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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?

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Acknowledgements

This research is fully supported financially by the Natural Sciences and Engineering Research Council of Canada, through Grant G121210818.

The authors would like to thank Safa Alsheyab, Jun Deng, Youri Kabanov and Michèle Vanmaele for several comments, fruitful discussions on the topic and/or for providing important and useful references.

The authors are very grateful to two anonymous referees and an anonymous AE for their several suggestions and comments that helped improve the paper. Any possible error is our sole responsibility.

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Correspondence to Tahir Choulli.

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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 5.125.14

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

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. □

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. □

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. □

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. □

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. □

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Choulli, T., Yansori, S. Log-optimal and numéraire portfolios for market models stopped at a random time. Finance Stoch (2022). https://doi.org/10.1007/s00780-022-00477-8

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Keywords

  • Random horizon
  • Log-optimal portfolio
  • Numéraire portfolio
  • Deflators
  • Informational risks
  • Utility
  • Progressive enlargement of filtration
  • Asymmetric information
  • Semimartingales and predictable characteristics

Mathematics Subject Classification (2020)

  • 60G48
  • 60H05
  • 91B16
  • 91G10
  • 91G40
  • 94A17

JEL Classification

  • D82
  • D89
  • G11
  • G14