star \mu +\sum \Phi ( \vert \widehat{Wf_{m}}\vert )(1-a)\in {\mathcal{A}}^{+}_{\mathrm{loc}}( \mathbb{F}). \end{aligned}$$

Using Jacod [31, Proposition III.3.68] again, we conclude that $(f_{m}W)\star (\mu -\nu )$ is a well-defined $\mathbb{F}$-local martingale. Similar arguments allow us to deduce that $(Wg_{m})\star \mu$ is a well-defined $\mathbb{F}$-local martingale iff $\Phi (\vert Wg_{m}\vert )\star \mu \in {\mathcal{A}}^{+}_{\mathrm{loc}}( \mathbb{F})$. To prove this, we use (E.3) and write

$$ {\frac{{\Phi (\vert Wg_{m}\vert )}}{{2}}}\leq \max (1,\vert g_{m} \vert )\vert g_{m}\vert \Phi (\vert W-\widehat{Wf_{m}}\vert )+\max \big(( \widehat{Wf_{m}})^{2}, \vert \widehat{Wf_{m}}\vert \big)\Phi ( \vert g_{m}\vert ).$$

Then by combining this with the first condition in (E.5) and (E.6), the fact that $\vert g_{m}\vert \leq{G}_{-}^{-1}\widetilde{G}+f_{m}\leq 2{G}_{-}^{-1}$ on $\{\Delta S\neq0\}$, the local boundedness of $\widehat{Wf_{m}}$ and $\sqrt{g_{m}^{2}\star \mu}\in {\mathcal{A}}^{+}_{\mathrm{loc}}(\mathbb{F})$, we deduce that $\Phi (\vert Wg_{m}\vert )\star \mu \in {\mathcal{A}}^{+}_{\mathrm{loc}}( \mathbb{F})$. Hence the proof of the lemma is complete as soon as we prove (E.7).

To prove (E.7), we first remark that $1+G_{-}^{-1} \Delta m>0$ implies that

$$ 1-(\widehat{f_{m}}-a)(1-a)^{-1}I_{\{a< 1\}}>0, $$

or equivalently $\{a<1\}\subseteq \{\widehat{f_{m}}<1\}$. Secondly, we consider the sequence

$$ T_{n}:=\inf \big\{ t\geq 0 : K_{t}:=\big((\widehat{f_{m}})_{t}-a_{t} \big)(1-a_{t})^{-1}I_{\{a_{t}< 1\}}>1-n^{-1}\big\} $$

for any $n\geq 2$ and remark that this sequence of $\mathbb{F}$-stopping times increases to infinity. Furthermore, on $[\!\![0, T_{n}[\!\![$, we have $Y\leq (1-K)^{-1}\leq n$. This proves that the predictable process $Y$ is $\mathbb{F}$-prelocally bounded, and so (E.7) follows from Lenglart’s lemma; see Dellacherie and Meyer [25, VIII.11]. This ends the proof of the lemma. □

5.4 E.4 Proof of Lemma 5.13

Let $\theta _{1}$ and $\theta _{2}$ be two elements of $\mathcal{L}(S,\mathbb{F})$ such that for any $\theta \in{\mathcal{L}}(S,\mathbb{F})$, and $i = 1,2$,

$$\begin{aligned} (\theta -{\theta}_{i})^{\mathrm{tr}}(b-c{\theta}_{i})+\int \left ({ \frac{{(\theta -{\theta}_{i})^{\mathrm{tr}}x}}{{1+{\theta}_{i}^{\mathrm{tr}}x}}}-( \theta -{\theta}_{i})^{\mathrm{tr}}h(x)\right )F(dx)\leq 0. \end{aligned}$$

(E.9)

By considering $\theta =\theta _{3-i}$ for (E.9) and adding the resulting two inequalities, we get

$$\begin{aligned} (\theta _{1}-{\theta}_{2})^{\mathrm{tr}}c({\theta}_{1}-{\theta}_{2})+ \int{ \frac{{({\theta}_{2}^{\mathrm{tr}}x- {\theta}_{1}^{\mathrm{tr}}x)^{2}}}{{(1+{\theta}_{1}^{\mathrm{tr}}x)(1+{\theta}_{2}^{\mathrm{tr}}x)}}}F(dx) \leq 0\qquad (P\otimes A)\mbox{-a.e.} \end{aligned}$$

Thus on the one hand, we deduce that $c\theta _{1}=c\theta _{2}$ $(P\otimes A)$-a.e. and $\theta _{1}^{\mathrm{tr}}x=\theta _{2}^{\mathrm{tr}}x$ $(P\otimes A \otimes F)$-a.e. On the other hand, using these two facts and again putting $\theta =\theta _{3-i}$ in (E.9), we conclude that $\theta _{1}^{\mathrm{tr}}b=\theta _{2}^{\mathrm{tr}}b$ $(P\otimes A)$-a.e. This ends the proof of the lemma. □

5.5 E.5 Proof of Lemma 5.14

This proof is achieved in three parts.

(a) To simplify the notation, we put

Hence we get $\Delta{\widetilde{L}}^{\mathbb{F}}=f(1-a+\widehat{f})^{-1}I_{\{\Delta S \neq0\}}-1+(1-a+\widehat{f})^{-1}I_{\{\Delta S=0\}}$ and

$$\begin{aligned} &\Delta{\widetilde{L}}^{\mathbb{F}}-\ln (1+\Delta{\widetilde{L}}^{ \mathbb{F}}) \\ &= \left (f-1-\ln f+{ \frac{{(f-1)(a-\widehat{f})}}{{1-a+\widehat{f}}}}\right )I_{\{\Delta S \neq0\}} +\left ({\frac{{a-\widehat{f}}}{{1-a+\widehat{f}}}}+\ln (1-a+ \widehat{f})\right ). \end{aligned}$$

Then by combining this with Definition 4.1, ${\widetilde{G}}=G_{-}(f_{m}(\Delta S)+g_{m}(\Delta S))$ on $\{ \Delta S\neq0\}$, and $\Delta {\widetilde{V}}^{\mathbb{F}}=a-{\widehat{f}}$, we derive

(E.10)

Now note that $\vert f_{m}\vert +\vert g_{m}\vert \leq 3G_{-}^{-1}$, $(1-\Delta {\widetilde{V}}^{\mathbb{F}})^{-1}$ is locally bounded due to its predictability and $\Delta {\widetilde{V}}^{\mathbb{F}}<1$ and both processes $(f-1-\ln f)\star \nu$ and

are locally bounded. These latter remarks imply that

$$ \Big(g_{m} \big(f-1-\ln f+(f-1)(a-\widehat{f})(1-a+\widehat{f})^{-1} \big)\Big)\star \mu $$

is an $\mathbb{F}$-local martingale with finite variation. Thus by combining all these facts with (E.10), we obtain

Hence, by combining this with , (5.17) and (5.18), we derive

This proves (5.44), and the proof of (a) is completed.

(b) Here, we calculate . To this end, we use (5.2) to get

$$ \Delta m=G_{-}(f_{m}+g_{m}-1)I_{\{\Delta S\neq0\}}+G_{-}\bigg( \Delta m^{\perp}-{\frac{{\widehat{f_{m}}-a}}{{1-a}}}\bigg)I_{\{ \Delta S=0\}}$$

and derive

The local boundedness of $m$ and of $G_{-}^{-1}$ imply that . Hence all the first three processes on the RHS term of the above equality are locally integrable, while the last term is a local martingale with finite variation due to the local boundedness of $(\widehat{f}-a)/(1-a+\widehat{f})=\Delta {\widetilde{V}}^{\mathbb{F}}/(1- \Delta {\widetilde{V}}^{\mathbb{F}})$. Thus (b) follows immediately from combining these remarks with the above equality, $a-\widehat{f}=\Delta {\widetilde{V}}^{\mathbb{F}}$, $^{p,\mathbb{F}}(I_{\{\Delta S=0\}})=1-a$ and

$$ \bigg({\frac{{\widehat{f}-a}}{{1-a+\widehat{f}}}}(f_{m}-1)\bigg) \star \nu =\sum {\frac{{\widehat{f}-a}}{{1-a+\widehat{f}}}}({ \widehat{f}}_{m}-a).$$

(c) Clearly, we have $E[V_{T}]=E[V^{p,\mathbb{F}}_{T}]$ for any nondecreasing process $V$, while (5.47) follows directly from (5.10) and (5.6). It remains to prove (5.46). To this end, we recall that $\Delta{\widetilde{K}}^{\mathbb{G}}= ({\widetilde{\Gamma}}(1+{ \widetilde{\varphi}}^{\mathrm{tr}}\Delta S)^{-1}-1 )I_{]\!\!]0,\tau ]\!\!]}$, see (5.39) for details, and we use Definition 4.1 to derive

which is a nondecreasing and integrable process. Thus all processes above have integrable variations. By applying Lemma A.3 to the processes on the RHS of the latter equality and using the fact that $\widetilde{G}=G_{-}(f_{m}(\Delta S)+g_{m}(\Delta S))$ on $\{\Delta S\neq0\}$, (5.46) follows immediately. □