No-arbitrage up to random horizon for quasi-left-continuous models

Abstract

This paper studies the impact, on no-arbitrage conditions, of stopping the price process at an arbitrary random time. As price processes, we consider the class of quasi-left-continuous semimartingales, i.e., semimartingales that do not jump at predictable stopping times. We focus on the condition of no unbounded profit with bounded risk (called NUPBR), also known in the literature as no arbitrage of the first kind. The first principal result describes all the pairs of quasi-left-continuous market models and random times for which the resulting stopped model fulfils NUPBR. Furthermore, for a subclass of quasi-left-continuous local martingales, we construct explicitly martingale deflators, i.e., strictly positive local martingales whose product with the price process stopped at a random time is a local martingale. The second principal result characterises the random times that preserve NUPBR under stopping for any quasi-left-continuous model. The analysis carried out in the paper is based on new stochastic developments in the theory of progressive enlargements of filtrations.

Appendices

Appendix A: Deflator in terms of predictable characteristics

For undefined notations, we refer the reader to Sect. 5.1 (see also [23, Chapter III] and [24, Chapter 2]). Here, we consider an $(\mathbb{H}, Q)$-semimartingale $X$. We denote by $\mu _{X}$ the random measure associated to the jumps of $X$, and by $\nu _{X}$ its $(\mathbb{H},Q)$-compensator. Suppose that the canonical decomposition of the model $(X,\mathbb{H}, Q)$ takes the form

$$ X=X_{0}+X^{c}+h\star (\mu _{X}-\nu _{X})+b\hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}A+(x-h)\star \mu _{X},$$

where $h$ is the truncation function. This, by virtue of Sect. 5.1, is equivalent to saying that the predictable characteristics of this model are $(b,c,F,A)$. The elements of this quadruplet naturally depend on the model itself, and we write $b^{X,\mathbb{H},Q}$ instead of $b$ and so on. However, here we opt for simplified notations as there is no risk of confusion.

Theorem A.1

Consider the model $(X,\mathbb{H},Q)$ defined above with its predictable characteristics $(b,c,F,A)$. Then $(X,\mathbb{H},Q)$ satisfies NUPBR if and only if there exists a pair $(\beta , f)$ consisting of an ℍ-predictable process and a $\widetilde{\mathcal{P}}(\mathbb{H})$-measurable functional satisfying

$$\begin{aligned} &f>0 \quad Q\otimes \mu _{X}\textit{-a.e.},\quad \beta ^{\mathrm{tr}}c\beta \hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}A+\sqrt{(f-1)^{2}\star \mu _{X}}\in {\mathcal{A}}^{+}_{\mathrm{loc}}(\mathbb{H},Q),\hskip1cm \end{aligned}$$

(A.1)

$$\begin{aligned} &\displaystyle \int \vert xf(x)-h(x)\vert F(dx)< +\infty \quad Q\otimes A\textit{-a.e.},\quad \textit{and} \end{aligned}$$

(A.2)

$$\begin{aligned} &b + c\beta + \displaystyle \int \Bigl(x f(x)- h(x)\Bigr)F(dx) = 0 \quad Q\otimes A\textit{-a.e.} \end{aligned}$$

(A.3)

Proof

This follows by combining Proposition 2.3 and [11, Lemma 2.4]. □

Appendix B: $\mathbb{G}$-localisation versus $\mathbb{F}$-localisation

We now present results which are important for the proofs of Sect. 5.3, and are the most innovative results of the appendix.

Lemma B.1

The following assertions hold:

(a) If $H^{\mathbb{G}}$ is a $\widetilde{\mathcal{P}}(\mathbb{G})$-measurable functional, then there exist a $\widetilde{\mathcal{P}}(\mathbb{F})$-measurable functional $H^{\mathbb{F}}$ and a $\mathcal{B}(\mathbb{R}_{+})\otimes \widetilde{\mathcal{P}}(\mathbb{F})$-measurable functional $K^{\mathbb{F}}: {\mathbb{R}}_{+}\times \Omega \times {\mathbb{R}}_{+} \times {\mathbb{R}}^{d} \rightarrow \mathbb{R}$ such that

$$\begin{aligned} H^{\mathbb{G}}(\omega ,t,x) = H^{\mathbb{F}}(\omega ,t,x) I_{]\!]0,\tau [\![}+ K^{\mathbb{F}}\big(\tau (\omega ),\omega ,t,x\big)I_{[\![\tau ,+\infty ]\!]}. \end{aligned}$$

(b) If furthermore $H^{\mathbb{G}}>0$ (respectively $H^{\mathbb{G}}\leq 1$), then we can choose $H^{\mathbb{F}}>0$ (respectively $H^{\mathbb{F}}\leq 1$).

(c) If $H^{\mathbb{G}}$ is an $\widetilde{\mathcal{O}}(\mathbb{G})$-measurable functional, then there exist an $\widetilde{\mathcal{O}}(\mathbb{G})$-measurable functional $H^{(1)}(\omega ,t,x)$, a $\widetilde{\mathrm{Prog}}(\mathbb{F})$-measurable functional $H^{(2)}(\omega ,t,x)$, and a $\mathcal{B}(\mathbb{R}_{+})\otimes \widetilde{\mathcal{O}}(\mathbb{F})$-measurable functional $H^{(3)}(v,\omega ,t,x)$ such that

$$ H^{\mathbb{G}}(\omega ,t,x) = H^{(1)}(\omega ,t,x)I_{]\!]0,\tau ]\!]}+H^{(2)}(\omega ,t,x)I_{]\!]\tau [\![} +H^{(3)}\big(\tau (\omega ),\omega ,t,x\big)I_{[\![\tau ,+\infty ]\!]}. $$

If furthermore $0< H^{\mathbb{G}}$ (respectively $H^{\mathbb{G}}\leq 1$), then all $H^{(i)}$ can be chosen such that $0< H^{(i)}$ (respectively $H^{(i)}\leq 1$), $i=1,2,3$.

(d) For any $\mathbb{F}$-stopping time $T$ and any positive $\mathcal{G}_{T}$-measurable random variable $Y^{\mathbb{G}}$, there exist two positive $\mathcal{F}_{T}$-measurable random variables $Y^{(1)}$ and $Y^{(2)}$ satisfying

$$\begin{aligned} Y^{\mathbb{G}}I_{\{ T\leq \tau \}}=Y^{(1)}I_{\{T< \tau \}}+Y^{(2)}I_{\{\tau =T\}}. \end{aligned}$$

Proof

Both (a) and (c) follow directly from mimicking Jeulin (see [25, Lemma 4.4 (b) and Proposition 5.3 (b), respectively]), and their proofs are omitted. Assertion (d) follows easily from applying (c) to $H^{\mathbb{G}}:=\ ^{o,\mathbb{G}}(Y^{\mathbb{G}})I_{]\!]0,\tau [\![}$ and $H^{\mathbb{G}}_{T}=Y^{\mathbb{G}}I_{\{T\leq \tau \}}$. Thus the rest of this proof focuses on (b). Suppose that $H^{\mathbb{G}}>0$ (resp. $H^{\mathbb{G}}\leq 1$). Then thanks to (a), there exists a $\mathcal{B}(\mathbb{R}_{+})\otimes \widetilde{\mathcal{P}}(\mathbb{F})$-measurable functional $H'$ such that $H^{\mathbb{G}}=H'$ on $]\!]0,\tau [\![$. This implies that $]\!]0,\tau [\![\subset \{H'>0\}$ (resp. $]\!]0,\tau [\![\subset \{H'\leq 1\}$). By considering $H^{\mathbb{F}}=(H')^{+}+I_{\{H'\leq 0\}}$ (resp. $H^{\mathbb{F}}=\min (1, H')$), (b) follows immediately. □

Proposition B.2

Let $\Phi _{\alpha }(\cdot )$ be given by (5.8). Then the following hold:

(a) Let $f$ be a ${\widetilde{\mathcal{P}}(\mathbb{H})}$-measurable functional. Then

$$\begin{aligned} \sqrt{(f-1)^{2}\star \mu } \in {\mathcal{A}}^{+}_{\mathrm{loc}}(\mathbb{H})\quad \textit{if and only if}\quad \Phi _{\alpha }(f)\star \mu \in {\mathcal{A}}^{+}_{\mathrm{loc}}(\mathbb{H}). \end{aligned}$$

(b) Let $(\sigma ^{\mathbb{G}}_{n})$ be a sequence of $\mathbb{G}$-stopping times that increases to $+\infty$. Then there exists a nondecreasing sequence $(\sigma ^{\mathbb{F}}_{n})_{n\geq 1}$ of $\mathbb{F}$-stopping times satisfying

$$ \sigma ^{\mathbb{G}}_{n}\wedge \tau = \sigma ^{\mathbb{F}}_{n}\wedge \tau ,\quad \sigma _{\infty }^{\mathbb{F}}:=\sup _{n} \sigma ^{\mathbb{F}}_{n}\geq R\ \ P\textit{-a.s.}, $$

(B.1)

and

$$ Z_{\sigma _{\infty }^{\mathbb{F}}-}=0\quad P\textit{-a.s.} \quad \textit{on }\Sigma \cap \{\sigma _{\infty }^{\mathbb{F}}< +\infty \}, $$

(B.2)

where $\Sigma :=\bigcap _{n\geq 1}\{\sigma _{n}^{\mathbb{F}}<\sigma _{\infty }^{\mathbb{F}}\}$ and $R:=\inf \{t\geq 0: Z_{t}=0\}$.

(c) Let $V$ be an $\mathbb{F}$-predictable nondecreasing process with values in $[0,+\infty ]$. Then $V^{\tau }\in {\mathcal{A}}_{\mathrm{loc}}^{+}(\mathbb{G})$ if and only if $I_{\{ Z_{-}\geq \delta \}} \hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}V\in {\mathcal{A}}_{\mathrm{loc}}^{+}(\mathbb{F})$ for any $\delta >0$.

(d) Let $k$ be a nonnegative $\widetilde{\mathcal{P}}(\mathbb{F})$-measurable functional. Then $kI_{[\![0,\tau [\![}\star \mu \in {\mathcal{A}}^{+}_{\mathrm{loc}}(\mathbb{G})$ if and only if for all $\delta >0$, $kI_{\{Z_{-}\geq \delta \}}\widetilde{Z}\star \mu \in {\mathcal{A}}^{+}_{\mathrm{loc}}(\mathbb{F})$ if and only if for any $\delta >0$, $kI_{\{Z_{-}\geq \delta \}}(Z_{-}+f_{m})\star \mu \in {\mathcal{A}}^{+}_{\mathrm{loc}}(\mathbb{F})$.

(e) Let $f$ be a $\widetilde{\mathcal{P}}(\mathbb{F})$-measurable functional. Then the following are equivalent:

(e.1) $\sqrt{(f-1)^{2}I_{[\![0,\tau [\![}\star \mu } \in {\mathcal{A}}^{+}_{\mathrm{loc}}(\mathbb{G})$.

(e.2) $\Phi _{\alpha }(f)I_{\{Z_{-}\geq \delta \}}{\widetilde{Z}}\star \mu \in {\mathcal{A}}^{+}_{\mathrm{loc}}(\mathbb{F})$ for all $\delta >0$.

(e.3) $\Phi _{\alpha }(f)I_{\{Z_{-}\geq \delta \}}(Z_{-}+f_{m})\star \mu \in {\mathcal{A}}^{+}_{\mathrm{loc}}(\mathbb{F})$ for all $\delta >0$.

(f) Let $\phi ^{\mathbb{F}}$ be an $\mathbb{F}$-predictable process with values in $[0,+\infty ]$ such that $P\otimes A$-a.e., $[\![0,\tau [\![\subset \{\phi ^{\mathbb{F}}<+\infty \}$. Then $P\otimes A$-a.e., $\{Z_{-}>0\}\subset \{\phi ^{\mathbb{F}}<+\infty \}$.

(g) Let $\phi ^{\mathbb{F}}$ be an $\mathbb{F}$-predictable process with values in $[0,+\infty ]$ such that $P\otimes A$-a.e., $[\![0,\tau [\![\subset \{\phi ^{\mathbb{F}}=0\}$. Then $P\otimes A$-a.e., $\{Z_{-}>0\}\subset \{\phi ^{\mathbb{F}}=0\}$.

Proof

(a) Put $W:=(f-1)^{2}\star \mu = W_{1} + W_{2}$ with $W_{1} := (f-1)^{2}I_{\{|f-1|\leq \alpha \}}\star \mu$, $W_{2}:= (f-1)^{2}I_{\{|f-1|> \alpha \}}\star \mu$ and $W_{2}' := |f-1|I_{\{|f-1|>\alpha \}}\star \mu$. Note that

$$\begin{aligned} \sqrt{W} =\sqrt{W_{1}+W_{2}}\leq \sqrt{W_{1}} + \sqrt{W_{2}} \leq \sqrt{W_{1}} + W_{2}'. \end{aligned}$$

Therefore $\sqrt{W_{1}}, W_{2}' \in {\mathcal{A}}^{+}_{\mathrm{loc}}$ imply that $\sqrt{W}$ is locally integrable. Conversely, if $\sqrt{W}\in {\mathcal{A}}^{+}_{\mathrm{loc}}$, then $\sqrt{W_{1}}$ and $\sqrt{W_{2}}$ are both locally integrable. Since $W_{1}$ is locally bounded and has finite variation, $W_{1}$ is locally integrable. In the following, we focus on the proof of the local integrability of $W_{2}'$. Denote

$$ \tau _{n} := \inf \{t\geq 0: V_{t} >n\}, \ \ V:=W_{2}. $$

It is easy to see that $(\tau _{n})$ increases to infinity and $V_{-}\leq n$ on the set $[\![0,\tau _{n} [\![$. On the set $\{\Delta V >0\}$, we have $\Delta V \geq \alpha ^{2}$. By using the elementary inequality

$$\sqrt{1 + {n}/{\alpha ^{2}}} - \sqrt{{n}/{\alpha ^{2}}} \leq \sqrt{1 + x} - \sqrt{x} \leq 1$$

when $0\leq x \leq {n}/{\alpha ^{2}}$, we have

$$\begin{aligned} \sqrt{V_{-} + \Delta V} - \sqrt{V_{-}} \geq \beta _{n} \sqrt{\Delta V}\qquad \text{on}\ [\![0,\tau _{n}[\![ \end{aligned}$$

and

$$\begin{aligned} (W_{2}')^{\tau _{n}} =& \left( \sum \sqrt{\Delta V}\right) ^{\tau _{n}} \leq \beta _{n}^{-1}\left( \sum \Delta \sqrt{ V}\right) ^{\tau _{n}} = \beta _{n}^{-1}\big(\sqrt{W_{2}}\big)^{\tau _{n}}\in {\mathcal{A}}^{+}_{\mathrm{loc}}(\mathbb{H}), \end{aligned}$$

where $\beta _{n}:= \sqrt{1 + {n}/{\alpha ^{2}}} - \sqrt{{n}/{\alpha ^{2}}}$. Therefore $W_{2}' \in (\mathcal{A}^{+}_{\mathrm{loc}}(\mathbb{H}))_{\mathrm{loc}}=\mathcal{A}^{+}_{\mathrm{loc}}(\mathbb{H})$.

(b) Due to [14, XX.75 b)], there exists a sequence of $\mathbb{F}$-stopping times $(\sigma _{n}^{\mathbb{F} })$ such that

$$\begin{aligned} \sigma _{n}^{\mathbb{G}}\wedge \tau = \sigma _{n}^{\mathbb{F}}\wedge \tau . \end{aligned}$$

(B.3)

By putting $\sigma _{n}:= \sup _{k\leq n} \sigma _{k}^{\mathbb{F}}$, we prove that

$$\begin{aligned} \sigma _{n}^{\mathbb{G}}\wedge \tau = \sigma _{n}\wedge \tau , \end{aligned}$$

(B.4)

or equivalently that $\{\sigma _{n}^{\mathbb{F}}\wedge \tau < \sigma _{n}\wedge \tau \}$ is negligible. Due to (B.3) and as $(\sigma _{n}^{\mathbb{G}})$ is nondecreasing, we get $\{\sigma _{n}^{\mathbb{F}} < \tau \}= \{\sigma _{n}^{\mathbb{G}} < \tau \} \subset \bigcap _{i=1}^{n} \{\sigma _{i}^{\mathbb{G}} =\sigma _{i}^{\mathbb{F}}\} \subset \{\sigma _{n}^{\mathbb{F}}= \sigma _{n}\}$. This implies that $\{\sigma _{n}^{\mathbb{F}}\wedge \tau < \sigma _{n}\wedge \tau \} = \{ \sigma _{n}^{\mathbb{F}} < \tau , \sigma _{n}^{\mathbb{F}}< \sigma _{n}\} =\emptyset$, and the proof of (B.4) is completed. Thus without loss of generality, we assume that the sequence $(\sigma _{n}^{\mathbb{F}})$ is nondecreasing. By taking limits in (B.3), we obtain $\tau = \sigma _{\infty }^{\mathbb{F}} \wedge \tau$ $P$-a.s. which is equivalent to $\sigma _{\infty }^{\mathbb{F}} \geq \tau$ $P$-a.s. Since ${R}$ is the smallest $\mathbb{F}$-stopping time dominating $\tau$ a.s., we obtain $\sigma _{\infty }^{\mathbb{F}} \geq {R} \geq \tau$ $P$-a.s. This completes the proof of (B.1).

On the set $\Sigma$, it is easy to show that

$$\begin{aligned} I_{{[\![0,\sigma ^{\mathbb{F}}_{n} [\![} } \longrightarrow I_{{[\![0,\sigma ^{\mathbb{F}}_{\infty }]\!]} } \qquad \text{as }n \to +\infty . \end{aligned}$$

Then thanks again to (B.3), by taking $\mathbb{F}$-predictable projections and letting $n$ go to $+\infty$, we obtain

$$\begin{aligned} Z_{-} = Z_{-}I_{{[\![0,\sigma _{\infty }^{\mathbb{F}} ]\!]} } \quad \text{on }\Sigma . \end{aligned}$$

Hence (B.2) follows immediately, and the proof of (b) is completed.

(c) Suppose that $V^{\tau }\in {\mathcal{A}^{+}_{\mathrm{loc}}}(\mathbb{G})$. Then there exists a sequence $(\sigma ^{\mathbb{G}}_{n})$ of $\mathbb{G}$-stopping times increasing to infinity such that $V^{\tau \wedge \sigma ^{\mathbb{G}}_{n}}\in {\mathcal{A}^{+}}(\mathbb{G})$. Consider a nondecreasing sequence $(\sigma _{n}^{\mathbb{F}})_{n\geq 1}$ of $\mathbb{F}$-stopping times satisfying (B.1) and (B.2) (its existence is guaranteed by (b)) and $\sigma _{\infty }^{\mathbb{F}}:=\sup _{n} \sigma _{n}^{\mathbb{F}}$. Therefore, for any fixed $\delta >0$,

$$\begin{aligned} W^{n}:=Z_{-} I_{\{Z_{-} \geq \delta \} }\hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}V^{\sigma _{n}^{\mathbb{F}}} \in \mathcal{A}_{\mathrm{loc}}^{+}(\mathbb{F}), \end{aligned}$$

or, equivalently, this process is càdlàg predictable with finite values. Thus, it is obvious that proving that the $\mathbb{F}$-predictable and nondecreasing process

$$\begin{aligned} W:=I_{\{Z_{-} \geq \delta \} }\hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}V\ \ \text{ is c\`{a}dl\`{a}g with finite values} \end{aligned}$$

(B.5)

is sufficient to prove its $\mathbb{F}$-local integrability. To prove (B.5), we consider the random time $\tau ^{\delta }$ defined by

$$ \tau ^{\delta }:= \sup \{t\geq 0 : Z_{t-} \geq \delta \}. $$

Then it is clear that $I_{[\![\tau ^{\delta },+\infty ]\!]}\hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}W\equiv 0$ and $P$-a.s. on $\{\tau ^{\delta }<+\infty \}$, we have

$$ \tau ^{\delta }\leq {R} \leq \sigma _{\infty }^{\mathbb{F}}\ \ \ \text{and}\ \ \ \ Z_{\tau ^{\delta }-} \geq \delta . $$

The proof of (B.5) will be achieved by considering the three sets $\{\sigma _{\infty }^{\mathbb{F}} = \infty \}$, $\Sigma \cap \{\sigma _{\infty }^{\mathbb{F}}<+\infty \}$ and $\Sigma ^{c}\cap \{\sigma _{\infty }^{\mathbb{F}}<+\infty \}$. It is obvious that (B.5) holds on $\{\sigma _{\infty }^{\mathbb{F}} = \infty \}$. From (B.2), we deduce that $\tau ^{\delta }< \sigma _{\infty }^{\mathbb{F}}$ $P$-a.s. on $\Sigma \cap \{\sigma _{\infty }^{\mathbb{F}}<+\infty \}$. Since $W$ is supported on $[\![0, \tau ^{\delta }[\![$, (B.5) follows immediately on the set $\Sigma \cap \{\sigma _{\infty }^{\mathbb{F}}<+\infty \}$. Finally, on the set

$$ \Sigma ^{c} \cap \{\sigma _{\infty }^{\mathbb{F}}< +\infty \} = \bigg(\bigcup _{n\geq 1} \{\sigma _{n}^{\mathbb{F}} = \sigma _{\infty }^{\mathbb{F}}\}\bigg) \cap \{\sigma _{\infty }^{\mathbb{F}}< +\infty \}, $$

the sequence $(\sigma _{n}^{\mathbb{F}})$ increases stationarily to $\sigma _{\infty }^{\mathbb{F}}$, and thus (B.5) holds on this set. This completes the proof of (B.5), and hence $I_{\{Z_{-} \geq \delta \} }Z_{-}\hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}V$ is locally integrable for any $\delta >0$.

Conversely, if $I_{\{Z_{-}\geq \delta \}}\hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}V \in {\mathcal{A}}^{+}_{\mathrm{loc}}(\mathbb{F})$, there exists a sequence of $\mathbb{F}$-stopping times $(\tau _{n})_{n\geq 1}$ that increases to infinity and $(I_{\{Z_{-}\geq \delta \}}\hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}V)^{\tau _{n}}\in {\mathcal{A}}^{+}(\mathbb{F})$. Then

$$\begin{aligned} E\left[ I_{\{Z_{-}\geq \delta \}} I_{[\![0,\tau [\![}\hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}V_{\tau _{n}}\right] =E\left[ I_{\{Z_{-}\geq \delta \}} Z_{-}\hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}V_{\tau _{n}}\right] < +\infty . \end{aligned}$$

This proves that $I_{\{Z_{-}\geq \delta \}} I_{[\![0,\tau [\![}\hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}V$ is $\mathbb{G}$-locally integrable for any $\delta >0$. Since $Z_{-}^{-1}I_{[\![0,\tau [\![}$ is $\mathbb{G}$-locally bounded, there exists a family of $\mathbb{G}$-stopping times $(\tau _{\delta })_{\delta >0}$ that increases to infinity when $\delta$ decreases to zero, and

$$ [\![0,\tau \wedge \tau _{\delta }[\![\subset \{Z_{-}\geq \delta \}.$$

This implies that the process $(I_{[\![0,\tau [\![}\hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}V)^{\tau _{\delta }}$ is $\mathbb{G}$-locally integrable, and (c) follows.

(d) Let us first observe the easy fact that $kI_{[\![0,\tau [\![}\star \mu \in {\mathcal{A}}^{+}_{\mathrm{loc}}(\mathbb{G})$ if and only if $Z_{-}kI_{[\![0,\tau [\![}\star \nu ^{\mathbb{G}}= kI_{[\![0,\tau [\![}(Z_{-}+f_{m})\star \nu \in {\mathcal{A}}^{+}_{\mathrm{loc}}(\mathbb{G})$. By combining this fact with (c) applied to $V:= k(Z_{-}+f_{m})I_{\{Z_{-}>0\}}\star \nu$, we deduce that $kI_{[\![0,\tau [\![}\star \mu \in {\mathcal{A}}^{+}_{\mathrm{loc}}(\mathbb{G})$ if and only if $k(Z_{-}+f_{m})I_{\{Z_{-}\geq \delta \}}\star \nu \in {\mathcal{A}}^{+}_{\mathrm{loc}}(\mathbb{F})$ for any $\delta >0$, which is equivalent to $k(Z_{-}+f_{m})I_{\{Z_{-}\geq \delta \}}\star \mu \in {\mathcal{A}}^{+}_{\mathrm{loc}}(\mathbb{F})$ for $\delta >0$. The equivalence between $k(Z_{-}+f_{m})I_{\{Z_{-}\geq \delta \}}\star \mu \in {\mathcal{A}}^{+}_{\mathrm{loc}}(\mathbb{F})$ and $k{\widetilde{Z}}I_{\{Z_{-}\geq \delta \}}\star \mu \in {\mathcal{A}}^{+}_{\mathrm{loc}}(\mathbb{F})$ follows from

$$ E [k(Z_{-}+f_{m})I_{\{Z_{-}\geq \delta \}}\star \mu _{\sigma }]=E [k\widetilde{Z}I_{\{Z_{-}\geq \delta \}}\star \mu _{\sigma }], $$

for any $\mathbb{F}$-stopping time $\sigma$. This ends the proof of (d).

(e) The proof of (e) follows from combining (a) and (d).

(f) Suppose that $[\![0,\tau [\![\subset \{\phi ^{\mathbb{F}}<+\infty \}$, or equivalently $I_{[\![0,\tau [\![}\leq I_{\{\phi ^{\mathbb{F}}<+\infty \}}$. By taking $\mathbb{F}$-predictable projections on both sides, we get $Z_{-}\leq I_{\{\phi ^{\mathbb{F}}<+\infty \}}$, and hence $\{Z_{-}>0\}\subset \{\phi ^{\mathbb{F}}<+\infty \}$. This proves (f).

(g) Suppose that $\phi ^{\mathbb{F}}I_{[\![0,\tau [\![}=0$ $P\otimes A$-a.e. Then by taking $\mathbb{F}$-predictable projections, we get $\phi ^{\mathbb{F}}Z_{-}=0$ $P\otimes A$-a.e. This proves (f). □