No-arbitrage under a class of honest times

Abstract

This paper quantifies the interplay between the no-arbitrage notion of no unbounded profit with bounded risk (NUPBR) and additional progressive information generated by a random time. This study complements the one of Aksamit et al. (Finance Stoch. 21:1103–1139, 2017) in which the authors have studied similar topics for the model stopped at the random time, while here we deal with the question of what happens after the random time. Given that the existing literature proves that NUPBR is always violated after honest times that avoid stopping times in a continuous filtration, we propose here a new class of honest times for which NUPBR can be preserved for some models. For these honest times, we obtain two principal results. The first result characterizes the pairs of initial market and honest time for which the resulting model preserves NUPBR, while the second result characterizes honest times that do not affect NUPBR of any quasi-left-continuous model (i.e., in which the asset price process has no predictable jump times). Furthermore, we construct explicitly local martingale deflators for a large class of models.

Appendices

Appendix A: Predictable characteristics and deflators

This section contains two lemmas, two propositions and a theorem. The two lemmas are new, while most of the rest of the results of this section are elaborated in [4], and we refer the reader to the appendix of that paper for details. Here, we consider a probability $Q$, a filtration ℍ and an $(\mathbb{H}, Q)$-quasi-left-continuous semimartingale $X$. To this process, we associate the random measure of its jumps, denoted by $\mu_{X}$, and its $(\mathbb{H},Q)$-compensator $\nu_{X}$. We suppose that $X$ has the canonical decomposition

$$X=X_{0}+X^{c}+h\star(\mu_{X}-\nu_{X})+(x-h)\star \mu_{X}+b\hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}A. $$

Here $h(x):=xI_{\{\vert x\vert\leq1\}}$, and $h\star(\mu_{X}-\nu _{X})$ represents the unique pure jump $(\mathbb{H},Q)$-local martingale with jumps taking the form $h(\Delta S)I_{\{\Delta S\not =0\}}$. We also suppose that we have $\nu_{X}(dt,dx)=F_{t}( dx)dA_{t}$, and we denote by $c$ the matrix such that $\langle X^{c}\rangle^{\mathbb{H}}=c\hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}A$. The quadruplet $(b, c, F, A)$ is the predictable characteristics of $X$ under $(\mathbb{H},Q)$.

Lemma 1

The $\mathbb{F}$-predictable characteristics of $S ^{(0)}$ defined in (4.12) are

$$\begin{aligned} b^{(0)} &:=b+\int h(x)\big(\psi(x)-1\big)F(dx),\quad \qquad c^{(0)}:=c, \\ F^{(0)}(dx) &:= \psi(x) F(dx), \qquad\qquad \qquad \qquad\qquad A^{(0)}:=I_{\{Z_{-}< 1\}}\hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}A, \end{aligned}$$

where $\psi$ is defined in (4.1).

Proof

Recall the canonical decomposition of $S$ under $(\mathbb{F}, P)$ as

$$ S=S_{0}+S^{c}+h\star(\mu-\nu)+b\hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}A+(x-h)\star\mu. $$

(A.1)

Remark that the random measure associated to the jumps of $S^{(0)}$ and its $\mathbb{F}$-compensator are given by

$$\mu^{(0)}(dt,dx)= I_{\{\widetilde{Z}_{t}< 1, Z_{t-}< 1\}} \mu(dt,dx),\quad\nu^{(0)} (dt,dx)= \psi(x) I_{\{ Z_{t-}< 1\}} \nu(dt,dx). $$

Then it is clear that

$$ I_{\{ Z_{-}< 1\}}h \star(\mu- \nu)=h\star(\mu ^{(0)} - \nu^{(0)})+ I_{\{ Z_{-}< 1=\widetilde{Z}\}} h\star\mu+(\psi-1)h I_{\{ Z_{-}< 1\}}\star\nu. $$

Therefore, by using the above equation and (A.1), we derive

$$\begin{aligned} S^{(0)} =& I_{\{ Z_{-}< 1\}} \hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}S-x I_{\{ Z_{-}< 1=\widetilde{Z}\}}\star\mu\\ =&I_{\{ Z_{-}< 1\}} \hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}S^{c}+h I_{\{ Z_{-}< 1 \}}\star(\mu-\nu) + (x-h-xI_{\{ \widetilde{Z}=1\}})I_{\{ Z_{-}< 1 \}} \star\mu\\ & + I_{\{ Z_{-}< 1 \}} b \hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}A\\ =& I_{\{ Z_{-}< 1\}} \hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}S^{c}+h \star(\mu^{(0)} - \nu^{(0)}) +(x-h)\star\mu^{(0)} \\ & +I_{\{ Z_{-}< 1 \}} \bigg( b +\int h(x)\big(\psi(x)-1\big)F(dx)\bigg)\hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}A. \end{aligned}$$

As a result, the lemma follows immediately from the above canonical decomposition of $(S^{(0)},\mathbb{F})$. □

Lemma 2

Let $S^{(1)}$ be given as in Proposition (4.3). Suppose that the process $Z^{(\psi)}:= {\mathcal{E}} ((\psi-1)I_{\{\psi>0\}} \star(\mu- \nu))$ is a uniformly integrable martingale and define $Q:= Z_{\infty}^{(\psi)} \cdot P$. Then the following assertions hold:

(a) The $(P,\mathbb{F})$-predictable characteristics of $S^{(1)}$ are

$$\begin{aligned} b^{(1)}&:=b-\int h(x)I_{\{\psi(x)=0\}}F(dx),\qquad\quad c^{(1)}:=c, \\ F^{(1)}(dx)&:=I_{\{\psi(t,x)>0\}}F(dx), \qquad \qquad \qquad \quad A^{(1)}:=I_{\{Z_{-}< 1\}}\hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}A. \end{aligned}$$

(b) The $(Q,\mathbb{F})$-predictable characteristics of $S^{(1)}$ are

$$\begin{aligned} b^{(1,Q)}&:=b-\int h(x)\big(\psi(x)-1\big)F(dx),\, \qquad c^{(1,Q)}:=c,\\ F^{(1,Q)}(dx)&:=\psi(x)F(dx), \qquad \qquad \qquad \qquad \quad A^{(1,Q)}:=I_{\{Z_{-}< 1\}}\hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}A. \end{aligned}$$

Proof

(a) The random measure of the jumps of $S^{(1)}$ and its $\mathbb {F}$-compensator are given by

$$\mu^{(1)}= I_{\{\psi>0, Z_{-}< 1\}}\star\mu,\quad \nu^{(1)} =I_{\{\psi>0, Z_{-}< 1\}}\star\nu. $$

Due to

$$hI_{\{ Z_{-}< 1\}} \star(\mu-\nu)= h\star(\mu ^{(1)} -\nu^{(1)} )+hI_{\{\psi=0, Z_{-}< 1\}}\star(\mu-\nu) $$

and (A.1), we derive

$$\begin{aligned} S^{(1)} =&I_{\{ Z_{-}< 1\}}\hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}S -xI_{\{\psi=0, Z_{-}< 1\}}\star\mu\\ =&I_{\{ Z_{-}< 1\}}\hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}S^{c} + I_{\{ Z_{-}< 1\}}h \star( \mu-\nu) +I_{\{ Z_{-}< 1\}}(x-h) \star\mu \\ & + I_{\{ Z_{-}< 1\}} b\hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}A -xI_{\{\psi=0, Z_{-}< 1\}} \star\mu\\ =& I_{\{ Z_{-}< 1\}}\hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}S^{c} + h \star( \mu^{(1)} -\nu^{(1)}) +hI_{\{\psi=0, Z_{-}< 1\}} \star(\mu-\nu)\\ & + I_{\{ Z_{-}< 1\}}(x-h) \star\mu+ I_{\{ Z_{-}< 1\}} b\hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}A-xI_{\{\psi =0, Z_{-}< 1\}} \star\mu. \end{aligned}$$

After simplifications, we obtain

$$\begin{aligned} S^{(1)} =&I_{\{ Z_{-}< 1\}}\hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}S^{c} + h \star( \mu^{(1)} -\nu^{(1)})+(x-h)\star\mu^{(1)} \\ & +I_{\{ Z_{-}< 1\}} \left( b-\int h(x) I_{\{\psi=0\}}F(dx)\right) \hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}A . \end{aligned}$$

(A.2)

From this canonical decomposition of $(S^{(1)},\mathbb{F})$, (a) follows immediately.

(b) Since $\mu^{(1)}=I_{\{\psi>0, Z_{-}<1\}} \star\mu$ (see proof of (a)), for any $\widetilde{\mathcal{ P}} (\mathbb {F})$-measurable $H$, we calculate the $(Q,\mathbb{F})$-compensator of $H\star\mu^{(1)}$ as

$$\begin{aligned} (H\star \mu^{(1)})^{p, Q,\mathbb{F}} =& \big( (1+\Delta N^{(\psi)})\hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}(H\star\mu^{(1)})\big)^{p, \mathbb{F}}\\ =& \Big( \big(1+(\psi -1)I_{\{\psi>0 \}}\big)H\star\mu^{(1)} \Big)^{p, \mathbb{F}}= \psi HI_{\{ Z_{-}< 1\}} \star\nu, \end{aligned}$$

where $N^{(\psi)}:= (\psi-1) I_{\{\psi>0\}}\star(\mu-\nu)$. We deduce that the $(Q,\mathbb{F})$-compensator of the random measure $\mu^{(1)}$ is given by

$$\nu^{(1,Q)}= \psi I_{\{ Z_{-}< 1\}} \star\nu= \psi \star \nu^{(1)}. $$

We now remark that

$$ h \star(\mu^{(1)}-\nu^{(1)})= h \star( \mu ^{(1)}- \nu^{(1,Q)})+ h(\psi-1)\star\nu^{(1)}. $$

By inserting this in (A.2), we derive

$$\begin{aligned} S^{(0)} =&I_{\{ Z_{-}< 1\}}\hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}S^{c}+h\star(\mu^{(1)}-\nu ^{(1,Q)})+(x-h)\star\mu^{(1)}\\ & + \left( b-\!\!\int h(x)I_{\{\psi (x)=0 \}}F(dx)+\!\!\int h(x) \big(\psi(x)-1\big)I_{\{\psi>0 \}} F(dx)\right) I_{\{ Z_{-}< 1\}} \hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}A\\ =&I_{\{ Z_{-}< 1\}}\hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}S^{c}+h\star(\mu^{(1)}-\nu ^{(1,Q)}) +(x-h)\star\mu^{(1)}\\ & + \left( b +\int h(x) \big(\psi(x)-1\big) F(dx)\right) I_{\{ Z_{-}< 1\}} \hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}A. \end{aligned}$$

This is the canonical decomposition of $S^{(1)}$ under $(Q,\mathbb {F})$, since $S^{c}$ coincides with $S^{c,Q}$, the continuous local martingale part of $S$ under $Q$. So we have (b). □

Theorem 3

Let $(X,Q,\mathbb{H})$ be a quasi-left-continuous model and $(b , c, F , A)$ its predictable characteristics under $(\mathbb{H}, Q)$. Then $X$ satisfies NUPBR $(\mathbb{H}, Q)$ if and only if there exists a pair $(\beta, f)$ consisting of an ℍ-predictable process $\beta$ and a $\widetilde{\mathcal{P}}(\mathbb{H})$-measurable functional $f$ such that $f>0$,

$$\begin{aligned} & \beta^{\mathrm{tr}}c\beta \hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}A+ \sqrt{(f-1)^{2}\star\mu_{X}}\in \mathcal{A}^{+}_{\mathrm{loc}}(\mathbb{H},Q), \end{aligned}$$

(A.3)

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

(A.4)

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

(A.5)

See [4, Theorem A.1] for the proof.

Proposition 4

Let $X$ be an ℍ-semimartingale. Then the following assertions are equivalent:

(a) There exists a sequence $(T_{n})_{n\geq1}$ of ℍ-stopping times that increases to $+\infty$ and such that for each $n\geq1$, there exists a probability $Q_{n}$ on $(\Omega, \mathcal{H}_{T_{n}})$ such that $Q_{n}$ is equivalent to $P$ and $X^{T_{n}}$ satisfies NUPBR $(\mathbb{H})$ under $Q_{n}$.

(b) $X$ satisfies NUPBR $(\mathbb{H})$.

(c) There exists an ℍ-predictable process $\phi$ such that $0<\phi\leq1$ and $(\phi \hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}X)$ satisfies NUPBR $(\mathbb{H})$.

The proof of this proposition can be found in [4, Proposition 2.5].

Proposition 5

Suppose that $\tau$ is an honest time and let $H^{\mathbb{G}}$ be a $\widetilde{\mathcal{P}}(\mathbb {G})$-measurable functional. Then the following assertions hold:

(a) There exist two $\widetilde{\mathcal{P}}(\mathbb{F})$-measurable functionals $H^{\mathbb{F}}$ and $K^{\mathbb {F}}$ such that

$$H^{\mathbb{G}}(\omega ,t,x)=H^{\mathbb{F}}(\omega ,t,x)I_{〚0,\tau 〛}+K^{\mathbb{F}}(\omega ,t,x)I_{〛\tau ,+\mathrm{\infty }〚}.$$

(A.6)

(b) If furthermore $H^{\mathbb{G}}>0$ (respectively $H^{\mathbb {G}}\leq1$), then we can choose $K^{\mathbb{F}}>0$ (respectively $K^{\mathbb{F}}\leq1$) in (A.6).

Proof

The proofs of (a) and (b) follow from mimicking Jeulin’s proof [20, Proposition 5.3] and are omitted here. □

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

This subsection connects the $\mathbb{G}$-localisation and the $\mathbb {F}$-localisation for the part after $\tau$. This completes the analysis of [4] regarding the issue of local integrability under $\mathbb{F}$ and $\mathbb{G}$, where the part up to $\tau$ is fully discussed. There is a major difference between the current results and those of [4], which lies in the fact that the $\mathbb{G}$-localisation does not imply the $\mathbb{F}$-localisation, in general, for the case up to $\tau$. This is possible when $Z$ vanishes (or, equivalently, $P[R<+\infty]>0$), as the increasing sequence of $\mathbb{F}$-stopping times $(\sigma^{\mathbb{F}}_{n})$, associated to a sequence of $\mathbb{G}$-stopping times that increases to infinity, might not increase to infinity on $\{R<+\infty\}$. However, for the part after $\tau$, as long as $\tau$ is finite, we pass from $\mathbb {G}$-localisation to $\mathbb{F}$-localisation without any loss of information.

Proposition 1

The following assertions hold:

(a) If $\tau$ is a finite honest time and $(\sigma _{n}^{\mathbb{G}})_{n\geq1}$ is a sequence of finite $\mathbb{G}$-stopping times that increases to infinity, then there exists a sequence $(\sigma_{n}^{\mathbb{F}})_{n\geq1}$ of finite $\mathbb{F}$-stopping times that increases to infinity as well and

$$ \max(\sigma_{n}^{\mathbb{G}},\tau)=\max(\sigma_{n}^{\mathbb {F}},\tau)\quad P\textit{-a.s.} $$

(B.1)

(b) If $\tau\in\mathcal{H}$, then there exists a sequence $(\sigma_{n})_{n\geq1}$ of $\mathbb{F}$-stopping times that increases to infinity and

$$\{Z_{-}<1\}\cap 〚0,{\sigma }_n〛\subseteq \{1-Z_{-}\ge \frac{1}{n}\},\mathrm{\forall }n\ge 1.$$

(B.2)

Equivalently, $(1-Z_{-})^{-1}I_{\{Z_{-}<1\}}$ is $\mathbb{F}$-locally bounded when $\tau\in\mathcal{H}$.

Proof

(a) This proof boils down to proving the following result:

$$\begin{aligned} &\mbox{For any $\mathbb{G}$-stopping time $\sigma^{\mathbb{G}}$, there exists an $\mathbb{F}$-stopping time $\sigma^{\mathbb{F}}$ such that} \\ &\sigma^{\mathbb{G}}\vee\tau=\mbox{$\sigma^{\mathbb{F}}\vee\tau$ $P$-a.s.} \end{aligned}$$

(B.3)

Indeed, if (B.3) holds, then there exist $\mathbb {F}$-stopping times $(\sigma_{n})_{n\geq1}$ such that for any $n\geq1$, the pair $(\sigma_{n}^{\mathbb{G}},\sigma_{n})$ satisfies (B.1). Since $(\sigma_{n}^{\mathbb{G}})$ increases with $n$, by putting $\sigma_{n}^{\mathbb{F}}:=\sup_{1\leq k\leq n}\sigma _{k}$, we can easily prove that the pair $(\sigma_{n}^{\mathbb{G}},\sigma_{n}^{\mathbb{F}})$ satisfies (B.1) as well. Then (a) follows immediately by taking the limit in (B.1) and making use of $\tau<+\infty$ $P$-a.s. which implies that $\sup_{n\geq1}\sigma_{n}=\lim_{n\rightarrow +\infty}\sigma_{n}^{\mathbb{F}}=+\infty$ $P$-a.s.

It remains to prove (B.3). By applying Proposition B.2 (c) (given at the end of this proof and due to Barlow [9]) to the process $Y^{\mathbb{G}}:=I_{〚{\sigma }^{\mathbb{G}}\vee \tau ,+\mathrm{\infty }〚}$, we obtain the existence of an $\mathbb{F}$-progressively measurable process $K^{\mathbb{F}}$ such that

$$Y^{\mathbb{G}}=K^{\mathbb{F}}{I}_{[[\tau ,+\mathrm{\infty }〚}.$$

Then put $\sigma:=\inf\{t\geq0: K^{\mathbb{F}}_{t}=1\}$. This is an $\mathbb{F}$-stopping time, and due to $〚{\sigma }^{\mathbb{G}}\vee \tau ,+\mathrm{\infty }〚\subseteq \{K^{\mathbb{F}}=1\}$, we get

$$ \mbox{$\sigma\leq\tau\vee\sigma^{\mathbb{G}} \quad P$-a.s.} $$

(B.4)

By applying Proposition B.2, we deduce the existence of two double sequences of $\mathbb{F}$-stopping times $(\alpha_{nm})_{n,m\geq1}$ and $(\beta_{nm})_{n,m\geq1}$ satisfying the four assertions of Proposition B.2. As a result, we get

$$ \{\tau< \sigma^{\mathbb{G}}\}\subseteq\bigcup_{n,m\geq1}\{\tau\leq \alpha_{nm}\leq\sigma^{\mathbb{G}}< \beta_{nm}\}. $$

(B.5)

Thanks to Proposition B.2 (d), for $n,m\geq1$, $K^{\mathbb {F}}$ is càdlàg on $\{\tau\leq\alpha_{nm}\leq\sigma^{\mathbb {G}}<\beta_{nm}\}$ and $K^{\mathbb{F}}\equiv0$ on $〚\tau ,{\alpha }_{nm}〚$. This implies that

$$\{\tau\leq\alpha_{nm}\leq\sigma^{\mathbb {G}}< \beta_{nm}\}\subseteq\{\tau\leq\alpha_{nm}\leq\sigma\leq \sigma^{\mathbb{G}}< \beta_{nm}\}. $$

By combining this with $〚\sigma ,\sigma +ϵ〚\cap \{K^{\mathbb{F}}=1\}\ne \mathrm{\varnothing }$ for all $\epsilon>0$, we deduce that

$$ \{\tau\leq\alpha_{nm}\leq\sigma^{\mathbb{G}}< \beta_{nm}\} \subseteq\{\tau\leq\alpha_{nm}\leq\sigma=\sigma^{\mathbb {G}}< \beta_{nm}\}\subseteq\{\sigma=\sigma^{\mathbb{G}}\}. $$

A combination of this with (B.5) leads to

$$ \{\tau< \sigma^{\mathbb{G}}\}\subseteq\{\sigma=\sigma^{\mathbb{G}}\}. $$

Therefore, thanks to this latter inclusion and (B.4), we derive

$$\begin{aligned} \tau\vee\sigma^{\mathbb{G}} =&(\tau\vee\sigma^{\mathbb{G}})I_{\{\sigma^{\mathbb{G}}\leq\tau\} }+(\tau\vee\sigma^{\mathbb{G}})I_{\{\tau< \sigma^{\mathbb{G}}\}} = \tau I_{\{\sigma^{\mathbb{G}}\leq\tau\}}+(\tau\vee\sigma)I_{\{ \tau< \sigma^{\mathbb{G}}\}}\\ =&(\tau\vee\sigma)I_{\{\sigma^{\mathbb{G}}\leq\tau\}}+(\tau\vee \sigma)I_{\{\tau< \sigma^{\mathbb{G}}\}}=\tau\vee\sigma. \end{aligned}$$

This proves (B.3), and the proof of (a) is complete.

(b) Since $\tau\in\mathcal{H}$, the process ${(1-Z_{-})}^{-1}{I}_{〛\tau ,+\mathrm{\infty }〚}$ is $\mathbb{G}$-locally bounded due to Lemma 2.6 (b). Thus, on the one hand, there exists a sequence $(\sigma_{n}^{\mathbb {G}})_{n\geq1}$ of $\mathbb{G}$-stopping times that increases to infinity and

$$〛\tau ,+\mathrm{\infty }〚\cap 〚0,{\sigma }_n^{\mathbb{G}}]]\subseteq \{1-Z_{-}\ge 1/n\}.$$

(B.6)

On the other hand, thanks to (a), there exists a sequence $(\sigma _{n})_{n\geq1}$ of $\mathbb{F}$-stopping times that increases to infinity and satisfies (B.1). By inserting this in (B.6), we get

$$〛\tau ,+\mathrm{\infty }〚\cap 〚0,{\sigma }_n〛\subseteq \{1-Z_{-}\ge 1/n\}.$$

By taking the $\mathbb{F}$-predictable projection on both sides, we get

$$0\le (1-Z_{-})I_{〚0,{\sigma }_n〛}\le I_{\{1-Z_{-}\ge 1/n\}}.$$

This implies (B.2) and achieves the proof of (b). □

Proposition 2

Suppose that $\tau$ is an honest time. Then the following hold:

(a) There exist two double sequences $(\alpha_{n,m})_{n,m\geq 1}$ and $(\beta_{n,m})_{n,m\geq1}$ of $\mathbb{F}$-stopping times such that $\alpha_{n,m}\leq\beta_{n,m}$ P-a.s. for all $n,m\geq1$ and

$$〛\tau ,+\mathrm{\infty }〚\subseteq \{Z<1\}\subseteq \bigcup _{n,m\ge 1}〚{\alpha }_{n,m},{\beta }_{n,m}〚.$$

(b) For any $n,m\geq1$, $\{\tau>\alpha_{nm}\}\subseteq\{ \tau\geq\beta_{nm}\}$ $P$-a.s.

(c) For any $\mathbb{G}$-optional process $Y^{\mathbb{G}}$, there exists an $\mathbb{F}$-progressively measurable process $K^{\mathbb{F}}$ such that

$$Y^{\mathbb{G}}{I}_{〚\tau ,+\mathrm{\infty }〚}=K^{\mathbb{F}}{I}_{〚\tau ,+\mathrm{\infty }〚}.$$

(B.7)

(d) For any $\mathbb{G}$-optional càdlàg process $Y^{\mathbb{G}}$ which is constant on $〚{\beta }_{n,m},+\mathrm{\infty }〚$ and $Y^{\mathbb{G}}=0$ on $〚0,{\alpha }_{n,m}〚$, there exists an $\mathbb{F}$-progressively measurable process $K^{\mathbb{F}}$ that is càdlàg and satisfies (B.7).

Proof

For the proof, we refer the reader to [9]. In fact, (a) and (b) are exactly Lemma 4.1 (iv) and Lemma 4.1 (ii) in [9], respectively (c) follows from Proposition 4.3, and (d) is a combination of Proposition 4.3 and Lemma 4.4 (ii), of the same paper. □

The next result addresses the $\mathbb{G}$-local integrability involving the random measures that is vital for the proof of Theorem 2.12.

Proposition 3

Suppose that $\tau\in{\mathcal{H}}$. Let $\Phi_{\alpha}(\cdot)$ (for $\alpha>0$) be defined in (4.4). Then the following properties hold:

(a) Let $f$ be a real-valued and $\widetilde{\mathcal{P}}(\mathbb {H})$-measurable functional. Then $\sqrt{(f-1)^{2}\star\mu}$ belongs to $\mathcal{A}^{+}_{\mathrm {loc}}(\mathbb{H})$ if and only if $\Phi_{\alpha}(f)\star\mu\in\mathcal{A}^{+}_{\mathrm{loc}}(\mathbb{H})$.

(b) Let $f$ be a real-valued and $\widetilde{\mathcal{P}}(\mathbb {H})$-measurable functional. Then we have that $\sqrt{{(f-1)}^2{I}_{〛\tau ,+\mathrm{\infty }〚}\star \mu }$ is in $\mathcal{A}^{+}_{\mathrm{loc}}(\mathbb{G})$ if and only if $\Phi_{\alpha}(f)(1-Z_{-}-f_{m}) I_{\{ Z_{-}<1\}}\star{\mu}$ is in $\mathcal{A}^{+}_{\mathrm{loc}}(\mathbb{F})$.

(c) Let $\phi$ be a nonnegative and $\mathbb{F}$-predictable process. Then $(P\otimes A)$-a.e., we have $〛\tau ,+\mathrm{\infty }〚\subseteq \{\varphi <+\mathrm{\infty }\}$ if and only if $\{Z_{-}<1\}\subseteq\{\phi<+\infty\}$.

(d) Let $\theta$ be an $\mathbb{F}$-predictable process. Then $(P\otimes A)$-a.e., $〛\tau ,+\mathrm{\infty }〚\subseteq \{\theta =0\}$ if and only if $\{Z_{-}<1\}\subseteq\{\theta=0\}$.

Proof

(a) This is borrowed from [4] (see Proposition C.3 (a) there).

(b) Thanks to (a), we deduce that $\sqrt{{(f-1)}^2{I}_{〛\tau ,+\mathrm{\infty }〚}\star \mu }\in {\mathcal{A}}_{\mathrm{loc}}^{+}(\mathbb{G})$ if and only if $\Phi_{\alpha }(f)\star\mu^{\mathbb{G}}\in\mathcal{A}^{+}_{\mathrm{loc}}(\mathbb{G})$ if and only if

$${\mathrm{\Phi }}_{\alpha }(f)(1-\frac{f_m}{1-Z_{-}})I_{〛\tau ,+\mathrm{\infty }〚}\star \nu ={\mathrm{\Phi }}_{\alpha }(f)\star {\nu }^{\mathbb{G}}\in {\mathcal{A}}_{\mathrm{loc}}^{+}(\mathbb{G}).$$

Then by directly applying Lemma 3.2 (d) to the pair

$$(\varphi, V):=\big( (1-f_{m}(1-Z_{-})^{-1})I_{\{ Z_{-}< 1\}},\ \Phi_{\alpha}(f)\star\nu\big), $$

the proof of (b) follows immediately.

(c) Suppose that $(P\otimes A)$-a.e., we have $〛\tau ,+\mathrm{\infty }〚\subseteq \{\varphi <+\mathrm{\infty }\}$. This is equivalent to $I_{〛\tau ,+\mathrm{\infty }〚}\le I_{\{\varphi <+\mathrm{\infty }\}}$ $(P\otimes A)$-a.e. Then by taking the $\mathbb{F}$-predictable projection on both sides, we get $1-Z_{-}\leq I_{\{\phi<+\infty\}}$ $(P\otimes A)$-a.e. This obviously proves that $〛\tau ,+\mathrm{\infty }〚\subseteq \{\varphi <+\mathrm{\infty }\}$ implies $\{Z_{-}<1\}\subseteq\{\phi<+\infty\}$. The converse follows from $〛\tau ,+\mathrm{\infty }〚\subseteq \{Z_{-}<1\}$.

(d) The proof of this mimics the proof of (c) and is omitted. □

Appendix C: Proofs for Lemmas 3.1 and 3.2

Proof of Lemma 3.1

(a) From Lemma 2.6, the process

$$I_{〛\tau ,+\mathrm{\infty }〚}\cdot V-I_{〛\tau ,+\mathrm{\infty }〚}\cdot V^{p,\mathbb{F}}+I_{〛\tau ,+\mathrm{\infty }〚}{(1-Z_{-})}^{-1}\cdot {〈V,m〉}^{\mathbb{F}}$$

is a $\mathbb{G}$-local martingale, hence

$$\begin{array}{rcl}{(I_{〛\tau ,+\mathrm{\infty }〚}\cdot V)}^{p,\mathbb{G}}& =& I_{]]\tau ,+\mathrm{\infty }〚}\cdot V^{p,\mathbb{F}}-I_{〛\tau ,+\mathrm{\infty }〚}{(1-Z_{-})}^{-1}\cdot {〈V,m〉}^{\mathbb{F}}\\ & =& I_{〛\tau ,+\mathrm{\infty }〚}\cdot V^{p,\mathbb{F}}-I_{〛\tau ,+\mathrm{\infty }〚}{(1-Z_{-})}^{-1}\cdot {(\mathrm{\Delta }m\cdot V)}^{p,\mathbb{F}}\\ & =& I_{〛\tau ,+\mathrm{\infty }〚}{(1-Z_{-})}^{-1}\cdot {((1-Z_{-}-\mathrm{\Delta }m)\cdot V)}^{p,\mathbb{F}},\end{array}$$

where the second equality follows from Yoeurp’s lemma. This proves (3.1). The equality (3.2) follows immediately from (3.1) by taking the jumps on both sides and using $\Delta(K^{p,\mathbb {H}})=\ ^{p,\mathbb{H}}(\Delta K)$ when both terms exist.

(b) By applying (3.2) for $V_{\epsilon,\delta}$ given by

$$ V_{\epsilon,\delta}:=\sum(\Delta M) (1-\widetilde{Z})^{-1}I_{\{\vert \Delta M\vert\geq\epsilon,\ 1-\widetilde{Z}\geq\delta\}} \in\mathcal{A}_{\mathrm{loc}}(\mathbb{F}), $$

we get that on $〛\tau ,+\mathrm{\infty }〚$,

$$ ^{p,{\mathbb{G}}}\big(\Delta M (1-\widetilde{Z})^{-1}I_{\{\vert\Delta M\vert\geq\epsilon,\ { 1-\widetilde{Z}\geq\delta}\}}\big)= {(1-Z_{-})^{-1}}\ ^{p,{\mathbb{F}}}(I_{\{\vert\Delta M\vert\geq \epsilon,\ 1-\widetilde{Z}\geq\delta\}}{\Delta M}). $$

Then the first equality in (3.3) follows by letting $\epsilon$ and $\delta$ go to zero, and we get on $〛\tau ,+\mathrm{\infty }〚$ that

$$ ^{p,{\mathbb{G}}}\left( \frac{{\Delta M}}{{ 1-\widetilde{Z}}}\right) = {(1-Z_{-})^{-1}}\ ^{p,{\mathbb{F}}}({\Delta M}\,I_{\{ 1-\widetilde {Z}>0\}})= {(1-Z_{-})^{-1}}\ ^{p,{\mathbb{F}}}({\Delta M}\,I_{\{\widetilde{Z}< 1\}}). $$

To prove the second equality in (3.3), we calculate that on $〛\tau ,+\mathrm{\infty }〚$,

$$\begin{aligned} ^{p,{\mathbb{G}}}\left( \frac{1}{{ 1-\widetilde{Z}}}\right) =&(1-Z_{-})^{-1}+(1-Z_{-})^{-1}\ ^{p,{\mathbb{G}}}\left( \frac{{\Delta m}}{{1-\widetilde{Z}}}\right) \\ =&(1-Z_{-})^{-1}+(1-Z_{-})^{-2}\ ^{p,{\mathbb{F}}}({{\Delta m}}I_{\{ 1-\widetilde{Z}>0\}})\\ =&(1-Z_{-})^{-1}\ ^{p,{\mathbb{F}}}(I_{\{\widetilde{Z}< 1\}}). \end{aligned}$$

The second equality is due to (3.2), and the third follows from combining ${}^{p,{\mathbb{F}}}(\Delta m)=0$ and $\Delta m=\widetilde{Z}-Z_{-}$. This proves (b).

(c) The proof of (3.4) follows immediately from (b) and the fact that the thin process ${}^{p,{\mathbb{F}}}(I_{\{ \widetilde{Z}<1\}}{\Delta M}\,)$ may take nonzero values on countably many predictable stopping times only, on which $\Delta M$ already vanishes. This completes the proof. □

Proof of Lemma 3.2

For proving (a) and (b), let $V$ be an $\mathbb{F}$-adapted process with finite variation and denote by Var$(V)$ its variation. Then we obtain

$$\mathrm{Var}(U)={(1-\tilde{Z})}^{-1}{I}_{〛\tau ,+\mathrm{\infty }〚}\cdot \mathrm{Var}(V).$$

Therefore, since $1-{\widetilde{Z}}_{t}=P[\tau< t|\mathcal{F}_{t}]\leq1-Z_{t}$, for any bounded nonnegative and $\mathbb{F}$-optional process $\phi$ such that $\phi \hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}{\mathrm{Var}}(V)\in\mathcal{A}^{+}(\mathbb{F})$, we obtain

$$\begin{aligned} E\big[\big(\phi \hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}{\mathrm{Var}}(U)\big)_{\infty}\big] =&\displaystyle E\left[ \int_{0}^{\infty} \frac{{ \phi_{t} I_{\{ t>\tau\}}}}{{1- Z_{t}}}d {\mathrm{Var}}(V)_{t}\right] \\ = &E\left[ \displaystyle\int_{0}^{\infty} \frac{{\phi_{t} P[\tau < t|\mathcal{F}_{t}]}}{{1- Z_{t}}}I_{\{ Z_{t}< 1\}}d {\mathrm{Var}}(V)_{t}\right ] \\ \leq& E\big[\big(\phi \hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}{\mathrm{Var}}(V)\big)_{\infty}\big]. \end{aligned}$$

(C.1)

As a result, by taking $\varphi =I_{〚0,\sigma 〚}$ in (C.1) for an $\mathbb{F}$-stopping time $\sigma$ such that Var$(V)^{\sigma-}\in\mathcal{A}^{+}(\mathbb{F})$, we get $E[{\mathrm {Var}}(U)_{\sigma-}]\leq E[{\mathrm{Var}}(V)_{\sigma-}]$. This proves that the process $U$ has finite variation and hence is well defined as well. It is clear that $U$ is $\mathbb{G}$-adapted, while its being càdlàg follows immediately from (C.1). This proves (a).

To prove (b), we assume that $V\in\mathcal{A}_{\mathrm{loc}}(\mathbb{F})$ and consider a sequence $(\vartheta_{n})_{n\geq1}$ of $\mathbb {F}$-stopping times that increases to $+\infty$ such that ${\mathrm {Var}}(V)^{\vartheta_{n}}$ belongs to $\mathcal{A}^{+}(\mathbb{F})$. Then by choosing $\varphi =I_{〚0,{\vartheta }_n〛}$ in (C.1), we conclude that $U$ belongs to $\mathcal{A}_{\mathrm {loc}}(\mathbb{G})$ whenever $V$ belongs to $\mathcal{A}_{\mathrm {loc}}(\mathbb{F})$. For the case when $V\in\mathcal{A}(\mathbb{G})$, it is enough to take $\phi=1$ in (C.1) and conclude that $U\in\mathcal{A}(\mathbb{G})$. To prove (3.5), for any $n\geq1$, we put

$$U_n:={(1-\tilde{Z})}^{-1}{I}_{〛\tau ,+\mathrm{\infty }〚}{I}_{\{\tilde{Z}\le 1-\frac{1}{n}\}}\cdot V.$$

Then thanks to (3.1), we derive

$$U^{p,\mathbb{G}}=\underset{n\to +\mathrm{\infty }}{lim}{\left(U_n\right)}^{p,\mathbb{G}}=\underset{n\to +\mathrm{\infty }}{lim}{(1-Z_{-})}^{-1}{I}_{〛\tau ,+\mathrm{\infty }〚}\cdot {(I_{\{\tilde{Z}\le 1-\frac{1}{n}\}}\cdot V)}^{p,\mathbb{F}}.$$

This clearly implies (3.5).

(c) It is enough to prove the assertion for the case when $V$ is nondecreasing. Then $(1-\widetilde{Z})\hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}V\in\mathcal{A}^{+}_{\mathrm{loc}}(\mathbb{F})$ implies $I_{〛\tau ,+\mathrm{\infty }〚}\cdot V\in {\mathcal{A}}_{\mathrm{loc}}^{+}(\mathbb{G})$. For the converse, suppose $I_{〛\tau ,+\mathrm{\infty }〚}\cdot V\in {\mathcal{A}}_{\mathrm{loc}}^{+}(\mathbb{G})$. Then there exists a sequence $(\sigma_{n}^{\mathbb{G}})$ of $\mathbb {G}$-stopping times that increases to infinity and such that ${(I_{〛\tau ,+\mathrm{\infty }〚}\cdot V)}^{{\sigma }_n^{\mathbb{G}}}\in {\mathcal{A}}^{+}(\mathbb{G})$. Thanks to Proposition B.1 (a), we obtain a sequence $(\sigma^{\mathbb{F}}_{n})_{n\geq1}$ of $\mathbb{F}$-stopping times that increases to infinity and $\sigma^{\mathbb{G}}_{n}\vee\tau=\tau \vee\sigma_{n}^{\mathbb{F}}$. Therefore, we get ${(I_{〛\tau ,+\mathrm{\infty }〚}\cdot V)}^{{\sigma }_n^{\mathbb{G}}}\equiv {(I_{〛\tau ,+\mathrm{\infty }〚}\cdot V)}^{{\sigma }_n^{\mathbb{F}}}$ and hence

$$E[(1-\tilde{Z})\cdot V_{{\sigma }_n^{\mathbb{F}}}]=E[I_{〛\tau ,+\mathrm{\infty }〚}\cdot V_{{\sigma }_n^{\mathbb{F}}}]=E[I_{〛\tau ,+\mathrm{\infty }〚}\cdot V_{{\sigma }_n^{\mathbb{G}}}]<+\mathrm{\infty }.$$

(C.2)

This proves that the process $(1-\widetilde{Z})\hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}V$ belongs to $\mathcal{A}^{+}_{\mathrm{loc}}(\mathbb{F})$, and the proof of (c) is achieved.

(d) The proof of this follows all the steps of the proof of (c), except that (C.2) takes the form

$$E[(1-Z_{-})\phi \cdot V_{{\sigma }_n^{\mathbb{F}}}]=E[I_{〛\tau ,+\mathrm{\infty }〚}\phi \cdot V_{{\sigma }_n^{\mathbb{F}}}]=E[I_{〛\tau ,+\mathrm{\infty }〚}\phi \cdot V_{{\sigma }_n^{\mathbb{G}}}]<+\mathrm{\infty }$$

instead, due to the predictability of $V$. This proves that $I_{〛\tau ,+\mathrm{\infty }〚}\phi \cdot V\in {\mathcal{A}}_{\mathrm{loc}}^{+}(\mathbb{G})$ if and only if $(1-Z_{-})\varphi \hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}V\in\mathcal{A}^{+}_{\mathrm {loc}}(\mathbb{F})$, while the equivalence $(1-Z_{-})\varphi \hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}V\in\mathcal{A}^{+}_{\mathrm{loc}}(\mathbb{F})$ if and only if $I_{\{Z_{-}<1\}}\varphi \hspace{0.5pt}{\mathbf{\cdot}}\hspace{0.5pt}V\in\mathcal{A}^{+}_{\mathrm{loc}}(\mathbb{F})$ follows from the $\mathbb{F}$-local boundedness of $(1-Z_{-})^{p}I_{\{ Z_{-}<1\}}$ for any real number $p$ (see Proposition B.1 (b) for details). This ends the proof. □