3.1. (Classic) NUPBR and its drawbacks

The classic definition of the AOA concept NUPBR is due to Delbaen and Schachermayer [Reference Delbaen and Schachermayer6, Reference Delbaen and Schachermayer8] under the assumption $S^1\equiv1$ . It stipulates that the set

(3.1) \begin{equation}\Big\{ H\bullet S^{2:d}_\infty \,:\, \hbox{$H\in L(S^{2:d})$, $H\bullet S^{2:d} \ge -1$, $\displaystyle H\bullet S^{2:d}_\infty \,:\!=\, \lim_{t\to\infty} H\bullet S^{2:d}_t$ exists a.s.} \Big\}\end{equation}

is bounded in $L^0$ . Because $S^1\equiv1$ , we have $\{ H\bullet S^{2:d} \,:\, H\in L(S^{2:d}) \}=\{ \psi\bullet S \,:\, \psi\in L(S) \}$ , and by Theorem 2.1 with $\eta\equiv e^1$ , this set equals $\{ V(\varphi) \,:\, \varphi\in\Theta^{\mathrm{sf}}(S), V_0(\varphi) = 0 \}$ . So we have

(3.2) \begin{equation}\{ H\bullet S^{2:d} \,:\, H\in L(S^{2:d}), H\bullet S^{2:d} \ge -1 \}=\mathcal{G}^1(S) - 1,\end{equation}

and thus classic NUPBR precisely captures the idea that in a model with $S^1 \equiv 1$ , the final values of self-financing strategies with initial wealth 1 and non-negative wealth throughout should remain in a bounded set. Because the definition of classic NUPBR involves only $S^{2:d}$ , one sometimes forgets to explicitly mention $S^1\equiv 1$ and talks only about classic NUPBR for a process Y (which then replaces $S^{2:d}$ in (3.1)). The next definition extends this, as done in [Reference Bálint and Schweizer5, Subsection 3.1], to the non-classic modelling framework considered here.

Definition 3.1. A semimartingale $S \in \mathcal{S}^d$ or the market S satisfies NUPBR if the set

(3.3) \begin{equation} \mathcal{G}_\infty^1(S) \,:\!=\, \Big\{ X_\infty \,:\, \hbox{$X \in \mathcal{G}^1(S)$, $\displaystyle X_\infty \,:\!=\, \lim_{t \to \infty} X_t$ exists a.s.} \Big\} \end{equation}

is bounded in $L^0$ .

As shown in [Reference Bálint and Schweizer5, Subsection 3.1] (where NUPBR was called $\textrm{NUPBR}_\infty$ ), NUPBR for a process $Y\in\mathcal{S}^d$ and classic NUPBR for $(1,Y) \in \mathcal{S}^{d+1}$ coincide. But NUPBR (in the sense of Definition 3.1) for Y is not the same as classic NUPBR for Y—whether or not we have a riskless first asset identically equal to 1 makes a crucial difference.

Unfortunately, NUPBR is not discounting-invariant with respect to $\mathcal{S}^1_{++}$ . This was pointed out in [Reference Bálint and Schweizer5, Example 3.2], and we reproduce a variant of that example here.

Example 3.1. Let $W^1$ and $W^2$ be Brownian motions with constant correlation $\varrho\in({-}1,1)$ . Take $d=2$ and let $S^i = \mathcal{E}(\sigma_i W^i + m_i t)$ for $i=1,2$ be a geometric Brownian motion with parameters $m_i\in\mathbb{R}$ , $\sigma_i\ge0$ , and $\sigma_1^2+\sigma_2^2 \ne 0$ . Then clearly both $S^1$ and $S^2$ are in $\mathcal{G}^1(S)\cap\mathcal{S}^1_{++}$ . We choose the parameters such that $S^2/S^1$ is a martingale, and write it as $S^2/S^1 = \mathcal{E}(\gamma B)$ for some Brownian motion B and some $\gamma\ne0$ . Then

\begin{equation*}\lim_{t\to\infty} \frac{S^2_t}{S^1_t} = \lim_{t\to\infty} e^{\gamma B_t - \frac{1}{2} \gamma^2 t} = 0\qquad\hbox{a.s.}\end{equation*}

by the law of large numbers for Brownian motion. Because $S^2/S^1$ is a martingale, we see that $S/S^1 = (1,S^2/S^1)$ satisfies classic NFLVR and hence also classic NUPBR. But $S/S^2 = (S^1/S^2,1)$ does not satisfy classic NUPBR because $S^1_t/S^2_t\to\infty$ a.s. as $t\to\infty$ .

On the positive side, we have the following result.

Proof. Fix $D \in {\mathcal{S}}^{\textrm{conv}}$ . By Proposition 2.3, we have $\mathcal{G}^1(DS) = ({D}/{D_0}) \mathcal{G}^1(S)$ . As D converges to a $(0,\infty)$ -valued random variable $D_\infty$ , this shows that

\begin{align*} \mathcal{G}^1_\infty(DS) &= \frac{1}{D_0} \Big \{ (DX)_\infty \,:\, \hbox{$X \in \mathcal{G}^1(S)$, $\displaystyle (DX)_\infty \,:\!=\, \lim_{t \to \infty} (DX)_t$ exists a.s.} \Big \} \\[2pt] &= \frac{1}{D_0} \Big \{ D_\infty X_\infty \,:\, \hbox{$X \in \mathcal{G}^1(S)$, $\displaystyle X_\infty \,:\!=\, \lim_{t \to \infty} X_t$ exists a.s.} \Big \} \\[2pt] &= \frac{D_\infty}{D_0} \mathcal{G}_\infty^1(S). \end{align*}

Now the statement is obvious.

In the preceding argument, the convergence of D to a $(0,\infty)$ -valued random variable was crucial. The next example shows that this is not merely a shortcoming of the proof; in fact, NUPBR is, in general, not discounting-invariant with respect to $\mathcal{S}_{++}^{\textrm{unif}}$ .

Example 3.2. Define $T \,:\!=\, \pi/2$ and let $Y'$ be a continuous one-dimensional semimartingale on [0, T] with respect to some filtration $\mathbb{F}'$ . Moreover, assume that $S' \,:\!=\, (1, Y')$ does not satisfy NUPBR on [0, T]. Define $\,\tilde{\!Y}_t \,:\!=\, Y'_{\arctan(t)}$ . Then

\begin{equation*}\,\tilde{\!S}_t \,:\!=\, (1, \,\tilde{\!Y}_t) = S'_{\arctan(t)}, \qquad t \in [0, \infty),\end{equation*}

is a semimartingale with respect to $(\tilde{\mathcal{F}}_t) \,:\!=\, (\mathcal{F}'_{\arctan(t)})$ ; see [Reference Jacod14, Corollary 10.12]. Moreover, the change of variables formula for stochastic integrals (see [Reference Jacod14, Proposition 10.21], or [Reference Revuz and Yor18, Proposition V.(1.5)] for an English version) shows that

\begin{align*} (\tilde\vartheta \bullet \,\tilde{\!Y})_{\tan} &= \tilde\vartheta_{\tan} \bullet \,\tilde{\!Y}_{\tan} \qquad\hbox{for $\tilde\vartheta \in L(\,\tilde{\!Y})$}, \\ (\vartheta' \bullet Y')_{\arctan} &= \vartheta'_{\arctan} \bullet Y'_{\arctan} \qquad\hbox{for $\vartheta' \in L(Y')$}. \end{align*}

Hence using $\Theta^{\mathrm{sf}}(\,\tilde{\!S}) \cong L^0(\mathcal{F}_0)\times L(\,\tilde{\!Y})$ , $\Theta^{\mathrm{sf}}(S') \cong L^0(\mathcal{F}'_0)\times L(Y')$ and setting $\vartheta' \,:\!=\, \tilde \vartheta_{\tan}$ so that $\tilde\vartheta = \vartheta'_{\arctan}$ , we obtain

(3.4) \begin{align} \mathcal{G}^1(\,\tilde{\!S}) &= \{ 1+ \tilde \vartheta \bullet \,\tilde{\!Y} \,:\, \tilde \vartheta \in L(\,\tilde{\!Y}) \} \cap \mathcal{S}^1_+ \nonumber \\ &= \{1+ (\vartheta' \bullet Y')_{\arctan} \,:\, \vartheta' \in L(Y') \} \cap \mathcal{S}^1_+ \nonumber \\ &= \{ X_{\arctan}' \,:\, X' \in \mathcal{G}^1(S') \}.\end{align}

Because $S'$ is continuous on [0, T], all the limits in $\mathcal{G}^1(\,\tilde{\!S})$ therefore exist.

Now take $D\in\mathcal{S}_{++}^{\textrm{unif}}\setminus{\mathcal{S}}^{\textrm{conv}}$ , e.g. $D \,:\!=\, 2 + \sin$ , and define $S = D \,\tilde{\!S}$ . Then the oscillation of D implies that $D\,\tilde{\!X}$ with $\,\tilde{\!X} \in \mathcal{G}^1(\,\tilde{\!S})$ can converge only on $\{\,\tilde{\!X}_\infty = 0 \}$ , and in this case, the limit must be 0 as D is pathwise bounded. Hence $\mathcal{G}^1(S) = \mathcal{G}^1(D\,\tilde{\!S}) = ({D}/{D_0})\mathcal{G}^1(\,\tilde{\!S})$ from Proposition 2.3 yields $\mathcal{G}^1_\infty(S) \subseteq \{0\}$ , so S trivially satisfies NUPBR. But by (3.4), $({1}/{D}) S = \,\tilde{\!S}$ like $S'$ does not satisfy NUPBR.

Note that in this example, we have NUPBR for $S = (D,D\,\tilde{\!Y})$ , which is not a classic model as $S^1 = D \not\equiv 1$ . See also Remark 3.2.

Example 3.2 highlights a major problem with NUPBR as an AOA condition on $[0,\infty)$ . If we take a discounter D which is a.s. uniformly bounded in t, then S and DS are of the same magnitude, and so are the value processes in $\mathcal{G}^1(S)$ and $\mathcal{G}^1(DS)$ by (2.5) in Proposition 2.3. So if unbounded profits are possible for S, the same should happen for DS, and vice versa. However, Example 3.2 shows that this fails for NUPBR—unless we impose the stronger requirement that D should converge a.s.

3.2. A new version of NUPBR with good discounting invariance

The source of the problem in Example 3.2 is that when defining NUPBR, we require the limits in (3.3) to exist. Recall that we want to put bounds on ‘terminal values’ of self-financing value processes. So if we cannot use a limit as $t\to\infty$ to define such terminal values, it is natural to consider $\limsup$ or $\liminf$ as alternatives, and this directly leads to our next definition.

Definition 3.2. A semimartingale $S \in \mathcal{S}^d$ satisfies $\underline{{NUPBR}}$ (or $\overline{{NUPBR}}$ ) if the set

\begin{equation*}\underline{\mathcal{G}^1_\infty}(S)\,:\!=\, \Big \{\kern-1.5pt\liminf_{t \to \infty} X_t \,:\, X \in \mathcal{G}^1(S) \Big \}\qquad \Big(\text{respectively} \, \overline{\mathcal{G}^1_\infty}(S) \,:\!=\, \Big \{\kern-1.5pt\limsup_{t \to \infty} X_t \,:\, X \in \mathcal{G}^1(S) \Big \}\Big)\end{equation*}

is bounded in $L^0$ .

We immediately get a nice property of these two concepts.

It is immediate from the definition that

(3.5) \begin{equation} {\overline{\textrm{NUPBR}}} \quad\Longrightarrow\quad {\underline{\textrm{NUPBR}}} \quad\Longrightarrow\quad \textrm{NUPBR}{}.\end{equation}

By Example 3.2, the converse of the second implication in (3.5) does not hold in general; see the discussion in Remark 3.2. Hence ${\underline{\textrm{NUPBR}}}$ is, in general, genuinely stronger than NUPBR. We show below that the first implication in (3.5) is an equivalence whenever $\mathcal{G}^1(S) \cap \mathcal{S}_{++}^{\textrm{unif}} \neq \varnothing$ . Moreover, we shall see that under $\mathcal{G}^1(S) \cap {\mathcal{S}}^{\textrm{conv}} \neq \varnothing$ and NUPBR, all the limits in $\mathcal{G}^1(S)$ exist, and in this case, we have equivalence everywhere in (3.5). To prove these statements, more work is needed.

3.3. The supermartingale numeraire

In the following, we want to characterise ${\underline{\textrm{NUPBR}}}$ and ${\overline{\textrm{NUPBR}}}$ in terms of a dual object for $\mathcal{G}^1(S)$ . We shall see that this is closely related to the question of whether $\mathcal{G}^1(S)$ contains a process that is in some sense better than any other element in $\mathcal{G}^1(S)$ . While this is already interesting on its own, the existence of such an ‘optimal’ process also allows us to prove the statements at the end of Subsection 3.2. In addition, the result obtained plays a key role when we find a dual object for S instead of $\mathcal{G}^1(S)$ in Section 4.

If it exists, the supermartingale numeraire $V^{\textrm{sn}}$ for $\mathcal{G} \subseteq \mathcal{G}^1(S)$ is optimal in $\mathcal{G}$ as follows. For any $X \in \mathcal{G}$ , the ratio $X/V^{\textrm{sn}}$ is a $\mathbb{P}$ -supermartingale, hence decreasing on average, and so $V^{\textrm{sn}}$ grows on average more quickly than any other self-financing value process $X\in\mathcal{G}$ . By the Doob–Meyer decomposition, $X/V^{\textrm{sn}}$ can be written as $X/V^{\textrm{sn}} = M + A$ , where $M\in {\mathcal{M}}_\textrm{loc}(\mathbb{P})$ can be interpreted as a noise term and A is a decreasing predictable process which describes a signal. For $X=V^{\textrm{sn}}$ , we get $M\equiv0$ and $A\equiv1$ . If in addition $X\in\mathcal{S}^1_{++}$ , we can also decompose $X/V^{\textrm{sn}}$ multiplicatively; see [Reference Jacod14, Corollary 6.21].

For the classic model, the supermartingale numeraire was introduced and has been extensively studied by Karatzas and Kardaras [Reference Karatzas and Kardaras15] under the name numeraire portfolio. For ease of reference, we formulate their main result [Reference Karatzas and Kardaras15, Theorem 4.12] in our notation. We define

\begin{equation*}\big( \mathcal{G}^1(S) \cap \mathcal{S}^1_{++} \big)_\infty\,:\!=\,\Big\{ X_\infty \,:\, \hbox{$X \in \mathcal{G}^1(S) \cap \mathcal{S}^1_{++}$, $\displaystyle X_\infty \,:\!=\, \lim_{t \to \infty} X_t$ exists a.s.} \Big\}.\end{equation*}

Recall that $S^1\equiv1$ implies $\mathcal{G}^1(S)\cap\mathcal{S}^1_{++} \ne \varnothing$ .

The proof of this result is long and uses semimartingale characteristics. We refer to [Reference Karatzas and Kardaras15, Theorem 4.12] for more details, but we give three comments because their theorem looks different at first glance.

First, Karatzas and Kardaras [Reference Karatzas and Kardaras15] prove the existence of the supermartingale numeraire for any predictable closed convex constraints on strategies; the above formulation is the special case of the natural constraints from [Reference Karatzas and Kardaras15, Example 2.7].

Second, at the beginning of [Reference Karatzas and Kardaras15] it is assumed that each $S^i$ belongs to $\mathcal{S}^1_{++}$ ; however, in Subsection 4.8 in [Reference Karatzas and Kardaras15] it is explained why this is not a loss of generality for their results.

Third, under the natural constraints, Definition 4.1 in [Reference Karatzas and Kardaras15] says that S satisfies NUPBR if and only if $(\mathcal{G}^1(S) \cap \mathcal{S}^1_{++})_\infty$ is bounded in $L^0$ . Because $S^1\equiv1$ , NUPBR for $S = (1,S^{2:d})$ and classic NUPBR (as in [Reference Karatzas and Kardaras15]) for $S^{2:d}$ are the same.

Thanks to our parametrisation of self-financing value processes in Theorem 2.1, the existence of the supermartingale numeraire for $\mathcal{G}^1(S)$ with general $S \in \mathcal{S}^d$ can be deduced from the special case in Theorem 3.1. The next theorem is one of our main results; it contains Theorem 3.1 because $1\equiv S^1$ is in $\mathcal{G}^1(S) \cap\mathcal{S}_{++}^{\textrm{unif}}$ .

Proof. (i) $\Rightarrow$ (ii): This is clear from (3.5).

(ii) $\Rightarrow$ (iii): Fix $V(\eta) \in \mathcal{G}^1(S) \cap \mathcal{S}_{++}^{\textrm{unif}}$ . By (2.6) in Proposition 2.3, we have

(3.6) \begin{equation}\mathcal{G}^1\bigg( 1, \frac{S}{V(\eta)} \bigg) = \frac{1}{V(\eta)}\mathcal{G}^1(S).\end{equation}

So $(1, S/V(\eta))$ satisfies ${\underline{\textrm{NUPBR}}}$ by Proposition 3.1 and hence also NUPBR by (3.5). By Theorem 3.1, the supermartingale numeraire $Y^{\textrm{SN}}$ for $\mathcal{G}^1(1,S/V(\eta)) \cap \mathcal{S}^1_{++}$ exists, and the limit $Y^{\textrm{SN}}_\infty$ exists and is finite, a.s. Moreover, $1/Y^{\textrm{SN}}$ is a non-negative supermartingale and thus converges a.s. to a finite (even integrable) random variable. Hence $Y^{\textrm{SN}}_\infty > 0$ , and thus $Y^{\textrm{SN}} \in {\mathcal{S}}^{\textrm{conv}}\subseteq\mathcal{S}_{++}^{\textrm{unif}}$ . Moreover, $Y^{\textrm{SN}}$ is by Lemma 3.1 even a supermartingale numeraire for $\mathcal{G}^1(1, S/V(\eta)) = ({1}/{V(\eta)}) \mathcal{G}^1(S)$ . Thus $V^{\textrm{sn}} \,:\!=\, Y^{\textrm{SN}} V(\eta)$ is in $\mathcal{G}^1(S) \cap \mathcal{S}_{++}^{\textrm{unif}}$ , and for every $X\in\mathcal{G}^1(S)$ , ${X}/{V(\eta) Y^{\textrm{SN}}} = X / V^{\textrm{sn}}$ is a $\mathbb{P}$ -supermartingale, as desired.

(iii) $\Rightarrow$ (i): For any $X \in \mathcal{G}^1(S)$ , the ratio $X/V^{\textrm{sn}}$ is a non-negative $\mathbb{P}$ -supermartingale and thus converges $\mathbb{P}$ -a.s. as $t\to\infty$ . (Note that X itself need not converge.) Hence

\begin{equation*} \limsup_{t \to \infty} X_t \le (V^{\textrm{sn}})^*_\infty \lim_{t \to \infty} \frac{X_t}{V^{\textrm{sn}}_t}\end{equation*}

together with Fatou’s lemma and $X_0/V^{\textrm{sn}}_0 = 1$ shows that the set $ ({1}/{(V^{\textrm{sn}})_\infty^*}) \overline{\mathcal{G}^1_\infty}(S)$ is even bounded in $L^1(\mathbb{P})$ and therefore also in $L^0(\mathbb{P})$ . So is then $\overline{\mathcal{G}^1_\infty}(S)$ because $V^{\textrm{sn}}\in\mathcal{S}_{++}^{\textrm{unif}}$ .

It remains to show the additional statements. Clearly, (i) implies (iv) by (3.5). Conversely, if S satisfies NUPBR and we take $V(\eta) \in \mathcal{G}^1(S) \cap {\mathcal{S}}^{\textrm{conv}}$ , then (3.6) shows that $(1, S/V(\eta))$ satisfies NUPBR, and the argument for (ii) $\Rightarrow$ (iii) yields the existence of the supermartingale numeraire $V^{\textrm{sn}} = V(\eta) Y^{\textrm{SN}}$ for $\mathcal{G}^1(S)$ with $Y^{\textrm{SN}}\in{\mathcal{S}}^{\textrm{conv}}$ . Because also $V(\eta) \in {\mathcal{S}}^{\textrm{conv}}$ , we get $V^{\textrm{sn}} \in {\mathcal{S}}^{\textrm{conv}}$ .

The next result is an easy consequence of Theorem 3.2 and will be the key to proving in Section 4 a dual characterisation of ${\overline{\textrm{NUPBR}}}$ in terms of S (instead of $\mathcal{G}^1(S)$ ).

Proof. By Theorem 3.2, $V^{\textrm{sn}}$ exists and is in ${\mathcal{S}}^{\textrm{conv}}$ . For any $a\in\mathcal{G}^1_\infty(S)$ with

(3.7) \begin{equation} V^{\textrm{sn}}_\infty \le a,\end{equation}

we have $a = X_\infty$ for some $X \in \mathcal{G}^1(S)$ such that the limit $X_\infty$ exists. Because $X/V^{\textrm{sn}}$ is a supermartingale, Fatou’s lemma yields ${\mathbb{E}}[ a/V^{\textrm{sn}}_\infty ] = {\mathbb{E}}[ X_\infty/V^{\textrm{sn}}_\infty ] \le 1$ . Clearly, this is possible only if we have equality in (3.7).

The converse of Corollary 3.3 is not true in general—not every self-financing value process $V(\varphi)$ whose final value is maximal in $\mathcal{G}_\infty^1(S)$ is a supermartingale numeraire.

3.4. Convergence of self-financing value processes at infinity

In Definition 3.3, we look at the set of those value processes $X\in\mathcal{G}^1(S)$ for which the limit $X_\infty$ is assumed to exist. We now study when convergence happens for all $X\in\mathcal{G}^1(S)$ .

In the previous result, we cannot replace the assumption $\mathcal{G}^1(S) \cap {\mathcal{S}}^{\textrm{conv}} \neq \varnothing$ by the weaker condition $\mathcal{G}^1(S) \cap \mathcal{S}_{++}^{\textrm{unif}} \neq \varnothing$ . Even if NUPBR is replaced by the stronger condition ${\overline{\textrm{NUPBR}}}$ , this does not work, as the following example shows.

In the classic setting of $S^1\equiv1$ , a result similar to Corollary 3.4 can be found in Theorem 3.3 in [Reference Delbaen and Schachermayer6]. Delbaen and Schachermayer [Reference Delbaen and Schachermayer6] assume NFLVR, but as already observed in the proof of Proposition 3.4 in [Reference Bálint and Schweizer5], a closer look at their arguments shows that they use only NUPBR. Since the existence of the supermartingale numeraire was not known at that time, Delbaen and Schachermayer [Reference Delbaen and Schachermayer6] had to mimic the proof of Doob’s martingale convergence theorem to show the existence of the limits. The main ingredient for their argument is Proposition 3.2 in [Reference Delbaen and Schachermayer6], which states that for $S \in \mathcal{S}^d$ with $S^1 \equiv 1$ and NUPBR (they assume NFLVR but use only NUPBR), the set (see (3.2))

\begin{equation*} \{ (H \bullet S^{2:d})^*_\infty \,:\, H \in L(S^{2:d}), H \bullet S^{2:d} \ge -1\} = \{ (X - 1)^*_\infty \,:\, X \in \mathcal{G}^1(S) \}\end{equation*}

is bounded in $L^0$ . We conclude this subsection by showing that such a result is true more generally. Unlike Corollary 3.4, the next result needs only $\mathcal{G}^1(S) \cap \mathcal{S}_{++}^{\textrm{unif}} \neq \varnothing$ . Observe also that $X^*-1 \le (X-1)^* \le X^* + 1$ for any RCLL process $X\ge0$ .

Proof. The ‘if’ part is clear from the definition of ${\underline{\textrm{NUPBR}}}$ . For the ‘only if’ part, by Theorem 3.2, the supermartingale numeraire $V^{\textrm{sn}}$ for $\mathcal{G}^1(S)$ exists and belongs to $\mathcal{S}_{++}^{\textrm{unif}}$ . In particular, $V^{\textrm{sn}}$ is bounded above by a finite-valued random variable. Consequently, it suffices to show that the set $\{ (X/V^{\textrm{sn}})^*_\infty \,:\, X \in \mathcal{G}^1(S) \}$ is bounded in $L^0$ . To this end, take any $X \in \mathcal{G}^1(S)$ and note that the ratio $U \,:\!=\, X/V^{\textrm{sn}}$ is a supermartingale with $U_0=1$ and non-negative, and hence is closable on the right by 0. For $\ell \in \mathbb{N}$ and $\tau \,:\!=\, \inf \{t \ge 0 \,:\, U_t > \ell \}$ , we therefore obtain from $U_\tau\ge\ell$ and the stopping theorem (see [Reference Dellacherie and Meyer9, Theorem VI.60]) that

\begin{equation*} \mathbb{P}[U^*_\infty > \ell] \le \mathbb{P}[\tau < \infty] \le \frac{1}{\ell} {\mathbb{E}}[U_\tau \textbf{1}_{\{\tau < \infty\}}] \le \frac{1}{\ell} {\mathbb{E}}[U_\tau] \le \frac{1}{\ell} {\mathbb{E}}[U_0] = \frac{1}{\ell} \longrightarrow 0 \qquad\hbox{as $\ell\to\infty$.}\end{equation*}

As $U = X/V^{\textrm{sn}}$ , this gives $\sup_{X\in\mathcal{G}^1(S)} P[(X/V^{\textrm{sn}})^*_\infty > \ell] \to 0$ for $\ell\to\infty$ as desired.

4.1. The main result

In this subsection, we establish in Theorem 4.1 our main result, a dual characterisation of the AOA concepts ${\underline{\textrm{NUPBR}}}$ and ${\overline{\textrm{NUPBR}}}$ . Theorem 3.2 has already characterised these in terms of a dual object for $\mathcal{G}^1(S)$ . Now we aim to find a dual object for S itself.

To apply the FTAP of Delbaen and Schachermayer [Reference Delbaen and Schachermayer8, Theorem 1.1], we briefly recall the definitions of (classic) NA and (classic) NFLVR. It should be noted that we use these AOA concepts only in the classic setting where $S^1 \equiv 1$ . For an extension of the definitions to general $S\in\mathcal{S}^d$ , we refer the reader to [Reference Bálint and Schweizer5, Subsection 3.1].

Remark 4.1. (1) Our definition of NA for a classic model $S=(1,X) \in \mathcal{S}^d$ is the same as the definition of NA for X in the sense of Delbaen and Schachermayer [Reference Delbaen and Schachermayer6, Definition 2.8]; see [Reference Bálint and Schweizer5, Subsection 3.1].

(2) Even though the definition of NFLVR for X in [Reference Delbaen and Schachermayer6, Definition 2.8] looks rather different at first glance, Delbaen and Schachermayer [Reference Delbaen and Schachermayer6, Corollaries 3.4 and 3.9] show that

\begin{equation*} \textrm{NA} + \textrm{NUPBR}{} \quad\Longleftrightarrow\quad \textrm{NFLVR}.\end{equation*}

Hence also our notion of NFLVR for S coincides with NFLVR for X in [Reference Delbaen and Schachermayer6].

Before we can state our main theorem, we need one more auxiliary result. As a special case, it contains a dual characterisation of NUPBR for the classic model, and thereby holds intrinsic interest.

Remark 4.2. If $S \in \mathcal{S}^d$ has the form

(4.1) \begin{equation} S = DM\end{equation}

with $M \in {\mathcal{M}}_\sigma(\mathbb{P})$ and $D \in \mathcal{S}_{++}^{\textrm{unif}}$ then S satisfies both ${\underline{\textrm{NUPBR}}}$ and ${\overline{\textrm{NUPBR}}}$ . Indeed, we have $\mathcal{G}^1(S) = ({D_0}/{D}) \mathcal{G}^1(M)$ , and $\mathcal{G}^1(M)$ contains only non-negative $\mathbb{Q}$ -supermartingales, as seen in the proof above. So $\mathcal{G}^1_\infty(M)$ is bounded in $L^0$ , and ${\underline{\textrm{NUPBR}}}$ and ${\overline{\textrm{NUPBR}}}$ follow like in Proposition 3.1 because $D\in\mathcal{S}_{++}^{\textrm{unif}}$ . The next result in particular shows conversely that S must be of the form (4.1) if it satisfies ${\underline{\textrm{NUPBR}}}$ and $\mathcal{G}^1(S) \cap \mathcal{S}_{++}^{\textrm{unif}} \ne \varnothing$ .

We can now formulate our main result. Comparisons with the literature are given in Subsection 4.4.

Proof. From the definitions, (i) $\Rightarrow$ (ii) is clear; see (3.5). Next, (ii) $\Rightarrow$ (iii) is from Theorem 3.2, which uses Theorem 3.1, and (iii) $\Rightarrow$ (iv) is also from Theorem 3.2. The equivalence (ii) $\Leftrightarrow$ (iv) is from Proposition 3.2 which uses Theorem 3.2. Under the additional assumption that $\mathcal{G}^1(S) \cap {\mathcal{S}}^{\textrm{conv}} \ne \varnothing$ , the equivalence (i) $\Leftrightarrow$ (viii) is from Theorem 3.2, and the equivalence (viii) $\Leftrightarrow$ (ix) is from Proposition 4.1. Hence it remains to show the equivalence of (i) and (v)–(vii).

(i) $\Rightarrow$ (v): Fix an arbitrary $V(\eta) \in \mathcal{G}^1(S) \cap \mathcal{S}_{++}^{\textrm{unif}}$ . As in the proof of (ii) $\Rightarrow$ (iii) in Theorem 3.2, $\,\tilde{\!S}\,:\!=\, (1, S/V(\eta))$ satisfies ${\overline{\textrm{NUPBR}}}$ and hence NUPBR. So Proposition 4.1 yields $Y \in \mathcal{G}^1(\,\tilde{\!S}) \cap {\mathcal{S}}^{\textrm{conv}}$ such that $(1,\,\tilde{\!S}/Y)$ satisfies NFLVR, and so the FTAP [Reference Delbaen and Schachermayer8, Theorem 1.1] yields the existence of $\mathbb{Q} \approx \mathbb{P}$ with

(4.2) \begin{equation} \tilde{\!S}/Y{} \in {\mathcal{M}}_\sigma(\mathbb{Q}). \end{equation}

The density process $\tilde{Z}$ of $\mathbb{Q}$ with respect to $\mathbb{P}$ is a strictly positive martingale on $[0, \infty]$ and thus belongs to ${\mathcal{S}}^{\textrm{conv}}$ . By the Bayes formula, by (4.2) and due to $\,\tilde{\!S}^1\equiv 1$ , we have

\begin{equation*} Z \,:\!=\, \tilde{Z}/Y{} \in {\mathcal{M}}_\sigma(\mathbb{P}) \cap {\mathcal{S}}^{\textrm{conv}}.\end{equation*}

Thus Corollary 3.5 in [Reference Ansel and Stricker1] yields $Z \in {\mathcal{M}}_\textrm{loc}(\mathbb{P})$ . Furthermore, the Bayes formula and (4.2) also show that, as desired,

\begin{equation*} Z (S/V(\eta)) = (\tilde{Z}/Y) \,\tilde{\!S}^{2:(d+1)} \in {\mathcal{M}}_\sigma(\mathbb{P}).\end{equation*}

(v) $\Rightarrow$ (vii): This follows by setting $D \,:\!=\, Z /V(\eta)$ .

(i) $\Rightarrow$ (vi): As seen in (i) $\Rightarrow$ (v), for any $V(\eta) \in \mathcal{G}^1(S) \cap \mathcal{S}_{++}^{\textrm{unif}}$ , the market $(1, S/V(\eta))$ satisfies NUPBR. So we need only to show that $V(\eta)$ can be chosen such that $(1, S/V(\eta))$ also satisfies NA. Theorem 3.2 gives the existence of the supermartingale numeraire $V^{\textrm{sn}}$ for $\mathcal{G}^1(S) = V^{\textrm{sn}} \mathcal{G}^1(1, S/V^{\textrm{sn}})$ , and so any process in $\mathcal{G}^1(1, S/V^{\textrm{sn}})$ is a supermartingale starting in 1. In consequence, 1 is maximal in $\mathcal{G}^1_\infty(1, S/V^{\textrm{sn}})$ and we can take $V(\eta) \,:\!=\, V^{\textrm{sn}}$ .

(vi) $\Rightarrow$ (vii): The FTAP [Reference Delbaen and Schachermayer8, Theorem 1.1] yields $\mathbb{Q} \approx \mathbb{P}$ with $S/V(\eta) \in {\mathcal{M}}_\sigma(\mathbb{Q})$ . The density process Z of $\mathbb{Q}$ with respect to $\mathbb{P}$ is a strictly positive martingale on $[0,\infty]$ and thus in ${\mathcal{S}}^{\textrm{conv}}$ . Hence, as before, $D \,:\!=\, Z/V(\eta) \in \mathcal{S}_{++}^{\textrm{unif}}$ , and the Bayes formula yields $DS \in {\mathcal{M}}_\sigma(\mathbb{P})$ .

(vii) $\Rightarrow$ (i): We can write $S = (1/D) M$ with $M\in{\mathcal{M}}_\sigma(\mathbb{P})$ and $1/D \in\mathcal{S}_{++}^{\textrm{unif}}$ like D. So Remark 4.2 shows that S satisfies ${\overline{\textrm{NUPBR}}}$ (and ${\underline{\textrm{NUPBR}}}$ ).

4.2. Maximal value processes

We now revisit our dual characterisation in Theorem 4.1. We already know that if $S \in \mathcal{S}^d$ with $\mathcal{G}^1(S) \cap \mathcal{S}_{++}^{\textrm{unif}} \neq \varnothing$ satisfies ${\underline{\textrm{NUPBR}}}$ , then for any $V(\eta) \in \mathcal{G}^1(S) \cap \mathcal{S}_{++}^{\textrm{unif}}$ , there exists by (v) a process $Z \in {\mathcal{M}}_\textrm{loc}(\mathbb{P}) \cap \mathcal{S}_{++}^{\textrm{unif}}$ such that $Z (S/V(\eta))$ is a $\sigma$ -martingale under $\mathbb{P}$ . We call such a Z a ( $\mathbb{P}$ -) $\sigma$ -martingale density for $S/V(\eta)$ . Moreover, (vi) states that for some $V(\eta)$ , this Z can be chosen to be a uniformly integrable martingale; then Z induces a probability measure $\mathbb{Q}\approx\mathbb{P}$ under which $S/V(\eta)$ becomes a $\sigma$ -martingale (and we then call $\mathbb{Q}$ an equivalent $\sigma$ -martingale measure for $S/V(\eta)$ ). In the following, our aim is to identify all $V(\eta)$ for which finding $\mathbb{Q}$ like that is possible.

Under the additional assumption $\mathcal{G}^1(S) \cap {\mathcal{S}}^{\textrm{conv}} \neq \varnothing$ , (ix) in Theorem 4.1 shows that Z can be chosen to be a uniformly integrable martingale whenever $V_\infty(\eta)$ is maximal in $\mathcal{G}_\infty^1(S)$ . We show below that this maximality already fully determines all $V(\eta)$ with the above desired property. However, as the limits in $\mathcal{G}^1(S)$ might not exist if we assume only $\mathcal{G}^1(S) \cap \mathcal{S}_{++}^{\textrm{unif}} \neq \varnothing$ instead of $\mathcal{G}^1(S) \cap {\mathcal{S}}^{\textrm{conv}} \neq \varnothing$ , we need to define maximality in a different way.

It is immediate from the definition that the $\mathbb{P}$ -supermartingale numeraire $V^{\textrm{sn}}$ is maximal in the sense of Definition 4.2 (with $\mathbb{Q} \,:\!=\, \mathbb{P}$ ). But while $V^{\textrm{sn}}$ is unique for a given $\mathbb{P}$ , several processes in $\mathcal{G}^1(S) \cap \mathcal{S}^1_{++}$ can be maximal in $\mathcal{G}^1(S)$ .

The next result is a dual characterisation of the maximality property in Definition 4.2 and answers the questions raised at the beginning of this subsection. If $V(\eta)$ is maximal in $\mathcal{G}^1(S)$ and also lies in $\mathcal{S}_{++}^{\textrm{unif}}$ , then $\mathcal{G}^1(S)\cap\mathcal{S}_{++}^{\textrm{unif}}\ne\varnothing$ , and the $\mathbb{Q}$ -supermartingale numeraire exists for some $\mathbb{Q}\approx\mathbb{P}$ . So S satisfies by Theorem 4.1 ${\underline{\textrm{NUPBR}}}$ under $\mathbb{Q}$ and hence also under $\mathbb{P}$ , and so we may assume in the next result that S satisfies ${\underline{\textrm{NUPBR}}}$ . We denote by $\overline{\mathcal{T}}$ the set of all (possibly infinite-valued) stopping times and set

\begin{equation*} \mathcal{T} \,:\!=\, \{ \tau \in \overline{\mathcal{T}} \,:\, \tau < \infty \}\end{equation*}

and

\begin{equation*}\mathcal{G}^1_\tau(S) \,:\!=\, \{ X_\tau \,:\, X\in\mathcal{G}^1(S) \}\qquad\hbox{for $\tau\in\mathcal{T}$.}\end{equation*}

If $X_\infty \,:\!=\, \lim_{t\to\infty} X_t$ exists a.s. for all $X\in\mathcal{G}^1(S)$ , then $\mathcal{G}^1_\tau(S)$ is also well defined for $\tau\in\overline{\mathcal{T}}$ .

4.3. An explicit example

In this subsection, we analyse in detail an example with $d=2$ assets given by correlated generalised geometric Brownian motions, and show how ${\underline{\textrm{NUPBR}}}$ can be characterised in terms of the model coefficients.

We start with two independent Brownian motions $B^1,B^2$ on $(\Omega, \mathcal{F}, \mathbb{P})$ , and denote by $\mathbb{F}$ the filtration they generate, made right-continuous and complete. This implies that all local martingales in $\mathbb{F}$ are continuous. We fix a predictable process $\varrho = (\varrho_t)_{t\ge 0}$ with values in $({-}1,1)$ , and define two new Brownian motions $W^1,W^2$ by $W^2 \,:\!=\, B^2$ and $W^1 = \int \varrho_s \,\mathrm{d} B^2_s + \int \sqrt{1-\varrho_s^2} \,\mathrm{d} B^1_s$ . If $\varrho$ is constant, then $W^1$ and $W^2$ have instantaneous correlation $\varrho$ and generate the same filtration $\mathbb{F}$ .

Our model for $S = (S^1,S^2)$ is given by

(4.3) \begin{equation}S^i=\mathcal{E}\bigg( \int \sigma^{(i)}_s \,\mathrm{d} W^i_s + \int \mu^{(i)}_s \,\mathrm{d} s \bigg)=\exp\bigg( \int\sigma^{(i)}_s \,\mathrm{d} W^i_s + \int \Big( \mu^{(i)}_s - \frac{1}{2} (\sigma^{(i)}_s)^2 \Big) \,\mathrm{d} s \bigg)\end{equation}

with predictable processes $\sigma^{(i)}, \mu^{(i)}$ satisfying

\begin{equation*}\int_0^t \big( (\sigma^{(i)}_s)^2 + | \mu^{(i)}_s | \big) \,\mathrm{d} s < \infty\qquad\hbox{a.s.{} for each $t\ge 0$.}\end{equation*}

We also assume that $\sigma^{(1)}\ge0$ and $\sigma^{(2)}>0$ . We impose the condition that

(4.4) \begin{equation}\sup\limits_{t\ge0} \bigg| \int_0^t \Big( \mu^{(1)}_s - \frac{1}{2} (\sigma^{(1)}_s)^2 \Big) \,\mathrm{d} s \bigg| < \infty\qquad\hbox{a.s.},\end{equation}

and note that this ensures that

(4.5) \begin{equation}\exp\bigg( \int \Big( \mu^{(1)}_s - \frac{1}{2} (\sigma^{(1)}_s)^2 \Big) \,\mathrm{d} s \bigg) \in \mathcal{S}_{++}^{\textrm{unif}}.\end{equation}

(Of course, we could equally well impose (4.4) and hence (4.5) for $i=2$ instead of $i=1$ .) Note that (4.4) excludes a model with constant coefficients as in Example 3.1, where we have already seen that problems come up.

The first part of the next auxiliary result is well known. Note that this does not use that $\mathbb{F}$ is generated by a Brownian motion.

Lemma 4.1. For any continuous local martingale $M = (M_t)_{t\ge0}$ , we have

(4.6) \begin{align}\{ [M]_\infty < \infty \}&=\Big\{ M_\infty = \hbox{$\displaystyle \lim_{t\to\infty} M_t\ exists\ in\ \mathbb{R}$} \Big\}\nonumber\\&=\Big\{ \sup\limits_{t\ge 0} M_t < \infty \Big\}=\Big\{ \inf_{t\ge 0} M_t > - \infty \Big\}\qquad {a.s.}\end{align}

As a consequence, $\mathcal{E}(M)$ is in $\mathcal{S}_{++}^{\textrm{unif}}$ if and only if $[M]_\infty < \infty$ a.s., and then $\mathcal{E}(M)$ is even in ${\mathcal{S}}^{\textrm{conv}}$ , and in particular has $\mathcal{E}(M)_\infty > 0$ a.s.

Proof. The first equality is from [Reference Revuz and Yor18, Proposition V.(1.8)], and the second and third equalities are from Remark 3 $^\circ$ after [Reference Revuz and Yor18, Proposition V.(1.8)]. Next, $\mathcal{E}(M) = \exp\!(M - \frac{1}{2} [M])$ together with (4.6) directly gives $\mathcal{E}(M)\in\mathcal{S}_{++}^{\textrm{unif}}$ whenever $[M]_\infty < \infty$ a.s. Conversely, $\mathcal{E}(M)\in\mathcal{S}_{++}^{\textrm{unif}}$ implies by (4.6) that $[\mathcal{E}(M)]_\infty < \infty$ a.s., and hence

\begin{equation*}[M]_\infty=\bigg[ \int \frac{1}{\mathcal{E}(M)} \,\mathrm{d} \mathcal{E}(M) \bigg]_\infty\le\Big( \inf_{t\ge 0} \mathcal{E}(M)_t \Big)^{-2} [\mathcal{E}(M)]_\infty<\infty\qquad \textrm{a.s.}\end{equation*}

Finally, any local martingale Z in $\mathcal{S}_{++}^{\textrm{unif}}$ is positive, and hence a supermartingale and thus a.s. convergent. So for local martingales, $Z\in\mathcal{S}_{++}^{\textrm{unif}}$ automatically implies $Z\in{\mathcal{S}}^{\textrm{conv}}$ .

Corollary 4.1. If we assume (4.4) holds then $S^1$ is in $\mathcal{S}_{++}^{\textrm{unif}}$ if and only if

(4.7) \begin{equation}\int_0^\infty (\sigma^{(1)}_s)^2 \,\mathrm{d} s < \infty\qquad {a.s.}\end{equation}

We are now ready for the main result of this subsection. Define the process

\begin{equation*}h \,:\!=\, (\sigma^{(2)} - \varrho\sigma^{(1)})^2 + (\sigma^{(1)})^2 (1-\varrho^2).\end{equation*}

Note that $h>0$ because $\sigma^{(1)}\ge 0$ and $\sigma^{(2)}>0$ . The meaning of h will become clear in the proof below.

Proof. By (4.4) and Corollary 4.1, (4.7) implies that $S^1\in\mathcal{S}_{++}^{\textrm{unif}}$ . By Proposition 3.1, ${\underline{\textrm{NUPBR}}}$ for S is then equivalent to ${\underline{\textrm{NUPBR}}}$ for $S/S^1 = (1,S^2/S^1)$ , and this motivates us to look at $X\,:\!=\, S^2/S^1$ in more detail. From the explicit expressions for $S^i$ in (4.3), it can be easily verified that

(4.9) \begin{equation}X = X_0 + M + A,\end{equation}

with

\begin{equation*}M = \int X \big( (\sigma^{(2)}-\varrho\sigma^{(1)}) \,\mathrm{d} B^2 - \sqrt{1-\varrho^2}\,\sigma^{(1)} \,\mathrm{d} B^1 \big),\end{equation*}

and hence

\begin{equation*}[M] = \int X^2 \big( (\sigma^{(2)}-\varrho\sigma^{(1)})^2 + (1-\varrho^2) (\sigma^{(1)})^2 \big) \,\mathrm{d} t = \int X^2 h \,\mathrm{d} t,\end{equation*}

and

\begin{equation*}A = \int X \bigg( \mu^{(2)} - \mu^{(1)} - \frac{1}{2} \big( (\sigma^{(2)})^2 - (\sigma^{(1)})^2 \big) + \frac{1}{2} h \bigg) \,\mathrm{d} t.\end{equation*}

This shows that X satisfies the so-called weak structure condition that $A\ll[M]$ ; in fact,

(4.10) \begin{equation}\frac{\mathrm{d} A}{\mathrm{d} [M]}=\frac{\mu^{(2)} - \mu^{(1)} - \frac{1}{2} ( (\sigma^{(2)})^2 - (\sigma^{(1)})^2 ) + \frac{1}{2} h}{X h}\,=\!:\,\lambda,\end{equation}

and therefore the so-called mean–variance tradeoff process K of X is given by

\begin{equation*}K = \int \lambda^2 \,\mathrm{d} [M] = \int \frac{(\mu^{(2)} - \mu^{(1)} - \frac{1}{2} ( (\sigma^{(2)})^2 - (\sigma^{(1)})^2 ) + \frac{1}{2} h)^2}{h} \,\mathrm{d} t.\end{equation*}

The process K is clearly null at 0, is increasing, and can take values in $[0,+\infty]$ . The condition (4.8) can then be rewritten as $K_\infty < \infty$ a.s.

(ii) $\Rightarrow$ (i): By Corollary 4.1, (4.7) ensures that $S^1\in\mathcal{S}_{++}^{\textrm{unif}}$ , which in turn implies that $\mathcal{G}^1(S) \cap\mathcal{S}_{++}^{\textrm{unif}} \ne \varnothing$ . Moreover, (4.8) yields $\int_0^\infty \lambda_s^2 \,\mathrm{d} [M]_s = K_\infty < \infty$ a.s., so $\hat Z \,:\!=\, \mathcal{E}({-} \lambda \bullet M )$ is a well-defined local $\mathbb{P}$ -martingale and in ${\mathcal{S}}^{\textrm{conv}}\subseteq\mathcal{S}_{++}^{\textrm{unif}}$ by the second part of Lemma 4.1. Finally, an easy application of the product rule together with (4.9) and (4.10) shows that $\hat Z X$ is a local $\mathbb{P}$ -martingale. So (v) in Theorem 4.1 is satisfied for $D\,:\!=\,\hat Z$ and $S/S^1 = (1,X)$ , and so $S/S^1$ satisfies ${\underline{\textrm{NUPBR}}}$ .

(i) $\Rightarrow$ (ii): Because S satisfies ${\underline{\textrm{NUPBR}}}$ and $\mathcal{G}^1(S) \cap \mathcal{S}_{++}^{\textrm{unif}}\ne\varnothing$ , the set $\{ Y^*_\infty \,:\, Y \in \mathcal{G}^1(S) \}$ is bounded in $L^0$ by Theorem 4.1(iv). Hence $\sup_{t\ge0} S^1_t < \infty$ a.s., and thanks to (4.3) and (4.5) (which follows from (4.4)), we also obtain $\sup_{t\ge0} \int_0^t \sigma^{(1)}_s \,\mathrm{d} W^1_s < \infty$ a.s. This implies (4.7) by Lemma 4.1, and then $S^1\in\mathcal{S}_{++}^{\textrm{unif}}$ by Corollary 4.1. So by Theorem 4.1(v) with $\eta\equiv e^1$ , hence $V(\eta) = S^1$ , there exists $Z\in{\mathcal{M}}_\textrm{loc}(\mathbb{P})\cap\mathcal{S}_{++}^{\textrm{unif}} \subseteq{\mathcal{S}}^{\textrm{conv}}$ such that $ZS/S^1 = Z(1,X) = (Z,ZX)$ is in ${\mathcal{M}}_\sigma(\mathbb{P})$ . Because $\mathcal{F}_0$ is trivial, we can take $Z_0=1$ , and because $X\ge0$ , $ZX\in{\mathcal{M}}_\sigma(\mathbb{P})$ is the same as $ZX\in{\mathcal{M}}_\textrm{loc}(\mathbb{P})$ . So we have a local $\mathbb{P}$ -martingale $Z>0$ for which ZX is also a local $\mathbb{P}$ -martingale, and this allows us to follow a familiar path as, for example, in the proof of Theorem 1 in [Reference Schweizer20].

Write $Z = \mathcal{E}(N)$ for a local $\mathbb{P}$ -martingale null at 0, and note that N is continuous because $\mathbb{F}$ is generated by a Brownian motion. Use the Galtchouk–Kunita–Watanabe decomposition to write $N = \nu \bullet M + L$ with $L\in{\mathcal{M}}_\textrm{loc}(\mathbb{P})$ strongly orthogonal to M; this means here $[M,L]\equiv0$ because M and L are both continuous. Now use the product rule, (4.9), and (4.10) to conclude that like ZX, also $\int Z\,\mathrm{d} A + \int Z \,\mathrm{d} [N,M] = \int Z(\lambda + \nu) \,\mathrm{d} [M]$ is a local $\mathbb{P}$ -martingale. This implies that $\lambda+\nu\equiv 0$ , and hence $N = -\lambda\bullet M + L$ , and so again by orthogonality of M and L that

\begin{equation*}K = [{-}\lambda \bullet M] \le [{-}\lambda \bullet M] + [L] = [N].\end{equation*}

But we also know that $Z = \mathcal{E}(N)$ is in $\mathcal{S}_{++}^{\textrm{unif}}$ ; so Lemma 4.1 yields $[N]_\infty < \infty$ a.s., and hence $K_\infty < \infty$ a.s., and this is (4.8).

We conclude this subsection with a few comments on the setting of Proposition 4.2.

Remark 4.5. (1) If $\sigma^{(1)}\equiv0\equiv\mu^{(1)}$ so that $S^1 \equiv 1$ then $h = (\sigma^{(2)})^2$ , (4.7) always holds, and (4.8) reduces to

\begin{equation*}\int_0^\infty \bigg( \frac{\mu^{(2)}_s}{\sigma^{(2)}_s} \bigg)^2 \,\mathrm{d} s < \infty\qquad\hbox{a.s.}\end{equation*}

If also $\mu^{(2)}$ and $\sigma^{(2)}$ are non-random and constant then (4.3) is the classic Black–Scholes model and satisfies ${\underline{\textrm{NUPBR}}}$ if and only if $\mu^{(2)}=0$ , i.e. if $S^2$ is a martingale. This is not surprising as ${\underline{\textrm{NUPBR}}}$ for $(1,S^2)$ is by Corollary 3.2 equivalent to classic NUPBR for $S^2$ .

(2) If $\sigma^{(1)} = \sigma^{(2)} = \sigma$ for a fixed process $\sigma>0$ then $h = 2\sigma^2 (1-\varrho)$ , and (4.8) reduces to

\begin{equation*}\int_0^\infty \frac{1}{2(1-\varrho_s)} \bigg( \frac{\mu^{(2)}_s - \mu^{(1)}_s}{\sigma_s} + \sigma_s (1-\varrho_s) \bigg)^2 \,\mathrm{d} s < \infty\qquad\hbox{a.s.}\end{equation*}

If we also impose (4.7) and assume that $\varrho\le 1-\delta$ for some constant $\delta>0$ , then (4.8) follows from

(4.11) \begin{equation}\int_0^\infty \bigg( \frac{\mu^{(2)}_s-\mu^{(1)}_s}{\sigma_s} \bigg)^2 \,\mathrm{d} s < \infty\qquad\hbox{a.s.}\end{equation}

(and this is even equivalent to (4.8) if, e.g., $\mu^{(2)}\ge \mu^{(1)}$ ). Equation (4.11) shows that, roughly speaking, in such a setting with identical volatilities for both assets, ${\underline{\textrm{NUPBR}}}$ for S forces $\mu^{(1)}\approx\mu^{(2)}$ at $\infty$ . But because of the correlation $\varrho$ , this does not necessarily imply $S^1 \approx S^2$ .

(3) By repeating the above analysis for a model on the (right-closed) interval [0, T], we could obtain a characterisation of ${\underline{\textrm{NUPBR}}}$ on [0, T] by replacing in (4.4) $\sup_{t\ge0}$ by $\sup_{0\le t\le T}$ and in (4.7) and (4.8) $\int_0^\infty$ by $\int_0^T$ . All these conditions are then automatically satisfied for any $T\in(0,\infty)$ if we assume, for instance, that $\mu^{(i)}$ , $\sigma^{(i)}$ and $\varrho$ are $\mathbb{P}$ -a.s. continuous (in s). This illustrates that ${\underline{\textrm{NUPBR}}}$ on $[0,\infty)$ is substantially stronger (or more restrictive) than ${\underline{\textrm{NUPBR}}}$ on [0, T] for every $T\in(0,\infty)$ . The same observation about NUPBR is made in [Reference Bálint and Schweizer3, Corollary 5.4]; see also the discussion at the end of Subsection 5.1.

(4) What can we say about NUPBR in this example? If we assume (4.4), then under each of (i) and (ii) in Proposition 4.2, S satisfies NUPBR and hence also NUPBR by (3.5). Moreover, $S^1$ is in $\mathcal{S}_{++}^{\textrm{unif}}$ by Corollary 4.1, so by Proposition 3.1, also $S/S^1 = (1,X)$ satisfies NUPBR and then also NUPBR. So $X = S^2/S^1$ satisfies classic NUPBR.

We can also obtain the last conclusion in a different way. If we assume (4.4) holds then in cases (i) and (ii), the proof of Proposition 4.2 yields a local martingale Z in $\mathcal{S}_{++}^{\textrm{unif}}$ , and hence in ${\mathcal{S}}^{\textrm{conv}}$ with $Z_\infty>0$ $\mathbb{P}$ -a.s., with the property that ZX is a local $\mathbb{P}$ -martingale. So ZY is a $\mathbb{P}$ -supermartingale for any $Y\in\mathcal{G}^1(1,X)$ , and Theorem 4.15 in [Reference Karatzas and Kardaras15] implies that X satisfies classic NUPBR. So $S/S^1 = (1,X)$ satisfies NUPBR.

4.4. Comparison with the literature

We conclude this section with a brief comparison to other AOA notions in the literature which are discounting-invariant in some form.

To the best of our knowledge, there is only one $\mathcal{S}^1_{++}$ -discounting-invariant FTAP for a process S on the right-open time interval $[0, \infty)$ . This is due to Bálint and Schweizer [Reference Bálint and Schweizer5]. The starting points of the present work and [Reference Bálint and Schweizer5] are similar. First a suitable class ${\mathcal{D}} \subseteq \mathcal{S}^1_{++}$ of discounters is fixed, and then an AOA concept which is discounting-invariant with respect to ${\mathcal{D}}$ is defined. Bálint and Schweizer [Reference Bálint and Schweizer5] use the largest class ${\mathcal{D}} = \mathcal{S}^1_{++}$ , whereas our work considers only the smaller set ${\mathcal{D}} = \mathcal{S}_{++}^{\textrm{unif}}$ . In both cases, the overall approach is in marked contrast to that of Delbaen and Schachermayer [Reference Delbaen and Schachermayer7], who first fix an AOA concept (NA or NFLVR) and then find a (relatively small) class of discounters that does not destroy the AOA property chosen; see [Reference Delbaen and Schachermayer7, Theorems 4.2 and 4.4].

The idea behind the AOA concept of DSV in [Reference Bálint and Schweizer5, Definition 2.8] is as follows. A priori, one fixes a reference strategy $\eta$ , which is a long-only (non-negative) strategy $\eta \in \Theta^{\mathrm{sf}}(S)$ satisfying $V(\eta) \in \mathcal{S}^1_{++}$ . Then $\textrm{DSV}(\eta)$ is defined as the zero strategy being strongly share maximal for  $\eta$ ; see [Reference Bálint and Schweizer5, Definition 2.6]. This approach is unique in the literature because share maximality compares two strategies via their numbers of shares, not via the value processes associated with the strategies. This is in contrast to the notion of strong value maximality, which appears in [Reference Herdegen11], where numeraire-independent no-arbitrage (NINA) is defined as the zero strategy being strongly value maximal.

Even though DSV and our notion of ${\overline{\textrm{NUPBR}}}$ look very different at first glance, statement (vii) in Theorem 4.1 also appears in the dual characterisation in [Reference Bálint and Schweizer5] and allows us to connect the two concepts. To apply Theorem 4.1 in [Reference Bálint and Schweizer5] and Theorem 4.1 at the same time, we assume that S is non-negative and that the reference strategy $\eta$ even satisfies $V(\eta) \in \mathcal{S}_{++}^{\textrm{unif}}$ . Then the connection between DSV and ${\overline{\textrm{NUPBR}}}$ is as follows.

Remark 4.6. (1) Under the assumptions of Corollary 4.2, $\textrm{DSV}(\eta)$ is more restrictive than NUPBR, and Example 3.2 shows that it can be strictly more restrictive. Indeed, the market $\,\tilde{\!S} \,:\!=\, DS$ with D and S as in Example 3.2 satisfies NUPBR, but ${\underline{\textrm{NUPBR}}}$ and thus also DSV for $\eta \equiv e_1\in\mathbb{R}^d$ (the buy-and-hold strategy of the first asset) fail.

On the other hand, Example 6.8 in [Reference Bálint and Schweizer5] shows that DSV for some reference strategy (even in the classic set-up) does not imply NUPBR. Thus, in general, we have

\begin{align*} \textrm{NUPBR}{} \quad\Longrightarrow\mskip-24mu/\mskip15mu\quad \textrm{DSV}(\eta), \\ \textrm{NUPBR}{} \quad\Longleftarrow\mskip-24mu/\mskip15mu\quad \textrm{DSV}(\eta). \end{align*}

But, of course, as seen in (3.5),

\begin{equation*} {\overline{\textrm{NUPBR}}} \quad\Longrightarrow\quad {\underline{\textrm{NUPBR}}} \quad\Longrightarrow\quad \textrm{NUPBR}{},\end{equation*}

and Theorem 3.2 gives equivalence if $\mathcal{G}^1(S) \cap {\mathcal{S}}^{\textrm{conv}} \neq \varnothing$ . In particular, this is the case for the classic set-up because $1\in\mathcal{G}^1(S)$ if $S^1\equiv 1$ . From this perspective, our concept of ${\overline{\textrm{NUPBR}}}$ (or ${\underline{\textrm{NUPBR}}}$ ) seems a more natural extension of the classic NUPBR condition in [Reference Delbaen and Schachermayer6], [Reference Delbaen and Schachermayer8], and [Reference Karatzas and Kardaras15]. However, in contrast to DSV, our AOA concepts are discounting-invariant with respect only to $\mathcal{S}_{++}^{\textrm{unif}}\subsetneqq\mathcal{S}^1_{++}$ , and Example 3.1 shows that this cannot be improved on $[0,\infty)$ .

(2) In the case of a right-closed time interval [0, T], Corollary 4.2 in particular shows that NUPBR is equivalent to NINA (see [Reference Herdegen11]), because the latter is simply strong value maximality of the zero strategy.

(3) In contrast to the dual characterisation in [Reference Bálint and Schweizer5, Theorem 4.1], (vii) of Theorem 4.1 tells us how at least one $\sigma$ -martingale discounter D looks—it has the form $D = Z/V(\eta)$ for some $Z\in{\mathcal{M}}_\textrm{loc}(\mathbb{P})$ and some $V(\eta)\in\mathcal{G}^1(S) \cap \mathcal{S}_{++}^{\textrm{unif}}$ . In [Reference Bálint and Schweizer5, Theorem 4.1, (v3)], we obtain only $D\in\mathcal{S}^1_{++}$ and $V(\eta)/D \in \mathcal{S}_{++}^{\textrm{unif}}$ for the reference strategy $\eta$ (which satisfies $V(\eta)\in\mathcal{S}^1_{++}$ ). In particular, whenever there exists $V(\eta) \in \mathcal{G}^1(S) \cap \mathcal{S}_{++}^{\textrm{unif}}$ such that $1/V(\eta)$ is locally bounded, we know that the discounter D in Theorem 4.1(vii) can be chosen to be a special semimartingale. This turns out to be very useful when we study random lower bounds in Subsection 5.3.

Remark 4.8. A setting without a tradable riskless asset or savings account is also discussed in [Reference Platen and Tappe17]. But apart from considering the simpler case [0, T] of a finite time horizon, this is also less general than our work for the following reasons.

(a) Characterisations of ‘NUPBR’ and ‘NFLVR’ by dual properties are discussed in [Reference Platen and Tappe17, Section 3]. We use quotation marks because, for instance, the property ‘NUPBR’ is defined there as $L^0$ -boundedness of the set

\begin{align*}\mathcal{I}^1_T(S)&\,:\!=\,\{ 1 + (\varphi\bullet S)_T \,:\, \varphi\in L(S), 1 + \varphi\bullet S \ge 0 \}\\&\phantom{:}\supsetneqq\{ 1 + (\varphi\bullet S)_T \,:\, \varphi\in \Theta^{\mathrm{sf}}(S), 1 + \varphi\bullet S \ge 0 \}=\mathcal{G}^1_T(S).\end{align*}

As explained before Proposition 2.3, $\mathcal{G}^1_T(S)$ has a financial meaning, but $\mathcal{I}^1_T(S)$ does not. Moreover, it can happen that S satisfies NUPBR (in our financial sense), but the purely mathematical condition ‘NUPBR’ fails; an example is given in [Reference Gebele10, Example 3.12].

(b) In [Reference Platen and Tappe17, Section 4] NUPBR and NFLVR (now without quotation marks) are studied for a market of dimension $d+1$ , consisting of $S\in\mathcal{S}^d_+$ and an extra tradable savings account B which is predictable and in $\textrm{FV}_{++}$ . This can be easily reduced to the classic set-up via Proposition 2.3.

5.1. NUPBR up to a stopping time

Recall that $\overline{\mathcal{T}}$ is the set of all $[0,\infty]$ -valued $\mathbb{F}$ -stopping-times and $\mathcal{T}\subseteq\overline{\mathcal{T}}$ consists of all finite-valued $\tau$ . The goal of this subsection is to make sense of NUPBR up to time $\tau$ for some $\tau \in \overline{\mathcal{T}}$ —we want to study AOA only on $[\![0, \tau]\!]$ , not necessarily on the entire space $[\![0,\infty]\!] = [\![ 0,\infty [\![\; = \Omega \times [0, \infty)$ . At first glance, for $\tau$ finite, it is tempting to define NUPBR up to time $\tau$ as $L^0$ -boundedness of the set

\begin{equation*} \mathcal{G}^1_\tau(S) \,:\!=\, \{X_\tau \,:\, X \in \mathcal{G}^1(S) \} =\{V_\tau(\varphi) \,:\, \varphi \in \Theta^{\mathrm{sf}}(S), V_0(\varphi)=1, V(\varphi) \ge 0 \}.\end{equation*}

It is immediate from (iv) in Theorem 4.1 that this set is bounded in $L^0$ under ${\underline{\textrm{NUPBR}}}$ and if $\mathcal{G}^1(S) \cap \mathcal{S}_{++}^{\textrm{unif}} \neq \varnothing$ . However, to study AOA on $[\![0,\tau]\!]$ , it does not make sense to insist on trading strategies that are self-financing and undefaultable on all of $[\![0,\infty]\!]$ , and we need for the self-financing property up to time $\tau$ only $S^\tau$ -integrability instead of S-integrability. So we consider the following strategies as self-financing on $[\![0,\tau]\!]$ .

The following result was pointed out to us by an unknown referee (and we gratefully acknowledge the referee’s insistence that it is true).

Lemma 5.1. We have

(5.1) \begin{equation}\Theta^{\mathrm{sf}}_{[\![0,\tau]\!]}(S)=\{ \varphi\ predictable \,:\, \varphi^\tau \in \Theta^{\mathrm{sf}}(S^\tau)\}.\end{equation}

Proof. By Definition 2.1, $\varphi^\tau\in\Theta^{\mathrm{sf}}(S^\tau)$ means (i) $\varphi^\tau\in L(S^\tau)$ , which is equivalent to $\varphi\in L(S^\tau)$ , and (ii) $\varphi^\tau \cdot S^\tau = \varphi^\tau_0 \cdot S^\tau_0 + \varphi^\tau \bullet S^\tau$ on $[\![ 0,\infty]\!]$ . The latter is equivalent to the two separate conditions

\begin{align*}\varphi\cdot S&= \varphi_0 \cdot S_0 + \varphi\bullet S\qquad\hbox{on $[\![ 0,\tau]\!]$,}\\\varphi_\tau \cdot S_\tau&=\varphi_0 \cdot S_0 + \varphi^\tau\bullet S^\tau\qquad\hbox{on $]\!] \tau, \infty]\!]$.}\end{align*}

But for $t>\tau$ , we have $(\varphi^\tau\bullet S^\tau)_t = \int_0^t \varphi^\tau_u \,\mathrm{d} S^\tau_u = \int_0^\tau \varphi^\tau_u \,\mathrm{d} S^\tau_u = \int_0^\tau \varphi_u \,\mathrm{d} S_u = (\varphi\bullet S)_\tau$ because  $S^\tau$ is constant after $\tau$ and $\varphi^\tau = \varphi$ , $S^\tau = S$ on $[\![ 0,\tau ]\!]$ . So the second condition simply says that $\varphi_\tau\cdot S_\tau = \varphi_0\cdot S_0 + (\varphi\bullet S)_\tau$ , and this already follows from the first condition. Therefore having (i) and (ii) from just above is equivalent to having (1) and (2) from Definition 5.1.

For each $\tau \in \overline{\mathcal{T}}$ , we consider the value processes

(5.2) \begin{equation}\mathcal{G}^1(S;\,\tau) \,:\!=\, \big\{ \big(V(\varphi)\big)^\tau \,:\, \hbox{$\varphi \in \Theta^{\mathrm{sf}}_{[\![0,\tau]\!]}(S)$, $V_0(\varphi) = 1$, $V(\varphi) \ge 0$ on $[\![0,\tau]\!]$} \big\}\end{equation}

and view them as reasonable investments up to time $\tau$ . Note that $X = X^\tau$ for any $X\in\mathcal{G}^1(S;\,\tau)$ , which is natural because we want to look at the market only up to time $\tau$ . As a consequence, we suggest the following AOA concepts up to time $\tau$ .

Definition 5.2. For $S \in \mathcal{S}^d$ and $\tau \in \overline{\mathcal{T}}$ , we say that S satisfies $\underline{\textit{NUPBR}}$ (respectively $\overline{\textit{NUPBR}}$ ) up to time $\tau$ ( $\underline{\textit{NUPBR}}_\tau$ respectively $\overline{\textit{NUPBR}}_\tau$ ) if the set

\begin{align*}\phantom{\Big(\hbox{resp.}}\,\underline{\mathcal{G}^1_{\tau}}(S;\,\tau)&\,:\!=\,\Big\{X_\tau \textbf{1}_{\{\tau < \infty\}} + \liminf_{t \to \infty} X_t \textbf{1}_{\{\tau = \infty\}} \,:\, X \in \mathcal{G}^1(S;\,\tau) \Big\}\\&\phantom{:}=\Big\{ \liminf_{t\to\infty} X^\tau_t \,:\, X \in \mathcal{G}^1(S;\,\tau) \Big\}\\\Big(\hbox{respectively}\ \overline{\mathcal{G}^1_{\tau}}(S;\,\tau)&\,:\!=\,\Big\{X_\tau \textbf{1}_{\{\tau < \infty\}} + \limsup_{t \to \infty} X_t \textbf{1}_{\{\tau = \infty\}} \,:\, X \in \mathcal{G}^1(S;\,\tau) \Big\}\\&\phantom{:}=\Big\{ \limsup_{t\to\infty} X^\tau_t \,:\, X \in \mathcal{G}^1(S;\,\tau) \Big\} \Big)\end{align*}

is bounded in $L^0$ .

As in Subsections 3.1 and 3.2, we want to look at ‘final values’ of processes $X\in\mathcal{G}^1(S;\,\tau)$ , and this is captured by using $\liminf$ (respectively $\limsup$ ) on $\{\tau = \infty\}$ . Note that in view of Lemma 5.1, we have $\underline{\mathcal{G}^1_{\tau}}(S;\,\tau) = \underline{\mathcal{G}^1_\infty}(S^\tau)$ .

We first provide a number of auxiliary results. The first is easy, and we only give the proof for completeness.

Proof. (i) Clearly $L(S^\tau)= L(S^\tau - S_0)$ , and $(S^i)^\tau - S^i_0= H^i \bullet S^i$ with $H^i \,:\!=\, \textbf{1}_{]\!]0, \tau]\!]}$ for all i. Now the second associativity in [Reference Shiryaev and Cherny21, Theorem 4.7] yields the claim.

(ii) For $\varphi \in L(S)$ , there exists a decomposition $S = S_0+M+A$ with $M \in {\mathcal{M}}_\textrm{loc}(\mathbb{P})$ and $A \in \textrm{FV}$ such that $\varphi \in L^1_\textrm{loc}(M) \cap L_\textrm{var}(A)$ . This implies $\varphi \textbf{1}_{]\!]0,\tau]\!]} \in L^1_\textrm{loc}(M) \cap L_\textrm{var}(A)$ and thus $\varphi \textbf{1}_{]\!]0,\tau]\!]} \in L(S)$ . Now we can use (i).

(iii) By (i) and (ii), we have for $\varphi \in L(S)$ that

\begin{equation*} \varphi \bullet S^\tau = (\varphi \textbf{1}_{]\!]0,\tau]\!]}) \bullet S = (\varphi^\tau \textbf{1}_{]\!]0,\tau]\!]}) \bullet S^\tau.\end{equation*}

So the first associativity in [Reference Shiryaev and Cherny21, Theorem 4.6] completes the proof.

Proof. Fix $\varphi \in \Theta^{\mathrm{sf}}(S)$ . Lemma 5.2 yields $\varphi \in L(S^\tau)$ , and $\varphi$ is in $\Theta^{\mathrm{sf}}_{[\![0, \tau]\!]}(S)$ because

\begin{align*} \varphi_0 \cdot S_0 + \varphi \bullet S^\tau = \varphi_0 \cdot S_0 + (\varphi \bullet S)^\tau = \big(V(\varphi)\big)^\tau \qquad\hbox{even on $[\![0,\infty]\!]$.} \end{align*}

From an economic point of view, it would be desirable to have ${\underline{\textrm{NUPBR}}}_\tau$ whenever ${\underline{\textrm{NUPBR}}}$ holds. Indeed, an agent who participates in the market S may stop trading at any time $\tau$ before reaching the time horizon. Consequently, a reasonable model should not only exclude arbitrage opportunities for those participants who continue trading until the (possibly infinite) end. However, the following example shows that even on the closed interval [0, T], this cannot hold in general.

Example 5.1. Consider $T=2$ and the deterministic price process $S = (S^1, S^2)$ with

\begin{align*} S_t^1 &\,:\!=\, 1- (t-1) \textbf{1}_{[1,2]}(t), \\ S_t^2 &\,:\!=\, 2 - t. \end{align*}

For any fixed $n \in \mathbb{N}$ , the constant strategy $\varphi \equiv (n, ({1-n})/{2})$ is self-financing for S, and its value process satisfies

\begin{equation*} V(\varphi) = nS^1 + \frac{1-n}{2}S^2 \ge 0.\end{equation*}

Since also $V_0(\varphi) = 1$ , this shows that $V(\varphi) \in \mathcal{G}^1(S)$ , and hence $ \mathcal{G}^1_1(S) \supseteq \{ ({n+1})/{2} \,:\, n \in \mathbb{N} \}$ is not bounded in $L^0$ . However, S satisfies NUPBR since $\mathcal{G}_2^1(S) = \{0\}$ .

Theorem 4.1 helps us to study how ${\underline{\textrm{NUPBR}}}$ behaves with respect to stopping S. For a given stopping time $\tau \in \overline{\mathcal{T}}$ , the set $L(S^\tau)$ of integrands might be strictly larger than L(S). So if we simply consider the set $\underline{\mathcal{G}^1_\infty}(S^\tau)$ , it is not clear whether this is bounded in $L^0$ whenever $\underline{\mathcal{G}^1_\infty}(S)$ is, i.e. when S satisfies ${\underline{\textrm{NUPBR}}}$ . However, with the dual characterisation, this can be answered easily.

We can now state the main result of this subsection.

Theorem 5.1 in particular shows for $\tau\equiv T$ that the condition ${\underline{\textrm{NUPBR}}}$ on $[0, \infty)$ for $S \in \mathcal{S}^d$ with $\mathcal{G}^1(S) \cap \mathcal{S}_{++}^{\textrm{unif}} \neq \varnothing$ implies NUPBR on any closed time interval [0, T] with $T>0$ and thus NA1—that is, ‘anything good costs something’; for a discussion, see, for example, [Reference Karatzas and Kardaras16, Subsection 2.2.2 on viability, and notes and complements for Chapter 2]. Therefore ${\underline{\textrm{NUPBR}}}$ excludes arbitrage opportunities in the sense of Schachermayer’s definition [Reference Schachermayer19] that ‘an arbitrage opportunity is the possibility to make a profit in a financial market without risk and without net investment of capital’. We emphasise that the converse is not true: Example 3.1 clearly shows that NA1, or equivalently NUPBR on any right-closed interval [0, T] with $T>0$ , can be strictly weaker than NUPBR on $[0, \infty)$ . In particular, NA1 is weaker than ${\underline{\textrm{NUPBR}}}$ on  $[0, \infty)$ .

5.2. Change of the market dimension

Fix $d, \,\tilde{\!d} \in \mathbb{N}$ . The goal of this subsection is to study how our AOA conditions behave under a change of the market dimension. More precisely, for a market $(S, \,\tilde{\!S}) \in \mathcal{S}^d \times\mathcal{S}^{\skew4\tilde{d}}$ , we are interested in the following questions.

1. (1) If the large market $(S,\,\tilde{\!S})$ satisfies ${\underline{\textrm{NUPBR}}}$ , what can we say about ${\underline{\textrm{NUPBR}}}$ for the small markets S and $\,\tilde{\!S}$ ?

2. (2) If the small markets S and $\,\tilde{\!S}$ each satisfy ${\underline{\textrm{NUPBR}}}$ , which additional assumptions are needed to establish ${\underline{\textrm{NUPBR}}}$ for the large market $(S, \,\tilde{\!S})$ ?

We start by investigating question (1). The next result lets us connect self-financing strategies of S and $\,\tilde{\!S}$ to a self-financing strategy for $(S,\,\tilde{\!S})$ . Its easy proof is omitted.