2. Basic notation and no-arbitrage concepts

In this section we introduce basic notation and the no-arbitrage concepts which we consider in this paper; see also [Reference Platen and Tappe55, Section 7]. Concerning the required theory of stochastic processes, we adopt the terminology of [Reference Jacod and Shiryaev39], where further details can be found.

From now on, let $T \in (0,\infty)$ be a finite time horizon, and let $(\Omega,\mathscr{F},(\mathscr{F}_t)_{t \in [0,T]},\mathbb{P})$ be a stochastic basis satisfying the usual conditions, such that $\mathscr{F}_0$ is $\mathbb{P}$ -trivial. We consider a market $\mathbb{S} = \{ S^i \,:\, i \in I \}$ consisting of nonnegative semimartingales $S^i \geq 0$ , the primary security accounts, for some index set $I \neq \emptyset$ . We denote by $\Delta(\mathbb{S})$ the set of all strategies for $\mathbb{S}$ ; these are the numbers of units of components of $\mathbb{S}$ held. For a strategy $\delta \in \Delta(\mathbb{S})$ we define the portfolio $S^{\delta} \,:\!=\, \delta \cdot S$ , where we recall that ‘ $\,\, \cdot \,$ ’ denotes the usual inner product in Euclidean space. Furthermore, we denote by $\Delta_{{\text{sf}}}(\mathbb{S})$ the set of all self-financing strategies for $\mathbb{S}$ , where changes in the portfolio value result only from changes in the primary security account values. Mathematically, this is expressed by the condition $S^{\delta} = S_0^{\delta} + \delta \bullet S$ , where ‘ $\,\, \bullet \,$ ’ denotes stochastic integration; see [Reference Jacod and Shiryaev39, Section I.4.d] for the stochastic integral with respect to a real-valued semimartingale, and [Reference Shiryaev and Cherny60] for vector stochastic integration. For $\alpha \geq 0$ and a strategy $\delta \in \Delta(\mathbb{S})$ we introduce the integral process $I^{\alpha,\delta} \,:\!=\, \alpha + \delta \bullet S$ . We call a process $B = (B_t)_{t \in [0,T]}$ a savings account (or a locally risk-free asset) if it is predictable, càdlàg, and of finite variation with $B_0 = 1$ and $B, B_- > 0$ . For another nonnegative semimartingale $Y \geq 0$ we agree on the notation

(2.1) \begin{align}\mathbb{S} Y \,:\!=\, \{ S^i Y \,:\, i \in I \}.\end{align}

Later on, we will often consider the discounted market $\mathbb{S} B^{-1}$ for some savings account B. For each $\alpha \geq 0$ we introduce the following sets of potential security processes:

• $\mathbb{I}_{\alpha}(\mathbb{S})$ consists of all integral processes $I^{\alpha,\delta}$ with $\delta \in \Delta(\mathbb{S})$ .

• $\mathbb{I}_{\alpha}^{{\text{adm}}}(\mathbb{S})$ consists of all admissible integral processes from $\mathbb{I}_{\alpha}(\mathbb{S})$ . Recall that a process X is called admissible if $X \geq -a$ for some constant $a \in \mathbb{R}_+$ .

• $\mathbb{I}_{\alpha}^+(\mathbb{S})$ consists of all nonnegative integral processes from $\mathbb{I}_{\alpha}(\mathbb{S})$ .

• $\mathbb{P}_{{\text{sf}},\alpha}(\mathbb{S})$ consists of all self-financing portfolios starting in $\alpha$ .

• $\mathbb{P}_{{\text{sf}},\alpha}^{{\text{adm}}}(\mathbb{S})$ consists of all admissible self-financing portfolios starting in $\alpha$ .

• $\mathbb{P}_{{\text{sf}},\alpha}^+(\mathbb{S})$ consists of all nonnegative self-financing portfolios starting in $\alpha$ .

In practice, under limited liability, market participants can only exploit forms of arbitrage with the nonnegative self-financing portfolio of their entire wealth, starting with their initial wealth $\alpha$ . This makes the set $\mathbb{P}_{{\text{sf}},\alpha}^+(\mathbb{S})$ rather special from a practical perspective, because it captures natural constraints that exist for exploiting possible forms of arbitrage. Most of the other sets of security processes introduced above remain arguably of a more purely theoretical nature. Now, let $(\mathbb{K}_{\alpha})_{\alpha \geq 0}$ be any of the above families of potential security processes. Denoting by $L^0 = L^0(\Omega,\mathscr{F}_T,\mathbb{P})$ the space of all equivalence classes of real-valued random variables, where two random variables are identified if they coincide $\mathbb{P}$ -almost surely, we define the family $(\mathscr{K}_{\alpha})_{\alpha \geq 0}$ of subsets of $L^0$ as

\begin{align*}\mathscr{K}_{\alpha} \,:\!=\, \{ X_T \,:\, X \in \mathbb{K}_{\alpha} \}, \quad \alpha \geq 0.\end{align*}

We may think of outcomes of trading strategies with initial value $\alpha$ . The definition above provides us with the families

\begin{align*}&(\mathscr{I}_{\alpha}(\mathbb{S}))_{\alpha \geq 0}, \qquad (\mathscr{I}_{\alpha}^{{\text{adm}}}(\mathbb{S}))_{\alpha \geq 0}, \qquad (\mathscr{I}_{\alpha}^+(\mathbb{S}))_{\alpha \geq 0},\\ &(\mathscr{P}_{{\text{sf}},\alpha}(\mathbb{S}))_{\alpha \geq 0}, \qquad (\mathscr{P}_{{\text{sf}},\alpha}^{{\text{adm}}}(\mathbb{S}))_{\alpha \geq 0}, \qquad (\mathscr{P}_{{\text{sf}},\alpha}^+(\mathbb{S}))_{\alpha \geq 0}\end{align*}

of outcomes of security processes. Denoting by $L_+^0$ the convex cone of all nonnegative random variables, and by $L^{\infty}$ the space of all bounded random variables, we define the convex cone $\mathscr{C} \subset L^{\infty}$ as

\begin{align*}\mathscr{C} \,:\!=\, (\mathscr{K}_0 - L_+^0) \cap L^{\infty}.\end{align*}

Moreover, we define the family $(\mathscr{B}_{\alpha})_{\alpha \geq 0}$ of subsets of $L_+^0$ as

\begin{align*}\mathscr{B}_{\alpha} \,:\!=\, (\mathscr{K}_{\alpha} - L_+^0) \cap L_+^0, \quad \alpha \geq 0.\end{align*}

We may think of all nonnegative elements which are equal to or below the outcome of a trading strategy with initial value $\alpha$ . Setting $\mathscr{B} \,:\!=\, \mathscr{B}_1$ , by [Reference Platen and Tappe55, Lemma 3.11] we have $\mathscr{B}_{\alpha} = \alpha \mathscr{B}$ for each $\alpha > 0$ , and hence for the upcoming no-arbitrage concepts it suffices to consider $\mathscr{K}_1$ rather than the family $(\mathscr{K}_{\alpha})_{\alpha > 0}$ . The Minkowski functional $p_{\mathscr{B}} \,:\, L^0 \to [0,\infty]$ is given by

\begin{align*}p_{\mathscr{B}}(\xi) = \inf \{ \alpha > 0 \,:\, \xi \in \mathscr{B}_{\alpha} \}, \quad \xi \in L^0.\end{align*}

Note that $p_{\mathscr{B}}(\xi)$ has the interpretation of the minimal superreplication price of $\xi$ . In the following definition we denote by $L_+^{\infty}$ the convex cone of all bounded, nonnegative random variables.

Remark 2.1. Note that the above list covers several well-known no-arbitrage concepts. Recall that an arbitrage opportunity is an element $X \in \mathscr{K}_0$ such that

\begin{align*}\mathbb{P}(X \geq 0) = 1 \quad \text{and} \quad \mathbb{P}(X > 0) > 0.\end{align*}

Therefore, the condition $\mathscr{K}_0 \cap L_+^0 = \{ 0 \}$ just means that there are no arbitrage opportunities, which explains the no-arbitrage concept NA. Often it is easy to find mathematical conditions which are sufficient for NA, but typically they fail to be necessary. There are two approaches to defining slightly stronger no-arbitrage concepts to overcome this problem:

• Observing that NA can equivalently be expressed as $\mathscr{C} \cap L_+^{\infty} = \{ 0 \}$ , we can impose stronger conditions of the type $\overline{\mathscr{C}}^{\tau} \cap L_+^{\infty} = \{ 0 \}$ , where the closure is taken with respect to some topology $\tau$ on $L^{\infty}$ . The no-arbitrage concepts NFL, NFLBR, and NFLVR introduced above are specific examples. Note that all these no-arbitrage concepts are related to $\mathscr{K}_0$ , which has the interpretation of outcomes of trading strategies with initial value zero.

• Another approach is to consider no-arbitrage concepts which are related to $(\mathscr{K}_{\alpha})_{\alpha > 0}$ , the outcomes of trading strategies with strictly positive initial value. Then the idea is that in a reasonable market it should not be possible to make unbounded profit when investing money in that market. With our notation, this means that the set $\mathscr{B}$ should be bounded in some sense, which is expressed by the no-arbitrage concepts NUPBR and NAA $_1$ . The no-arbitrage concept NA $_1$ means that a strictly positive payoff $\xi \in L_+^0 \setminus \{ 0 \}$ can only be replicated with strictly positive initial wealth.

The pricing under a given no-arbitrage concept is typically aligned to a so-called deflator. To introduce this concept, let $\mathbb{X}$ be a family of semimartingales. For a semimartingale $Z = (Z_t)_{t \in [0,T]}$ such that $Z,Z_- > 0$ , we use the following terminology:

1. 1. Z is an equivalent martingale deflator (EMD) for $\mathbb{X}$ if X Z is a martingale for all $X \in \mathbb{X}$ .

2. 2. Z is an equivalent local martingale deflator (ELMD) for $\mathbb{X}$ if X Z is a local martingale for all $X \in \mathbb{X}$ .

3. 3. Z is an equivalent $\sigma$ -martingale deflator (E $\mathit{\Sigma}$ MD) for $\mathbb{X}$ if X Z is a $\sigma$ -martingale for all $X \in \mathbb{X}$ .

A pricing rule can often also be characterized by a so-called pricing measure. More precisely, for an equivalent probability measure $\mathbb{Q} \approx \mathbb{P}$ on $(\Omega,\mathscr{F}_T)$ , we use the following terminology:

1. 1. $\mathbb{Q}$ is an equivalent martingale measure (EMM) for $\mathbb{X}$ if X is a $\mathbb{Q}$ -martingale for all $X \in \mathbb{X}$ .

2. 2. $\mathbb{Q}$ is an equivalent local martingale measure (ELMM) for $\mathbb{X}$ if X is a $\mathbb{Q}$ -local martingale for all $X \in \mathbb{X}$ .

3. 3. $\mathbb{Q}$ is an equivalent $\sigma$ -martingale measure (E $\mathit{\Sigma}$ MM) for $\mathbb{X}$ if X is a $\mathbb{Q}$ - $\sigma$ -martingale for all $X \in \mathbb{X}$ .

The required results about the concepts just introduced are presented in Appendix C. For this, the following two results are essential tools; Proposition 2.2 will also be useful for the study of dynamic trading strategies later on in Section 9. For what follows, we recall that $\mathscr{S}$ denotes the space of all semimartingales (see [Reference Jacod and Shiryaev39, Section I.4.c]), that $\mathscr{M}_{{\text{loc}}}$ denotes the space of all local martingales (see [Reference Jacod and Shiryaev39, Section I.1.e]), and that $\mathscr{M}_{\sigma}$ denotes the space of all $\sigma$ -martingales (see [Reference Jacod and Shiryaev39, Section III.6.e]).

Proof. Let $i \in \{ 1,\ldots,d \}$ be arbitrary. Using integration by parts (see [Reference Jacod and Shiryaev39, Definition I.4.45]), we have

\begin{align*}X_0^i Y_0 + X_-^i \bullet Y + Y_- \bullet X^i + [X^i,Y] = X^i Y \in \mathscr{M}_{\sigma}.\end{align*}

Since $Y \in \mathscr{M}_{\sigma}$ , we have $X_-^i \bullet Y \in \mathscr{M}_{\sigma}$ , and hence

\begin{align*}Y_- \bullet X^i + [X^i,Y] \in \mathscr{M}_{\sigma}.\end{align*}

By Lemma A.3 we have $H \in L([X,Y])$ and

\begin{align*}[H \bullet X,Y] = H \bullet [X,Y].\end{align*}

Since $Y_-$ is predictable and locally bounded, by Lemma A.2 we have

\begin{align*}Y_- \in L(H \bullet X), \quad Y_- H \in L(X), \quad H \in L(Y_- \bullet X)\end{align*}

and

\begin{align*}Y_- \bullet (H \bullet X) = (Y_- H) \bullet X = H \bullet (Y_- \bullet X).\end{align*}

Therefore, using integration by parts (see [Reference Jacod and Shiryaev39, Definition I.4.45]) again, we obtain

\begin{align*}(H \bullet X) Y &= (H \bullet X)_- \bullet Y + Y_- \bullet (H \bullet X) + [H \bullet X,Y]\\ &= (H \bullet X)_- \bullet Y + H \bullet (Y_- \bullet X) + H \bullet [X,Y]\\ &= (H \bullet X)_- \bullet Y + H \bullet ( Y_- \bullet X + [X,Y] ) \in \mathscr{M}_{\sigma},\end{align*}

completing the proof.

Proposition 2.2. Let $X \in \mathscr{S}^d$ and $Z \in \mathscr{S}$ be such that $X^i Z \in \mathscr{M}_{\sigma}$ for each $i = 1,\ldots,d$ . Then for every $H \in L(X)$ and every $Y \in \mathscr{M}_{\sigma}$ with $[Y,Z] = 0$ and

(2.2) \begin{align}H \cdot X = H_0 \cdot X_0 + H \bullet X + Y,\end{align}

the process Y is predictable, and we have $(H \cdot X) Z \in \mathscr{M}_{\sigma}$ .

Proof. By (2.2) we have $H \cdot X \in \mathscr{S}$ . Using integration by parts (see [Reference Jacod and Shiryaev39, Definition I.4.45]), we have

(2.3) \begin{equation}\begin{aligned}(H \cdot X - Y) Z &= (H_0 \cdot X_0 - Y_0) Z_0 + (H \cdot X - Y)_- \bullet Z\\ &\quad + Z_- \bullet (H \cdot X - Y ) + [H \cdot X - Y,Z].\end{aligned}\end{equation}

By (2.2) and Lemma A.3 we have $H \in L_{\text{var}}([X,Z])$ and

(2.4) \begin{align}[H \cdot X - Y,Z] = [H \bullet X,Z] = H \bullet [X,Z].\end{align}

Furthermore, we have

\begin{align*}(H \cdot X - Y)_- &= (H \cdot X - Y) - \Delta ( H \cdot X - Y ) = ( H \cdot X - Y ) - \Delta ( H \bullet X )\\ &= ( H \cdot X - Y ) - H \cdot \Delta X = H \cdot X_- - Y.\end{align*}

Therefore, we have

\begin{align*}\Delta Y = H \cdot X_- - (H \cdot X)_-,\end{align*}

showing that the process Y is predictable. By [Reference Jacod and Shiryaev39, Theorem I.4.52], the assumption $[Y,Z] = 0$ implies that

\begin{align*}\sum_{s \leq t} \Delta Y_s \Delta Z_s = 0, \quad t \in \mathbb{R}_+,\end{align*}

and hence we have $\Delta Y \in L(Z)$ with $\Delta Y \bullet Z = 0$ . Therefore, we also have $Y \in L(Z)$ with $Y \bullet Z = Y_- \bullet Z$ . Furthermore, by Lemma A.1 and [Reference Shiryaev and Cherny60, Theorem 4.7] we have $H X_- \in L(Z \textbf{1}_{\mathbb{R}^d})$ , $H \in L(X_- \bullet Z)$ , and

(2.5) \begin{equation}\begin{aligned}(H \cdot X - Y)_- \bullet Z &= (H \cdot X_- - Y) \bullet Z = (H X_-) \bullet (Z \textbf{1}_{\mathbb{R}^d}) - Y \bullet Z\\ &= H \bullet (X_- \bullet Z) - Y_- \bullet Z,\end{aligned}\end{equation}

where $X_- \bullet Z$ denotes the $\mathbb{R}^d$ -valued process with components $(X_- \bullet Z)^i = X_-^i \bullet Z$ for each $i=1,\ldots,d$ . Furthermore, by Lemma A.2 we have $H \in L(Z_- \bullet X)$ and

(2.6) \begin{align}Z_- \bullet (H \cdot X - Y) = Z_- \bullet (H \bullet X) = H \bullet (Z_- \bullet X),\end{align}

where $Z_- \bullet X$ denotes the $\mathbb{R}^d$ -valued process with components $(Z_- \bullet X)^i = Z_- \bullet X^i$ for each $i=1,\ldots,d$ . Consequently, using (2.3)–(2.6), integration by parts, and the assumption $[Y,Z] = 0$ , we deduce that

\begin{align*}(H \cdot X) Z &= (H \cdot X - Y) Z + YZ\\ &= (H_0 \cdot X_0 - Y_0) Z_0 + H \bullet \big( X_- \bullet Z + Z_- \bullet X + [X,Z] \big) - Y_- \bullet Z + YZ\\ &= (H_0 \cdot X_0) Z_0 + H \bullet (XZ) + Z_- \bullet Y \in \mathscr{M}_{\sigma},\end{align*}

where $XZ \in \mathscr{M}_{\sigma}^d$ denotes the $\mathbb{R}^d$ -valued $\sigma$ -martingale with components $X^i Z$ for each $i = 1,\ldots,d$ .

3. The fundamental theorems of asset pricing revisited

In this section we review the fundamental theorems of asset pricing in our present framework for the given no-arbitrage concepts, and present some extensions. The mathematical framework is that of Section 2; however, we now consider a finite market $\mathbb{S} = \{ S^1,\ldots,S^d \}$ with nonnegative semimartingales for some $d \in \mathbb{N}$ . Let us first formulate a theorem that links weaker no-arbitrage concepts related to NUPBR, which allow a richer modeling world than those we link later on to NFLVR.

If the previous conditions are fulfilled, then $\mathscr{I}_0^+(\mathbb{S})$ satisfies NFL, NFLBR, NFLVR, and NA.

Recall that the securities in the set $\mathscr{I}_1^+(\mathbb{S})$ represent the nonnegative sums of the initial security value one and integrals of strategies with respect to primary security accounts. Particularly important in this result is the fact that NUPBR is equivalent to the existence of an ELMD which is a local martingale.

Let us now link no-arbitrage concepts related to the stronger, widely assumed NFLVR condition.

Recall that the set $\mathscr{I}_0^{{\text{adm}}}(\mathbb{S})$ relates to admissible securities that are the integral of strategies with respect to primary security accounts and can get negative down to a certain level. It is important to note that NFLVR is equivalent to the existence of a deflator that is a true martingale, which underpins risk-neutral pricing.

4. The main results and their consequences

In this section we present our main results and show some of their consequences. We emphasize that the proofs of the main results presented in this section are straightforward, and essentially rely on the FTAPs revisited in Section 3. As in Section 3, we consider a finite market $\mathbb{S} = \{ S^1,\ldots,S^d \}$ with nonnegative semimartingales for some $d \in \mathbb{N}$ . Recall that a semimartingale Z with $Z,Z_- > 0$ is called a multiplicative special semimartingale if it admits a multiplicative decomposition

(4.1) \begin{align}Z = DC\end{align}

with a local martingale $D > 0$ and a predictable càdlàg process $C > 0$ of finite variation. Then by [Reference Jacod and Shiryaev39, Lemma III.3.6] we also have $D_- > 0$ , and hence $C_- > 0$ .

In the situation of Theorem 4.1 we call D the local martingale part and C the finite-variation part of the multiplicative decomposition (4.1).

Now we are ready to state our result about the no-arbitrage concept NUPBR under the two natural constraints of nonnegativity and self-financing, which are important in practice and give access to a wide range of financial market models.

Important aspects of the previous result are the fact that NUPBR is equivalent to NAA $_1$ and NA $_1$ , and the existence of a deflator which is a multiplicative special semimartingale.

The result is also in line with the concept of numéraire-independent no-arbitrage (NINA) from [Reference Herdegen34], which means (in the mathematical framework of [Reference Herdegen34]) that the zero strategy is strongly maximal for the strategy cone of all self-financing, undefaultable strategies; see [Reference Herdegen34, Definition 3.21] for the precise definition. Verbally, the definition says that every nonzero (nonnegative) contingent claim must have a positive (superreplication) price. Moreover, as the name NINA suggests, it does not assume the existence of a numéraire strategy; see the paragraphs after [Reference Herdegen34, Definition 3.21], which discuss the relationships between NINA and other no-arbitrage concepts. For the proof of the following result we will use [Reference Herdegen34, Theorem 4.10], where NINA is characterized for so-called numéraire markets. In (4.2) below, $\mathscr{S}$ denotes the space of all semimartingales.

Proof. First of all, note that for every savings account B we have

\begin{align*}\inf_{0 \leq t \leq T} \bigg( \sum_{i=1}^d |S_t^i| + |B_t| \bigg) \geq \inf_{0 \leq t \leq T} B_t > 0 \quad \text{$\mathbb{P}$-almost surely,}\end{align*}

showing that (4.2) is a market in the sense of [Reference Herdegen34, Definition 2.3]. Furthermore, the market (4.2) is a numéraire market in the sense of [Reference Herdegen34, Definition 2.10], because $e_{d+1} = (0,\ldots,0,1)$ is a numéraire strategy. This ensures that we may apply [Reference Herdegen34, Theorem 4.10] in the sequel.

(i) $\Rightarrow$ (ii): By [Reference Herdegen34, Theorem 4.10] there exists a semimartingale Z with $Z,Z_- > 0$ such that $(S^1 Z,\ldots,S^d Z,B Z)$ is a $\mathbb{P}$ -local-martingale representative; that is, we have $S^1 Z,\ldots,S^d Z,B Z \in \mathscr{M}_{{\text{loc}}}$ , showing that Z is an ELMD for $\mathbb{S}$ . Furthermore, the process Z is a multiplicative special semimartingale with multiplicative decomposition $Z = D B^{-1}$ , where $D = BZ$ .

(ii) $\Rightarrow$ (i): There exist a local martingale $D \in \mathscr{M}_{{\text{loc}}}$ with $D > 0$ and a savings account B such that $Z = D B^{-1}$ . Since Z is an ELMD for $\mathbb{S}$ , we have $S^1 Z, \ldots, S^d Z, BZ \in \mathscr{M}_{{\text{loc}}}$ ; that is, the process $(S^1 Z, \ldots, S^d Z, BZ)$ is $\mathbb{P}$ -local-martingale representative in the terminology of [Reference Herdegen34]. Therefore, by [Reference Herdegen34, Theorem 4.10], the market (4.2) satisfies NINA.

Next we present our result concerning the no-arbitrage concept NFLVR, which is more restrictive than NUPBR.

If the previous conditions are fulfilled, then the savings accounts B in (i)–(iv) can be chosen to be equal, and in (v) we can choose an ELMD Z for $\mathbb{S}$ with multiplicative decomposition $Z = D B^{-1}$ with this savings account B. Furthermore, $\mathscr{P}_{{\text{sf}},0}^{{\text{adm}}}(\mathbb{S} \cup \{ B \})$ satisfies NA, and $\mathscr{P}_{{\text{sf}},1}^{{\text{adm}}}(\mathbb{S} \cup \{ B \})$ satisfies NA $_1$ , NAA $_1$ , and NUPBR.

In the previous result, it is important to note that NFLVR is equivalent to the existence of an ELMM $\mathbb{Q} \approx \mathbb{P}$ , which underpins the practice of risk-neutral pricing. Furthermore, when NFLVR is imposed, NUPBR still holds. However, this is in general not true the other way around. Indeed, if there exists a deflator as in Theorem 4.2 which is a multiplicative special semimartingale, then NFLVR only holds true if the local martingale part is a true martingale, which then gives rise to the aforementioned measure change. This means that assuming NUPBR does not give away anything in modeling freedom, but postulating NFLVR, which is what most of the literature does, restricts the phenomena one can describe for a market model.

Before we proceed with the question of how to extend a market by a savings account, let us prepare an auxiliary result.

Proof. Using integration by parts (see [Reference Jacod and Shiryaev39, Definition I.4.45]), we have

\begin{align*}MA = M_0 A_0 + M_- \bullet A + A_- \bullet M + [M,A].\end{align*}

By [Reference Jacod and Shiryaev39, Proposition I.4.49.c] we have $[M,A] \in \mathscr{M}_{{\text{loc}}}$ , and hence $M_- \bullet A \in \mathscr{M}_{{\text{loc}}} \cap \mathscr{V}$ . Since this process is also predictable, by [Reference Jacod and Shiryaev39, Corollary I.3.16] we deduce that $M_- \bullet A = 0$ . Since $M > 0$ , by [Reference Jacod and Shiryaev39, Lemma III.3.6] we also have $M_- > 0$ , and it follows that $A = A_0$ up to an evanescent set.

As we have seen, NUPBR represents, in the sense described, the least restrictive no-arbitrage concept, and we may assume it for our next result. In the previous results, the savings account B could already be contained in the market $\mathbb{S}$ . As the next result shows, an arbitrage-free market can have at most one savings account.

In the situation of the previous results, the savings account B, and hence the ELMD $Z = D B^{-1}$ , do not need to be unique. However, as the following result shows, for a given local martingale $D > 0$ there is at most one suitable savings account fitting into the multiplicative decomposition $Z = D B^{-1}$ . For this result, we consider a general market $\mathbb{S} = \{ S^i \,:\, i \in I \}$ with an arbitrary index set $I \neq \emptyset$ .

Proof. We have $S^i D B^{-1} \in \mathscr{M}_{{\text{loc}}}$ and $S^i D \hat{B}^{-1} \in \mathscr{M}_{{\text{loc}}}$ . Note that $A \,:\!=\, B \hat{B}^{-1}$ is another savings account, and that

\begin{align*}(S^i D B^{-1}) A \in \mathscr{M}_{{\text{loc}}}.\end{align*}

Applying Lemma 4.2 gives us that $A = 1$ up to an evanescent set, and hence we have $B = \hat{B}$ up to an evanescent set.

5. Semimartingale term structure models

In this section we discuss the connections with the results of [Reference Döberlein and Schweizer22] concerning semimartingale term structure models. We consider a market $\mathbb{S} = \{ P^S \,:\, S \in [0,T] \}$ consisting of zero-coupon bonds. Assume that $P^S,P_-^S > 0$ and $P_S^S = 1$ for each $S \in [0,T]$ . We call the market $\mathbb{S}$ a semimartingale term structure model, or simply a term structure model. The following concepts of a generated term structure model and an implied savings account are provided in [Reference Döberlein and Schweizer22] in a risk-neutral framework. We recall and extend these notions to the present situation with the real-world probability measure as follows.

We close this section with the following result about the uniqueness of an implied savings account.

6. Equivalent local martingale deflators

In this section we clarify when a deflator which is a multiplicative special semimartingale exists. Recall from Theorem 4.2 and Remark 4.3 that this ensures that the market satisfies NUPBR for all its nonnegative, self-financing portfolios.

Let $\mathbb{S} = \{ S^i \,:\, i \in I \}$ be a financial market consisting of nonnegative semimartingales $S^i \geq 0$ with an arbitrary index set $I \neq \emptyset$ . We assume that for each $i \in I$ the semimartingale $S^i$ cannot revive from bankruptcy, which means that $S_t^i = 0$ for all $t \geq T_i$ , where $T_i$ denotes the bankruptcy time of $S^i$ , given by

\begin{align*}T_i \,:\!=\, \inf \{ t \in \mathbb{R}_+ \,:\, S_{t-}^i = 0 \text{ or } S_t^i = 0 \}.\end{align*}

Then we have

(6.1) \begin{align}S^i = S_0^i \mathscr{E}(X^i),\end{align}

where $X^i$ denotes the semimartingale

\begin{align*}X^i \,:\!=\, (S_-^i)^{-1} \textbf{1}_{[\![ 0,T_i ]\!]} \bullet S^i,\end{align*}

and where $\mathscr{E}$ denotes the stochastic exponential; see [Reference Jacod and Shiryaev39, Section I.4.f]. Note that $X_0^i = 0$ and $X^i = (X^i)^{T_i}$ . Before we proceed, we recall that $\mathscr{V}$ denotes the space of all càdlàg, adapted processes of locally finite variation starting in zero, and that $\mathscr{A}_{{\text{loc}}}$ denotes the space of all locally integrable processes from $\mathscr{V}$ ; see [Reference Jacod and Shiryaev39, Section I.3.a]. Furthermore, we recall that $\mathscr{S}$ denotes the space of all semimartingales, and that $\mathscr{S}_p$ denotes the space of all special semimartingales; see [Reference Jacod and Shiryaev39, Section I.4.c]. For each truncation function $h^i \in \mathscr{C}_t$ (see [Reference Jacod and Shiryaev39, Definition II.2.3]) we have

\begin{align*}X^i = X^i(h^i) + \breve{X}^i(h^i),\end{align*}

where $\breve{X}^i(h^i) \in \mathscr{V}$ and $X^i(h^i) \in \mathscr{S}_p$ are defined as

\begin{align*}\breve{X}^i(h^i) &\,:\!=\, \sum_{s \leq \bullet} [ \Delta X_s^i - h^i(\Delta X_s^i) ],\\ X^i(h^i) &\,:\!=\, X - \breve{X}^i(h^i).\end{align*}

Here for every optional process H we agree that $\sum_{s \leq \bullet} H_s$ denotes the process given by $\sum_{s \leq t} H_s$ for each $t \in \mathbb{R}_+$ , provided the series are convergent. We denote by

\begin{align*}X^i(h^i) = M^i(h^i) + A^i(h^i)\end{align*}

the canonical decomposition of the special semimartingale $X^i(h^i)$ . Then the semimartingale $X^i$ has the decomposition

(6.2) \begin{align}X^i = M^i(h^i) + A^i(h^i) + \breve{X}^i(h^i).\end{align}

Let Z be a multiplicative special semimartingale with multiplicative decomposition $Z = D B^{-1}$ for some $D \in \mathscr{M}_{{\text{loc}}}$ with $D_0 = 1$ and a savings account B. Then we have $D = \mathscr{E}({-}\Theta)$ for some $\Theta \in \mathscr{M}_{{\text{loc}}}$ with $\Theta_0 = 0$ and $\Delta \Theta < 1$ , and $B = \mathscr{E}(R)$ for some predictable process $R \in \mathscr{V}$ with $\Delta R > -1$ . Indeed, these two processes are given by the stochastic logarithms $\Theta = - D_-^{-1} \bullet D$ and $R = B_-^{-1} \bullet B$ ; see [Reference Jacod and Shiryaev39, Section II.8.a]. We call $\Theta$ the market price of risk and R the locally risk-free return or virtual short rate of the savings account B. In the subsequent results, all upcoming identities are meant up to an evanescent set. Furthermore, we recall that for each $A \in \mathscr{A}_{{\text{loc}}}$ the process $A^p$ denotes its predictable compensator; see [Reference Jacod and Shiryaev39, Section I.3.b].

We will provide the proof of Theorem 6.1 below. Now, let us assume that for each $i \in I$ the process $X^i$ is a special semimartingale, and consider its canonical decomposition

\begin{align*}X^i = M^i + A^i.\end{align*}

Furthermore, let us agree on the notation $\langle M,N \rangle \,:\!=\, [M,N]^p$ for all $M,N \in \mathscr{M}_{{\text{loc}}}$ with $[M,N] \in \mathscr{A}_{{\text{loc}}}$ . By [Reference Jacod and Shiryaev39, Proposition I.4.50.b], this is consistent with the definition of the predictable quadratic covariation. The proof of the following result is similar to that of Theorem 6.1; indeed, the arguments are even simpler.

Now we prepare the proof of Theorem 6.1 For this purpose, we provide some auxiliary results. The predictable process $\widetilde{R} \in \mathscr{V}$ given by

\begin{align*}\widetilde{R} \,:\!=\, \frac{1}{1 + \Delta R} \bullet R\end{align*}

satisfies $\Delta \widetilde{R} < 1$ , and we have $\widetilde{R}^c = R^c$ . Since the inverse of $({-}1,\infty) \to ({-}\infty,1)$ , $x \mapsto x/(1+x)$ is given by $({-}\infty,1) \to ({-}1,\infty)$ , $x \mapsto x/(1-x)$ , we have

\begin{align*}R = \frac{1}{1 - \Delta \widetilde{R}} \bullet \widetilde{R}.\end{align*}

Therefore, it follows that

(6.6) \begin{align}\Delta \widetilde{R} = \frac{\Delta R}{1 + \Delta R}, \quad \Delta R = \frac{\Delta \widetilde{R}}{1 - \Delta \widetilde{R}}, \quad (1 + \Delta R) (1 - \Delta \widetilde{R}) = 1, \quad R = \widetilde{R} + [R,\widetilde{R}],\end{align}

and by Yor’s formula (see [Reference Jacod and Shiryaev39, II.8.19]) we obtain

\begin{align*}B^{-1} = \mathscr{E}({-}\widetilde{R}).\end{align*}

Next we define the two processes

(6.7) \begin{align}\widetilde{\Theta} &\,:\!=\, \Theta - [\Theta,\widetilde{R}] = (1 - \Delta \widetilde{R}) \bullet \Theta,\end{align}

(6.8) \begin{align} Y &\,:\!=\, \widetilde{\Theta} + \widetilde{R}.\end{align}

Then we have $\widetilde{\Theta} \in \mathscr{M}_{{\text{loc}}}$ with $\widetilde{\Theta}_0 = 0$ and $\widetilde{\Theta}^c = \Theta^c$ , as well as $Y \in \mathscr{S}_p$ with $Y_0 = 0$ and $\Delta Y < 1$ . Furthermore, we have

\begin{align*}\Theta = \widetilde{\Theta} + [\widetilde{\Theta},R] = (1 + \Delta R) \bullet \widetilde{\Theta}\end{align*}

and

(6.9) \begin{align}\frac{\Delta Y - 1}{1 - \Delta \widetilde{R}} &= \Delta \Theta - 1.\end{align}

By Yor’s formula (see [Reference Jacod and Shiryaev39, II.8.19]) we obtain

(6.10) \begin{align}Z = D B^{-1} = \mathscr{E}({-}\Theta) \mathscr{E}({-}\widetilde{R}) = \mathscr{E}({-}Y).\end{align}

Proof. (i) $\Rightarrow$ (ii): This implication follows because $BZ = D \in \mathscr{M}_{{\text{loc}}}$ .

(ii) $\Rightarrow$ (iii): Let $i \in I$ and $h^i \in \mathscr{C}_t$ be arbitrary. By Lemma 6.1 we have $X^i - Y - [X^i,Y] \in \mathscr{M}_{{\text{loc}}}$ . Taking into account the decompositions (6.2) and (6.8), we obtain

\begin{align*}M^i(h^i) + A^i(h^i) + \breve{X}^i(h^i) - \widetilde{\Theta} - \widetilde{R} - [X^i,Y] \in \mathscr{M}_{{\text{loc}}},\end{align*}

which implies

\begin{align*}A^i(h^i) + \breve{X}^i(h^i) - \widetilde{R} - [X^i,Y] \in \mathscr{M}_{{\text{loc}}} \cap \mathscr{V}.\end{align*}

Taking into account [Reference Jacod and Shiryaev39, Lemmas I.3.10 and I.3.11], we have $[X^i,Y] - \breve{X}^i(h^i) \in \mathscr{A}_{{\text{loc}}}$ and

(6.12) \begin{align}A^i(h^i) - \widetilde{R} - ([X^i,Y] - \breve{X}^i(h^i))^p \in \mathscr{M}_{{\text{loc}}} \cap \mathscr{V}.\end{align}

Note that the process on the left-hand side of (6.12) is predictable. Hence, by [Reference Jacod and Shiryaev39, Corollary I.3.16], we obtain (6.11) up to an evanescent set.

(iii) $\Rightarrow$ (iv): This implication is obvious.

(iv) $\Rightarrow$ (i): Let $i \in I$ be arbitrary, and let $h^i \in \mathscr{C}_t$ be a truncation function such that we have $[X^i,Y] - \breve{X}^i(h^i) \in \mathscr{A}_{{\text{loc}}}$ and (6.11). Then by the decompositions (6.2) and (6.8) we have

\begin{align*}&X^i - Y - [X^i,Y] = M^i(h^i) + A^i(h^i) + \breve{X}^i(h^i) - \widetilde{\Theta} - \widetilde{R} - [X^i,Y]\\ &= M^i(h^i) + ( [ X^i,Y ] - \breve{X}^i(h^i) )^p - ([X^i,Y] - \breve{X}^i(h^i)) - \widetilde{\Theta} \in \mathscr{M}_{{\text{loc}}}.\end{align*}

Therefore, by Lemma 6.1 the process Z is an ELMD for $\mathbb{S}$ .

For what follows, we fix an index $i \in I$ and a truncation function $h^i \in \mathscr{C}_t$ . We define the two processes $B^i(h^i), C^i(h^i) \in \mathscr{V}$ as

\begin{align*}B^i(h^i) &\,:\!=\, [M^i(h^i),\widetilde{\Theta}] + [\breve{X}^i(h^i),Y] - \breve{X}^i(h^i),\\ C^i(h^i) &\,:\!=\, [ M^i(h^i) + \breve{X}^i(h^i), \Theta ] - \breve{X}^i(h^i).\end{align*}

Proof. By the decompositions (6.2) and (6.8) and by [Reference Jacod and Shiryaev39, Proposition I.4.49.a], we have

\begin{align*}[X^i,Y] - \breve{X}^i(h^i) &= [M^i(h^i),\widetilde{\Theta}] + [M^i(h^i),\widetilde{R}] + [A^i(h^i),\widetilde{\Theta}] + [A^i(h^i),\widetilde{R}]\\ &\quad + [\breve{X}^i(h^i),Y] - \breve{X}^i(h^i)\\ &= B^i(h^i) + [M^i(h^i),\widetilde{R}] + [A^i(h^i),\widetilde{\Theta}] + [A^i(h^i),\widetilde{R}].\end{align*}

Furthermore, by [Reference Jacod and Shiryaev39, Proposition I.4.49.c] we have $[M^i(h^i),\widetilde{R}], [A^i(h^i),\widetilde{\Theta}] \in \mathscr{M}_{{\text{loc}}}$ . Therefore, taking into account [Reference Jacod and Shiryaev39, Lemmas I.3.10 and I.3.11], we have $[X^i,Y] - \breve{X}^i(h^i) \in \mathscr{A}_{{\text{loc}}}$ if and only if $B^i(h^i) \in \mathscr{A}_{{\text{loc}}}$ , and in this case we have

\begin{align*}([X^i,Y] - \breve{X}^i(h^i))^p = [A^i(h^i),\widetilde{R}] + B^i(h^i)^p,\end{align*}

completing the proof.

Before we proceed, recall that each $A \in \mathscr{V}$ admits a unique decomposition $A = A^c + A^d$ into a continuous process $A^c \in \mathscr{V}$ and a purely discontinuous process $A^d \in \mathscr{V}$ , and that each purely discontinuous $A \in \mathscr{V}$ admits a unique decomposition $A = A^q + A^a$ into purely discontinuous processes $A^q,A^a \in \mathscr{V}$ such that $A^q$ is quasi-left-continuous and $A^a$ has only accessible jumps; see [Reference He, Wang and Yan33, Theorem 4.25].

Proof. By (6.7) we have

\begin{align*}B^i(h^i)^c = [M^i(h^i),\widetilde{\Theta}]^c = (1 - \Delta \widetilde{R}) \bullet [M^i(h^i),\Theta]^c = [M^i(h^i),\Theta]^c = C^i(h^i)^c.\end{align*}

Now we define the two purely discontinuous processes $I,J \in \mathscr{V}$ as

\begin{align*}I &\,:\!=\, [\breve{X}^i(h^i),Y] - \breve{X}^i(h^i),\\ J &\,:\!=\, [\breve{X}^i(h^i),\Theta] - \breve{X}^i(h^i).\end{align*}

Then by [Reference Jacod and Shiryaev39, Proposition I.4.49.a] and (6.9) we have

\begin{align*}I &= (\Delta Y - 1) \bullet \breve{X}^i(h^i),\\ J &= (\Delta \Theta - 1) \bullet \breve{X}^i(h^i) = \frac{\Delta Y - 1}{1 - \Delta \widetilde{R}} \bullet \breve{X}^i(h^i).\end{align*}

According to [Reference He, Wang and Yan33, Theorem 4.21], there exist an exhausting sequence $(T_n)_{n \in \mathbb{N}}$ of totally inaccessible stopping times and an exhausting sequence $(S_m)_{m \in \mathbb{N}}$ of predictable stopping times such that

\begin{align*}\{ \Delta \breve{X}^i(h^i) \neq 0 \} \subset \bigcup_{n \in \mathbb{N}} [\![ T_n ]\!] \cup \bigcup_{m \in \mathbb{N}} [\![ S_m ]\!].\end{align*}

Since the process $\widetilde{R}$ is predictable, by the construction in the proof of [Reference He, Wang and Yan33, Theorem 4.25] and [Reference Jacod and Shiryaev39, Proposition I.2.24] we have

\begin{align*}I^q = \sum_{n \in \mathbb{N}} \Delta I_{T_n} \textbf{1}_{[\![ T_n,\infty [\![} = \sum_{n \in \mathbb{N}} \Delta J_{T_n} \textbf{1}_{[\![ T_n,\infty [\![} = J^q.\end{align*}

Similarly, by (6.7) we obtain

\begin{align*}[M^i(h^i),\widetilde{\Theta}]^{dq} = \big( (1 - \Delta \widetilde{R}) \bullet [M^i(h^i),\Theta]^d \big)^q = [M^i(h^i),\Theta]^{dq}.\end{align*}

Consequently, we obtain

\begin{align*}B^i(h^i)^{dq} = [M^i(h^i),\widetilde{\Theta}]^{dq} + I^q = [M^i(h^i),\Theta]^{dq} + J^q = C^i(h^i)^{dq},\end{align*}

completing the proof.

Lemma 6.3. We have

(6.14) \begin{align}\Delta C^i(h^i) &= (1 + \Delta R) \Delta B^i(h^i),\end{align}

(6.15) \begin{align} \Delta B^i(h^i) = (1 - \Delta \widetilde{R}) \Delta C^i(h^i).\end{align}

Proof. By (6.6), (6.7), and (6.9) we obtain

\begin{align*}(1 + \Delta R) \Delta B^i(h^i) &= \frac{1}{1 - \Delta \widetilde{R}} \Delta ( [M^i(h^i),\widetilde{\Theta}] + [\breve{X}^i(h^i),Y] - \breve{X}^i(h^i) )\\ &= \frac{1}{1 - \Delta \widetilde{R}} ( (1 - \Delta \widetilde{R}) \Delta [M^i(h^i),\Theta] + (\Delta Y - 1) \Delta \breve{X}^i(h^i) )\\ &= \Delta M^i(h^i) \Delta \Theta + (\Delta \Theta - 1) \Delta \breve{X}^i(h^i)\\ &= \Delta ( [ M^i(h^i) + \breve{X}^i(h^i), \Theta ] - \breve{X}^i(h^i) ) = \Delta C^i(h^i),\end{align*}

showing (6.14). Now the identity (6.15) follows from (6.6).

Lemma 6.4. We have

\begin{align*}C^i(h^i)^d = B^i(h^i)^d + [R,B^i(h^i)]^d \quad and \quad B^i(h^i)^d = C^i(h^i)^d - [\widetilde{R},B^i(h^i)]^d.\end{align*}

Proof. Suppose that $B^i(h^i) \in \mathscr{A}_{{\text{loc}}}$ . By [Reference Jacod and Shiryaev39, Lemma I.3.10] we have

\begin{align*}{\text{Var}} [R,B^i(h^i)]^d &= \sum_{s \leq \bullet} | \Delta R_s \Delta B^i(h^i)_s | \leq {\text{Var}}(R) \sum_{s \leq \bullet} |\Delta B^i(h^i)_s|\\ &\leq {\text{Var}}(R) {\text{Var}}(B^i(h^i)) \in \mathscr{A}_{{\text{loc}}}.\end{align*}

Therefore, by Lemma 6.4, we deduce that $C^i(h^i) \in \mathscr{A}_{{\text{loc}}}$ . The converse implication is proven analogously.

Lemma 6.5. Suppose that $B^i(h^i) \in \mathscr{A}_{{\text{loc}}}$ , or equivalently $C^i(h^i) \in \mathscr{A}_{{\text{loc}}}$ . Then we have

(6.16) \begin{align}(B^i(h^i)^p)^c &= (C^i(h^i)^p)^c,\qquad\qquad\end{align}

(6.17) \begin{align}\Delta C^i(h^i)^p &= (1 + \Delta R) \Delta B^i(h^i)^p,\end{align}

(6.18) \begin{align}\Delta B^i(h^i)^p &= (1 - \Delta \widetilde{R}) \Delta C^i(h^i)^p.\end{align}

Proof. By [Reference He, Wang and Yan33, Theorem 7.14] and Lemma 6.2 we have

\begin{align*}(B^i(h^i)^p)^c = (B^i(h^i)^c + B^i(h^i)^{dq})^p = (C^i(h^i)^c + C^i(h^i)^{dq})^p = (C^i(h^i)^p)^c,\end{align*}

showing (6.16). Furthermore, by [Reference Jacod and Shiryaev39, I.3.21, I.2.30] and Lemma 6.3 we obtain

\begin{align*}\Delta C^i(h^i)^p &= {}^p [\Delta C^i(h^i)] = {}^p [(1 + \Delta R) \Delta B^i(h^i)] = (1 + \Delta R) \, {}^p [\Delta B^i(h^i)]\\ &= (1 + \Delta R) \Delta B^i(h^i)^p,\end{align*}

showing (6.17). Now the identity (6.18) is a consequence of (6.6).

Proposition 6.4. Suppose that $B^i(h^i) \in \mathscr{A}_{{\text{loc}}}$ , or equivalently $C^i(h^i) \in \mathscr{A}_{{\text{loc}}}$ . Then we have (6.13) if and only if

(6.19) \begin{align}A^i(h^i) - R = C^i(h^i)^p.\end{align}

Proof. We have (6.13) if and only if

(6.20) \begin{align}A^i(h^i)^c - R^c &= (B^i(h^i)^p)^c \quad \text{and}\end{align}

(6.21) \begin{align}\Delta A^i(h^i) - \Delta \widetilde{R} &= \Delta [A^i(h^i), \widetilde{R}] + \Delta B^i(h^i)^p.\end{align}

By virtue of (6.6), the condition (6.21) is equivalent to

(6.22) \begin{align}\Delta A^i(h^i) - \Delta R = \frac{\Delta B^i(h^i)^p}{1 - \Delta \widetilde{R}}.\end{align}

Furthermore, we have (6.19) if and only if

(6.23) \begin{align}A^i(h^i)^c - R^c &= (C^i(h^i)^p)^c \quad \text{and}\end{align}

(6.24) \begin{align}\Delta A^i(h^i) - \Delta R &= \Delta C^i(h^i)^p.\end{align}

Using Lemma 6.5, we obtain the equivalences (6.20) $\Leftrightarrow$ (6.23) and (6.22) $\Leftrightarrow$ (6.24), completing the proof.

Now, the proof of Theorem 6.1 is a consequence of Propositions 6.1, 6.2, 6.3, and 6.4

7. Jump-diffusion models with fixed times of discontinuities

In this section we study the existence of ELMDs for jump-diffusion models with fixed times of discontinuities. Let $\lambda$ be the Lebesgue measure on $(\mathbb{R}_+,\mathcal{B}(\mathbb{R}_+))$ , and let W be an $\mathbb{R}^m$ -valued standard Wiener process for some $m \in \mathbb{N}$ . Furthermore, let $\mu^c$ be a homogeneous Poisson random measure (see [Reference Jacod and Shiryaev39, Section II.1.c]) on some mark space $(E,\mathscr{E})$ , which we assume to be a Blackwell space. Then its predictable compensator (see [Reference Jacod and Shiryaev39, Theorem II.1.8]) is of the form $\nu^c = \lambda \otimes F$ for some $\sigma$ -finite measure F on the mark space $(E,\mathscr{E})$ . Let $\mu^d$ be another Poisson random measure on $(E,\mathscr{E})$ , and denote by $\nu^d$ its predictable compensator. We set $a_t \,:\!=\, \nu^d(\{ t \} \times E)$ for each $t \in \mathbb{R}_+$ , and define the set $J \subset \mathbb{R}_+$ as $J \,:\!=\, \{ t \in \mathbb{R}_+ \,:\, a_t > 0 \}$ . According to [Reference Jacod and Shiryaev39, Proposition II.1.17], the set J is countable, and we may assume that $a_t \leq 1$ for all $t \in \mathbb{R}_+$ . We assume that $\mu^d$ is purely discontinuous in the sense that $\nu^d(dt,dx) = \nu^d(dt,dx) \textbf{1}_J(t)$ . Let $\zeta^J$ be the measure on $(\mathbb{R}_+,\mathcal{B}(\mathbb{R}_+))$ given by $\zeta^J(B) = \sum_{k \in B \cap J} 1$ for each $B \in \mathcal{B}(\mathbb{R}_+)$ ; that is, $\zeta^J$ is the counting measure with support J.

Let $L_{{\text{loc}}}^1(\lambda)$ be the space of all predictable processes $\alpha \,:\, \Omega \times \mathbb{R}_+ \to \mathbb{R}$ such that $|\alpha| \bullet \lambda \in \mathscr{A}_{{\text{loc}}}$ , let $L_{{\text{loc}}}^1(\zeta^J)$ be the space of all predictable processes $\alpha \,:\, \Omega \times \mathbb{R}_+ \to \mathbb{R}$ such that $|\alpha| \bullet \zeta^J \in \mathscr{A}_{{\text{loc}}}$ , and let $L_{{\text{loc}}}^2(W)$ be the space of all predictable processes $\sigma \,:\, \Omega \times \mathbb{R}_+ \to \mathbb{R}^m$ such that $\| \sigma \|_{\mathbb{R}^m}^2 \bullet \lambda \in \mathscr{A}_{{\text{loc}}}$ . Furthermore, let $L_{{\text{loc}}}^2(\mu^c)$ be the space of all predictable processes $\gamma \,:\, \Omega \times \mathbb{R}_+ \times E \to \mathbb{R}$ such that $|\gamma|^2 * \nu^c \in \mathscr{A}_{{\text{loc}}}$ , and let $L_{{\text{loc}}}^2(\mu^d)$ be the space of all predictable processes $\delta \,:\, \Omega \times \mathbb{R}_+ \times E \to \mathbb{R}$ such that $(\delta - \widehat{\delta})^2 * \nu^d + \sum_{s \leq \bullet} (1 - a_s) \widehat{\delta}_s^2 \in \mathscr{A}_{{\text{loc}}}$ , where

\begin{align*}\widehat{\delta}_t \,:\!=\, \int_E \delta_t(x) \nu^d(\{ t \} \times dx), \quad t \in \mathbb{R}_+.\end{align*}

As in the previous section, we consider a financial market $\mathbb{S} = \{ S^i \,:\, i \in I \}$ with an arbitrary index set $I \neq \emptyset$ . We assume that for each $i \in I$ the semimartingale $S^i$ is given by

\begin{align*}S^i = S_0^i \mathscr{E} \big( \alpha^{i,c} \bullet \lambda + \alpha^{i,d} \bullet \zeta^J + \sigma^i \bullet W + \gamma^i * (\mu^c - \nu^c) + \delta^i * (\mu^d - \mu^d) \big)\end{align*}

with $\alpha^{i,c} \in L_{{\text{loc}}}^1(\lambda)$ , $\alpha^{i,d} \in L_{{\text{loc}}}^1(\zeta^J)$ , $\sigma^i \in L_{{\text{loc}}}^2(W)$ , $\gamma^i \in L_{{\text{loc}}}^2(\mu^c)$ such that $\gamma^i > -1$ , and $\delta^i \in L_{{\text{loc}}}^2(\mu^d)$ such that $\delta^i - \widehat{\delta}^i > -1$ . Here ‘ $*$ ’ denotes the stochastic integral with respect to a random measure; see [Reference Jacod and Shiryaev39, Section II.1.d]. Let Z be a multiplicative special semimartingale with multiplicative decomposition $Z = D B^{-1}$ , where

\begin{align*}D = \mathscr{E} \big( -\theta \bullet W - \psi * (\mu^c - \nu^c) - \phi * (\mu^d - \nu^d) \big) \quad \text{and} \quad B = \mathscr{E} \big( r^c \bullet \lambda + r^d \bullet \zeta^J \big),\end{align*}

with $\theta \in L_{{\text{loc}}}^2(W)$ , $\psi \in L_{{\text{loc}}}^2(\mu^c)$ such that $\psi < 1$ , and $\phi \in L_{{\text{loc}}}^2(\mu^d)$ such that $\phi - \widehat{\phi} < 1$ , as well as $r^c \in L_{{\text{loc}}}^1(\lambda)$ and $r^d \in L_{{\text{loc}}}^1(\zeta^J)$ .

Before we provide the proof of Theorem 7.1, let us prepare an auxiliary result.

Lemma 7.1. Let $\delta \in L_{{\text{loc}}}^2(\mu^d)$ be arbitrary, and define the purely discontinuous local martingale $M \,:\!=\, \delta * (\mu^d - \nu^d)$ . Then the predictable quadratic variation $\langle M,M \rangle$ is purely discontinuous, and we have

\begin{align*}\Delta \langle M,M \rangle_t = \int_E \delta_t(x)^2 \nu^d(\{ t \} \times dx) - \widehat{\delta}_t^2, \quad t \in \mathbb{R}_+.\end{align*}

Proof. According to [Reference Jacod and Shiryaev39, Theorem II.1.33.a] we have

\begin{align*}\langle M,M \rangle = (\delta - \widehat{\delta})^2 * \nu^d + \sum_{s \leq \bullet} (1 - a_s) \widehat{\delta}_s^2.\end{align*}

For each $t \in \mathbb{R}_+$ we obtain

\begin{align*}(\delta - \widehat{\delta})^2 * \nu_t^d &= \int_{[0,t] \times E} (\delta_s(x) - \widehat{\delta}_s)^2 \nu^d(ds,dx) = \sum_{s \leq t} \int_E (\delta_s(x) - \widehat{\delta}_s)^2 \nu^d(\{ s \} \times dx)\\ &= \sum_{s \leq t} \int_E \big( \delta_s(x)^2 - 2 \delta_s(x) \widehat{\delta}_s + \widehat{\delta}_s^2 \big) \nu^d(\{ s \} \times dx)\\ &= \sum_{s \leq t} \bigg( \int_E \delta_s(x)^2 \nu^d(\{ s \} \times dx) - 2 \widehat{\delta}_s^2 + \widehat{\delta}_s^2 a_s \bigg),\end{align*}

and hence

\begin{align*}\langle M,M \rangle = \sum_{s \leq \bullet} \bigg( \int_E \delta_s(x)^2 \nu^d(\{ s \} \times dx) - \widehat{\delta}_s^2 \bigg),\end{align*}

completing the proof.

For the upcoming proof of Theorem 7.1, we recall that $\mathscr{H}_{{\text{loc}}}^2$ denotes the space of all locally square-integrable martingales.

Proof of Theorem 7.1. We define the processes

\begin{align*}M^i &\,:\!=\, \sigma^i \bullet W + \gamma^i * (\mu^c - \nu^c) + \delta^i * (\mu^d - \nu^d), \quad i \in I,\\ A^i &\,:\!=\, \alpha^{i,c} \bullet \lambda + \alpha^{i,d} \bullet \zeta^J, \quad i \in I,\\ \Theta &\,:\!=\, \theta \bullet W + \psi * (\mu^c - \mu^c) + \phi * (\nu^d - \nu^d),\\ R &\,:\!=\, r^c \bullet \lambda + r^d \bullet \zeta^J.\end{align*}

According to [Reference Jacod and Shiryaev39, Theorem II.1.33.a] we have $M^i \in \mathscr{H}_{{\text{loc}}}^2$ for each $i \in I$ , and we have $\Theta \in \mathscr{H}_{{\text{loc}}}^2$ . Let $i \in I$ be arbitrary. By [Reference Jacod and Shiryaev39, Theorem I.4.50.b] we have $[M^i,\Theta] \in \mathscr{A}_{{\text{loc}}}$ . Furthermore, by [Reference Jacod and Shiryaev39, Theorem II.1.33.a] we obtain

\begin{align*}\langle M^i,\Theta \rangle^c = \big( \langle \sigma^i,\theta \rangle_{\mathbb{R}^m} + \langle \gamma^i, \psi \rangle_{L^2(F)} \big) \bullet \lambda,\end{align*}

and by Lemma 7.1 we have

\begin{align*}\Delta \langle M^i,\Theta \rangle_t = \int_E \delta_t^i(x) \phi_t(x) \nu^d(\{ t \} \times dx) - \widehat{\delta}_t^i \widehat{\phi}_t, \quad t \in \mathbb{R}_+.\end{align*}

Consequently, applying Theorem 6.2 completes the proof.

We conclude this section by considering the particular situation where the risky assets are given by finitely many diffusion processes. More precisely, consider a continuous market $\mathbb{S} = \{ S^1,\ldots,S^d \}$ where

\begin{align*}S^i = S_0^i \mathscr{E}(X^i), \quad i=1,\ldots,d,\end{align*}

and where the $\mathbb{R}^d$ -valued semimartingale $X = (X^1,\ldots,X^d)$ is an Itô process of the form

\begin{align*}X = \alpha \bullet \lambda + \sigma \bullet W.\end{align*}

Here we may regard $\sigma$ as an $\mathbb{R}^{d \times m}$ -valued process. Let Z be a multiplicative special semimartingale with multiplicative decomposition $Z = D B^{-1}$ , where

\begin{align*}D = \mathscr{E}({-}\theta \bullet W) \quad \text{and} \quad B = \mathscr{E}(r \bullet \lambda) = \exp(r \bullet \lambda)\end{align*}

with market price of risk $\theta \in L_{{\text{loc}}}^2(W)$ and short rate $r \in L_{{\text{loc}}}^1(\lambda)$ . As an immediate consequence of Theorem 7.1 we obtain the following result.

Corollary 7.1. Z is an ELMD for $\mathbb{S}$ if and only if

(7.3) \begin{align}\sigma \theta = \alpha - r \textbf{1}_{\mathbb{R}^d} \quad \lambda-a.e., \quad \mathbb{P}-a.e., \end{align}

where we agree on the notation $\textbf{1}_{\mathbb{R}^d} = (1,\ldots,1) \in \mathbb{R}^d$ .

• Let us fix a market price of risk $\theta$ . Then an appropriate short rate r satisfying (7.3) exists if and only if $\alpha - \sigma \theta \in {\text{lin}} \{ \textbf{1}_{\mathbb{R}^d} \}$ . In this case, the short rate is unique and given by

  \begin{align*}r \textbf{1}_{\mathbb{R}^d} = \alpha - \sigma \theta.\end{align*}
  This confirms the statement about the uniqueness of the savings account; see Proposition 4.2.

• Let us fix a short rate r. Then an appropriate market price $\theta$ is a solution of the $\mathbb{R}^d$ -valued linear equation (7.3). Depending on the structure of the matrix $\sigma$ , there may be several solutions.

8. Further examples

In this section we provide further examples which are related to our main results. The first example deals with the construction of arbitrage-free markets by using the real-world pricing formula.

Example 8.1. (Real-world pricing) We fix a savings account B and a local martingale $D > 0$ , and we define the multiplicative special semimartingale $Z \,:\!=\, D B^{-1}$ . Let $H^1,\ldots,H^d$ be nonnegative $\mathscr{F}_T$ -measurable contingent claims for some $d \in \mathbb{N}$ such that

(8.1) \begin{align}H^i Z_T \in L^1 \quad \text{for all $i=1,\ldots,d$.}\end{align}

We define the market $\mathbb{S} = \{ S^1,\ldots,S^d \}$ by the real-world pricing formula

(8.2) \begin{align}S_t^i \,:\!=\, Z_t^{-1} \mathbb{E} [ H^i Z_T | \mathscr{F}_t ], \quad t \in [0,T],\end{align}

for all $i=1,\ldots,d$ . Then Z is an ELMD for $\mathbb{S}$ , and by Theorem 4.2 the set $\mathscr{P}_{{\text{sf}},1}^+(\mathbb{S} \cup \{ B \})$ satisfies NUPBR. If $H^1 = 1$ , then the real-world pricing formula reads

(8.3) \begin{align}S_t^1 = Z_t^{-1} \mathbb{E} [ Z_T | \mathscr{F}_t ], \quad t \in [0,T],\end{align}

and the first primary security account is a zero-coupon bond with maturity date T.

We can derive the widely applied risk-neutral pricing formula from the more general real-world pricing formula as a special case.

Example 8.2. (Risk-neutral pricing) Suppose that in the setting of Example 8.1 the savings accounts B and $B^{-1}$ are bounded. By Theorem 4.4 the set $\mathscr{P}_{{\text{sf}},0}^{{\text{adm}}}(\mathbb{S} \cup \{ B \})$ satisfies NFLVR if and only if D is a true martingale. Suppose that this is the case. Then, also by Theorem 4.4, there exists an ELMM $\mathbb{Q} \approx \mathbb{P}$ for $\mathbb{S} B^{-1}$ , and its density process is given by $D / D_0$ . By (8.1) we have

\begin{align*}H^i B_T^{-1} \in L^1(\mathbb{Q}) \quad \text{for all $i=1,\ldots,d$.}\end{align*}

Hence, using Bayes’ rule (see [Reference Jacod and Shiryaev39, III.3.9]), we obtain the risk-neutral pricing formula

(8.4) \begin{align}S_t^i = B_t \, \mathbb{E}_{\mathbb{Q}} [ H^i B_T^{-1} | \mathscr{F}_t ], \quad t \in [0,T],\end{align}

for all $i=1,\ldots,d$ . If $H^1 = 1$ , then the real-world pricing formula reads

(8.5) \begin{align}S_t^1 = B_t \, \mathbb{E}_{\mathbb{Q}} [ B_T^{-1} | \mathscr{F}_t ], \quad t \in [0,T],\end{align}

and the first primary security account is a zero-coupon bond with maturity date T.

The real-world pricing formula (8.2) can also be used for pricing and hedging contingent claims, as follows.

In principle, the real-world prices of contingent claims can also be determined with the risk-neutral pricing formula, but only if the local martingale D is a true martingale. An example of a local martingale which is not a true martingale is the inverse of a squared Bessel process of dimension 4, which is given by the solution of the stochastic differential equation

(8.8) \begin{align}d D_t = - 2 D_t^{\frac{3}{2}} dW_t,\end{align}

where W is a real-valued standard Wiener process. This process plays an important role in the minimal market model; see, for example, [Reference Platen and Heath53, Section 13]. The following example shows that the real-world pricing formula also provides the forward measure pricing formula in a market with a zero-coupon bond.

Example 8.4. (Forward measure pricing formula) Let $\mathbb{S} = \{ P^T \}$ be a market consisting of a zero-coupon bond $P^T$ with maturity T. Let Z be an ELMD for $\mathbb{S}$ which is a multiplicative special semimartingale of the form $Z = D B^{-1}$ with a local martingale $D > 0$ and a savings account B. By Theorem 4.2 the set $\mathscr{P}_{{\text{sf}},1}^+(\{ P^T,B \})$ satisfies NUPBR. Now, let H be a nonnegative $\mathscr{F}_T$ -measurable contingent claim such that $H Z_T \in L^1$ . By Example 8.3 we know that the least expensive price process $\pi$ is given by the real-world pricing formula (8.6). Since Z is an ELMD for $\mathbb{S}$ , the process $P^T Z$ is a local martingale. If $P^T Z$ is a true martingale, then using Bayes’ rule (see [Reference Jacod and Shiryaev39, III.3.9]) the price process $\pi$ can be expressed by the forward measure pricing formula

(8.9) \begin{align}\pi_t = P_t^T \mathbb{E}_{\mathbb{Q}^T} [ H | \mathscr{F}_t ], \quad t \in [0,T],\end{align}

where $\mathbb{Q}^T \approx \mathbb{P}$ denotes the T-forward measure with density process $(P^T Z) / (P_0^T Z_0)$ . For practitioners it may be interesting to note that the convenient forward measure pricing formula (8.9) can still be applied when D is not a true martingale, which means that a risk-neutral measure $\mathbb{Q} \approx \mathbb{P}$ for the discounted bond $P^T / B$ might not exist; see also Example 5.3 below. If D is a true martingale, then the measure change $\mathbb{Q}^T \approx \mathbb{P}$ can be performed in the two steps $\mathbb{Q} \approx \mathbb{P}$ and $\mathbb{Q}^T \approx \mathbb{Q}$ with respective density processes $D / D_0$ and $(P^T B^{-1}) / P_0^T$ , which is usually done under the risk-neutral approach.

We continue the above example by calculating the least expensive zero-coupon bond price $P^T_0$ under the minimal market model with D as in (8.8).

Example 8.5. (A free lunch with vanishing risk) As in the previous example, we consider a market $\mathbb{S} = \{ P^T \}$ consisting of a zero-coupon bond $P^T$ with maturity T. Here we fix a deterministic savings account B; for example it could be $B_t = e^{rt}$ , $t \in [0,T]$ , for some constant interest rate r. Furthermore, we denote by $D > 0$ the strict local martingale given by the stochastic differential equation (8.8) with $D_0 = 1$ , and we specify the zero-coupon bond $P^T$ by the real-world pricing formula (8.3) with $Z = D B^{-1}$ , which here becomes

\begin{align*}P_t^T = \big( B_t B_T^{-1} \big) D_t^{-1} \mathbb{E} [ D_T | \mathscr{F}_t ], \quad t \in [0,T].\end{align*}

Then Z is an ELMD for $\mathbb{S}$ , and hence, by Theorem 4.2, the set $\mathscr{P}_{{\text{sf}},1}^+(\{ P^T,B \})$ satisfies NUPBR. However, since D is not a true martingale, we cannot apply Theorem 4.4, and, as a consequence, we cannot ensure that the set $\mathscr{P}_{{\text{sf}},0}^{{\text{adm}}}(\{ P^T,B \})$ satisfies NFLVR. However, the process $P^T Z$ is a true martingale, which allows us to use the convenient forward measure pricing formula (8.9) for pricing contingent claims H. In order to investigate our observation about the possible existence of a free lunch with vanishing risk, let us denote by $\widetilde{P}^T$ the price of a zero-coupon bond with maturity T computed formally with the risk-neutral pricing formula (8.5) under some putative risk-neutral probability measure $\mathbb{Q}$ . Since B is deterministic, we obtain

\begin{align*}\widetilde{P}^T = B / B_T\end{align*}

for every choice of the measure $\mathbb{Q}$ , and hence $P_0^T < \widetilde{P}_0^T$ , because by the formula (8.7.17) in [Reference Platen and Heath53] we have

\begin{align*}P_0^T = \frac{1}{B_T} \bigg[ 1 - \exp \bigg( - \frac{1}{2T} \bigg) \bigg],\end{align*}

which is less than the risk-neutral zero-coupon bond $\widetilde{P}_0^T = B_T^{-1}$ . In accordance with our previous observation, the difference between these two zero-coupon bond prices indicates the presence of some free lunch with vanishing risk. However, as we have seen, the NUPBR condition is still satisfied. By making $1/Z$ tradeable as numéraire portfolio (see [Reference Karatzas and Kardaras44, Reference Platen and Heath53]), one can hedge the least expensive zero-coupon bond. This allows us to exploit the identified free lunch with vanishing risk, which is caused by the strict local martingale property of D. From the practical perspective, this allows us to produce long-term zero-coupon bond-type payouts less expensively than under the NFLVR condition, which is likely to have significant impact on the pension and insurance industry (see e.g. [Reference Sun, Zhu and Platen63]) through the realistic modeling of the ELMD Z for $\mathbb{S}$ , which is the inverse of the numéraire portfolio $1/Z$ . The latter coincides, up to some subtleties, with the growth-optimal portfolio; see [Reference Karatzas and Kardaras44].

9. Dynamic trading strategies

For risk management purposes, trading strategies that do not need to be self-financing can be crucial; see, for example, [Reference Du and Platen24] and [Reference Schweizer59]. In this section we will construct such strategies which are in line with our findings from the previous sections. Of course, when looking for trading strategies which are not self-financing, not all strategies can be allowed. For the construction of reasonable strategies that do not need to be self-financing, the concept of a locally real-world mean self-financing dynamic trading strategy (see [Reference Du and Platen24]) turns out to be fruitful. In the risk-neutral context, an analogous notion has been introduced in [Reference Schweizer59].

As in the previous sections, we consider a finite market $\mathbb{S} = \{ S^1,\ldots,S^d \}$ with nonnegative semimartingales. We assume that there exists an ELMD Z for $\mathbb{S}$ which is a multiplicative special semimartingale of the form $Z = D B^{-1}$ with a local martingale $D > 0$ and a savings account B. According to Theorem 4.2, the set $\mathscr{P}_{{\text{sf}},1}^+(\mathbb{S} \cup \{ B \})$ satisfies NUPBR.

Definition 9.1. A dynamic trading strategy $\nu = (\delta,\eta)$ consists of a self-financing strategy $\delta \in \Delta_{{\text{sf}}}(\mathbb{S})$ and a real-valued optional process $\eta$ , which is integrable with respect to B, such that the portfolio

\begin{align*}V^{\nu} \,:\!=\, S^{\delta} + B^{\eta},\end{align*}

where we use the common notation $S^{\delta} \,:\!=\, \delta \cdot S$ and $B^{\eta} \,:\!=\, \eta \cdot B$ , satisfies

\begin{align*}V^{\nu} = S_0^{\delta} + \delta \bullet S + B^{\eta}.\end{align*}

In this case, we call $\delta$ the self-financing part of $\nu$ .

Definition 9.2. Let $\nu = (\delta,\eta)$ be a dynamic trading strategy. The profit-and-loss (P&L) process $C^{\nu}$ is defined as

\begin{align*}C^{\nu} \,:\!=\, B^{\eta} - B_0^{\eta} - \eta \bullet B.\end{align*}

Remark 9.1. Note that $C_0^{\nu} = 0$ and

(9.1) \begin{align}V^{\nu} = V_0^{\nu} + \delta \bullet S + \eta \bullet B + C^{\nu}.\end{align}

Thus, the P&L process $C^{\nu}$ monitors the cumulative inflow and outflow of extra capital. Note that $C^{\nu} = 0$ if and only if

(9.2) \begin{align}V^{\nu} = V_0^{\nu} + \delta \bullet S + \eta \bullet B.\end{align}

This in particular is satisfied if $\nu$ is self-financing in the market $\mathbb{S} \cup \{ B \}$ , but the condition (9.2) is a bit more general, because $\eta$ may be optional, which allows us to go beyond predictable processes. More precisely, we have $\nu \in \Delta_{{\text{sf}}}(\mathbb{S} \cup \{ B \})$ if and only if $\nu \in L((S,B))$ and Equation (9.2) is satisfied.

Note that every self-financing strategy $\nu$ is locally real-world mean self-financing. The following result shows how we can easily construct locally real-world mean self-financing strategies.

Proof. According to [Reference Jacod and Shiryaev39, Proposition I.4.49.a] we have

\begin{align*}B^{\eta} = B_0^{\eta} + B_- \bullet \eta + \eta \bullet B,\end{align*}

and hence $C^{\nu} = B_- \bullet \eta \in \mathscr{M}_{{\text{loc}}}$ , completing the proof.

Recall that Z denotes an ELMD for the financial market $\mathbb{S}$ . For every self-financing strategy $\nu$ , the corresponding self-financing portfolio $V^{\nu}$ has the property that $V^{\nu} Z$ is also a local martingale, which by virtue of Theorem 4.2 means that $\mathscr{P}_{{\text{sf}},1}^+(\mathbb{S} \cup \{ V^{\nu} \} \cup \{ B \} )$ also satisfies NUPBR. In this spirit, we wish to construct such dynamic trading strategies $\nu$ such that $V^{\nu} Z$ is also a local martingale. As we will see, this can be achieved by choosing locally real-world mean self-financing strategies which are orthogonal to the deflator Z. More precisely, we have the following result.

Recall that Z is an ELMD for $\mathscr{P}_{{\text{sf}},1}^+(\mathbb{S} \cup \{ B \} \})$ . Now, let us denote by $\mathscr{P}_{{\text{msf}},1}^+(\mathbb{S} \cup \{ B \})$ the set of all outcomes of trading strategies with initial value one which arise from a nonnegative self-financing portfolio $V^{\nu}$ or from a nonnegative locally real-world mean self-financing portfolio $V^{\nu}$ as in Theorem 9.1 Obviously, the set $\mathscr{P}_{{\text{msf}},1}^+(\mathbb{S} \cup \{ B \})$ is larger than $\mathscr{P}_{{\text{sf}},1}^+(\mathbb{S} \cup \{ B \})$ . However, using Theorem 9.1 we deduce that Z is even an ELMD for $\mathscr{P}_{{\text{msf}},1}^+(\mathbb{S} \cup \{ B \})$ .

Example 9.1. (Construction of locally real-world mean self-financing strategies.) Let us suppose we are in the situation of Corollary 7.1, where we consider diffusion models. Then we have $Z = \mathscr{E}(Y)$ , where

\begin{align*}Y = -\theta \bullet W - r \bullet \lambda\end{align*}

with an $\mathbb{R}^m$ -valued standard Wiener process process W, a market price of risk $\theta \in L_{{\text{loc}}}^2(W)$ , and a short rate $r \in L_{{\text{loc}}}^1(\lambda)$ . Let $\delta \in \Delta_{{\text{sf}}}(\mathbb{S})$ be a self-financing strategy, and let $\vartheta \in L_{{\text{loc}}}^2(W)$ be such that $V^{\nu} \geq 0$ , where $\nu = (\delta,\eta)$ denotes the dynamic trading strategy given by $\eta \,:\!=\, \vartheta \bullet W$ . We assume $\theta \perp \vartheta$ in the sense that $\theta \cdot \vartheta = 0$ . Since $Z = 1 + Z_- \bullet Y$ , we obtain

\begin{align*}[\eta,Z] = [\eta, Z_- \bullet Y] = - \langle \vartheta \bullet W, Z_- \theta \bullet W \rangle = - Z_- (\vartheta \cdot \theta) \bullet \lambda = 0.\end{align*}

Therefore, by Theorem 9.1, the strategy $\nu$ is locally real-world mean self-financing, and the process $V^{\nu} Z$ is a local martingale.

Example 9.2. (Pricing and hedging of contingent claims.) Consider the framework of Example 8.3, where we have constructed the least expensive price process $\pi$ for a nonnegative contingent claim H with maturity T by using the real-world pricing formula (8.6). Note that the claim H can be non-replicable; that is, it can happen that a self-financing portfolio $\nu = (\delta,\eta) \in \Delta_{{\text{sf}}}(\mathbb{S} \cup \{ B \})$ with $\pi = V^{\nu}$ does not exist. However, allowing dynamic trading strategies that do not need to be self-financing, it might be possible to find a locally real-world mean self-financing strategy $\nu = (\delta,\eta)$ as in Theorem 9.1 such that $\pi = V^{\nu}$ . Then, by the decomposition (9.1), the P&L process $C^{\nu}$ can also be regarded as the non-hedgeable part of the claim H. In any case, we can improve the lower bound (8.7) for the least expensive nonnegative portfolio which replicates H. Indeed, using our previous results, we even have

\begin{align*}\pi \leq V^{\nu},\end{align*}

where $\nu$ is any self-financing strategy or any locally real-world mean self-financing strategy as in Theorem 9.1 such that $V^{\nu} \geq 0$ and $V_T^{\nu} = H$ .

10. The semimartingale property of the primary security accounts

So far, we have assumed that the financial market consists of nonnegative semimartingales. Using the results from [Reference Kardaras and Platen48], we will show in this section that the primary security accounts must be semimartingales under the mild and natural condition that the market satisfies NUPBR for all its self-financing portfolios which are simple and without short positions. We also refer to [Reference Bálint and Schweizer9] for similar results.

Let $\mathbb{S} = \{ S^1,\ldots,S^d \}$ be a market consisting of nonnegative, adapted, càdlàg processes $S^1,\ldots,S^d$ . Furthermore, let B be a savings account. For what follows, we recall that ‘ $\,\, \bullet \,$ ’ denotes stochastic integration, whereas ‘ $\,\, \cdot \,$ ’ denotes the usual inner product in Euclidean space. A strategy $\nu = (\delta,\eta)$ for $\mathbb{S} \cup \{ B \}$ is called simple if the following hold:

1. (a) $\delta$ is of the form

   (10.1) \begin{align}\delta = \sum_{j=1}^n \Delta_j \textbf{1}_{]\!] \tau_{j-1},\tau_j ]\!]}\end{align}
   for some $n \in \mathbb{N}$ , where $0 = \tau_0 < \tau_1 < \ldots < \tau_n$ are finite stopping times, and $\Delta_j = (\Delta_j^i)_{i=1,\ldots,d}$ is $\mathscr{F}_{\tau_{j-1}}$ -measurable for each $j=1,\ldots,n$ , and

2. (b) $\eta$ is a real-valued optional process, which is integrable with respect to B.

For such a simple strategy we define the portfolio $V^{\nu} \,:\!=\, S^{\delta} + B^{\eta}$ , where we use the common notation $S^{\delta} \,:\!=\, \delta \cdot S$ and $B^{\eta} \,:\!=\, \eta \cdot B$ . The portfolio $V^{\nu}$ is called self-financing if

\begin{align*}V^{\nu} = V_0^{\nu} + \sum_{j=1}^n \Delta_j \cdot (S^{\tau_j} - S^{\tau_{j-1}}) + \eta \bullet B,\end{align*}

where $\tau_0,\ldots,\tau_n$ and $\Delta_1,\ldots,\Delta_n$ stem from the representation (10.1). We denote by $\mathscr{P}_{{\text{sf}},1,s}^{\nu \geq 0}(\mathbb{S} \cup \{ B \})$ the set of all outcomes of simple self-financing portfolios with initial value one such that $\delta^1,\ldots,\delta^d \geq 0$ and $\eta \geq 0$ . The latter condition means that no short selling is allowed. We say that a nonnegative semimartingale $S \geq 0$ cannot revive from bankruptcy if $S = 0$ on $[\![ \tau,\infty [\![$ , where $\tau \,:\!=\, \inf \{ t \in \mathbb{R}_+ \,:\, X_{t-} = 0 \text{ or } X_t = 0 \}$ .

For the proof of Theorem 10.1 we prepare some auxiliary results. Let B be a savings account. We consider the discounted market $\mathbb{X} \,:\!=\, \mathbb{S} B^{-1}$ . A strategy $\theta$ for $\mathbb{X}$ is called simple if it is of the form

(10.2) \begin{align}\theta = \sum_{j=1}^n \vartheta_j \textbf{1}_{[\![ \tau_{j-1},\tau_j [\![}\end{align}

for some $n \in \mathbb{N}$ , where $0 = \tau_0 < \tau_1 < \ldots < \tau_n$ are finite stopping times, and $\vartheta_j = (\vartheta_j^i)_{i=1,\ldots,d}$ is $\mathscr{F}_{\tau_{j-1}}$ -measurable for each $j=1,\ldots,n$ . For $x \in \mathbb{R}$ and such a simple strategy $\theta$ we define the integral process

\begin{align*}X^{x,\theta} \,:\!=\, x + \sum_{j=1}^n \vartheta_j \cdot (S^{\tau_j} - S^{\tau_{j-1}}),\end{align*}

where $\tau_0,\ldots,\tau_n$ and $\vartheta_1,\ldots,\vartheta_n$ stem from the representation (10.2). We denote by $\mathscr{I}_{1,s}^{\theta \geq 0}(\mathbb{X})$ the set of all outcomes of integral processes with $x=1$ and a simple strategy $\theta$ such that $\theta \geq 0$ and $X_-^{x,\theta} - \theta \cdot X_- \geq 0$ . Furthermore, let $\Delta_s(\mathbb{X})$ be the set of all simple strategies for $\mathbb{X}$ , and let $\Delta_{{\text{sf}},s}(\bar{\mathbb{X}})$ be the set of all simple self-financing strategies for $\bar{\mathbb{X}} \,:\!=\, \mathbb{X} \cup \{ 1 \}$ . Then we have the following result, which is similar to Lemma B.1.

Lemma 10.1. Suppose that $1 \notin \mathbb{X}$ . Then there is a bijection between $\mathbb{R} \times \Delta_s(\mathbb{X})$ and $\Delta_{{\text{sf}},s}(\bar{\mathbb{X}})$ , which is defined as follows:

1. 1. For $\delta \in \Delta_{{\text{sf}}}(\bar{\mathbb{X}})$ we assign

   (10.3) \begin{align}(\delta,\eta) = \nu \mapsto (x,\theta) \,:\!=\, (X_0^{\delta},\delta) \in \mathbb{R} \times \Delta(\mathbb{X}).\end{align}

2. 2. For $(x,\theta) \in \mathbb{R} \times \Delta(\mathbb{X})$ we assign

   (10.4) \begin{align}(x,\theta) \mapsto (\delta,\eta) = \nu = (\theta, X_-^{x,\theta} - \theta \cdot X_- ) \in \Delta_{{\text{sf}}}(\bar{\mathbb{X}}).\end{align}

Furthermore, for all $(x,\theta) \in \mathbb{R} \times \Delta_s(\mathbb{X})$ and the corresponding self-financing strategy $\nu \in \Delta_{{\text{sf}},s}(\bar{\mathbb{X}})$ we have

(10.5) \begin{align}\bar{X}^{\nu} = X^{x,\theta}.\end{align}

Now we are ready to provide the proof of Theorem 10.1.

11. Filtration enlargements

It can happen that some additional information arises which is not originally present in the market. Models with insider information have been widely studied in the literature; see, for example, [Reference Acciaio, Fontana and Kardaras1–Reference Aksamit and Jeanblanc4, Reference Fontana, Jeanblanc and Song28]. Mathematically, such additional information means that we consider an enlargement of the original filtration. More precisely, we consider a new filtration $\mathbb{G} = (\mathscr{G}_t)_{t \in \mathbb{R}_+}$ such that $\mathscr{F}_t \subset \mathscr{G}_t$ for all $t \in \mathbb{R}_+$ , where $\mathbb{F} = (\mathscr{F}_t)_{t \in \mathbb{R}_+}$ denotes the original filtration. Typically, there is also a $\mathbb{G}$ -stopping time $\tau$ involved, and the question arises of when the absence of arbitrage in a financial market $\mathbb{S} = \{ S^1,\ldots,S^d \}$ under the original filtration $\mathbb{F}$ implies the absence of arbitrage in the stopped market $\mathbb{S}^{\tau} = \{ S^{1,\tau}, \ldots, S^{d,\tau} \}$ under the enlarged filtration $\mathbb{G}$ . There exist several results for the case that the financial market is already discounted by some numéraire; see, for example, the aforementioned articles. By virtue of Proposition B.1, we can transfer most of these results to the situation which we consider in this paper.

As an illustration, consider the situation with a progressive filtration enlargement, which we briefly recall; see [Reference Acciaio, Fontana and Kardaras1, Section 1.3] for further details. Let $\tau \,:\, \Omega \to [0,\infty]$ be an $\mathscr{F}$ -measurable random time such that $\mathbb{P}(\tau = \infty) = 0$ . The progressively enlarged filtration $\mathbb{G} = (\mathscr{G}_t)_{t \in \mathbb{R}_+}$ is defined as

\begin{align*}\mathscr{G}_t \,:\!=\, \{ B \in \mathscr{F} \,:\, B \cap \{ \tau > t \} = B_t \cap \{ \tau > t \} \text{ for some } B_t \in \mathscr{F}_t \} \quad \text{for all $t \in \mathbb{R}_+$.}\end{align*}

Let Z be the Azéma supermartingale given by $Z_t = \mathbb{P}(\tau > t | \mathscr{F}_t)$ for all $t \in \mathbb{R}_+$ , and let A be the dual optional projection of $\textbf{1}_{[\![ \tau,\infty [\![}$ . Furthermore, we define

\begin{align*}\zeta \,:\!=\, \inf \{ t \in \mathbb{R}_+ \,:\, Z_t = 0 \},\end{align*}

the $\mathscr{F}_{\zeta}$ -measurable event $\Lambda \,\,:\,\!=\, \{ \tau < \infty, Z_{\zeta-} > 0, \Delta A_{\zeta} = 0 \}$ , and

\begin{align*}\eta \,:\!=\, \zeta_{\Lambda} \,:\!=\, \zeta \textbf{1}_{\Lambda} + \infty \textbf{1}_{\Omega \setminus \Lambda}.\end{align*}

The following result applies in the situation of Theorem 4.2.

12. Financial models in discrete time

Using our previous results for continuous-time models, we can also derive a no-arbitrage result for discrete-time models. This result is in accordance with the well-known result concerning the absence of arbitrage in discrete-time finance. In this section, we assume that a discrete filtration $(\mathscr{F}_k)_{k=0,\ldots,T}$ for some integer $T \in \mathbb{N}$ with $\mathscr{F}_0 = \{ \Omega,\emptyset \}$ is given, and we consider a finite market $\mathbb{S} = \{ S^1,\ldots,S^d \}$ consisting of nonnegative, adapted processes. As shown in [Reference Jacod and Shiryaev39, p. 14], this setting can be regarded as a particular case of the continuous-time framework which we have considered so far. Note that every $\mathbb{R}^d$ -valued predictable process $\delta$ belongs to $\Delta(\mathbb{S})$ , and that the stochastic integral $\delta \bullet S = (\delta \bullet S_t)_{t=0,\ldots,T}$ is given by

\begin{align*}\delta \bullet S_0 &= 0,\\ \delta \bullet S_t &= \sum_{k=1}^t \delta_k \cdot (S_{k} - S_{k-1}), \quad t=1,\ldots,T.\end{align*}

Proof. Let B be an arbitrary savings account.

(i) $\Rightarrow$ (iii): By Proposition B.1, the set $\mathscr{I}_0(\mathbb{S} B^{-1})$ also satisfies NA, and hence, by [Reference Kabanov and Stricker43, Theorem 1], the set $\mathscr{I}_0(\mathbb{S} B^{-1}) - L_+^0$ is closed in $L^0$ . Therefore, by [Reference Platen and Tappe55, Corollary 5.9], the set $\mathscr{I}_0(\mathbb{S} B^{-1})$ also satisfies NFLVR. By [Reference Platen and Tappe55, Proposition 7.27] it follows that $\mathscr{I}_1(\mathbb{S} B^{-1})$ satisfies NA $_1$ , and hence NUPBR. Of course, the subset $\mathscr{I}_1^+(\mathbb{S} B^{-1})$ also satisfies NUPBR. Therefore, by Proposition B.1 the set $\mathscr{P}_{{\text{sf}},1}^+(\mathbb{S} \cup \{ B \})$ satisfies NUPBR. Hence by Theorem 4.2 there exists a local martingale $D > 0$ such that $Z = D B^{-1}$ is an ELMD for $\mathbb{S}$ . By [Reference Jacod and Shiryaev38, Theorem 1] the process D is a generalized martingale. Therefore, for each $t=0,\ldots,T$ we have $\mathbb{P}$ -almost surely

\begin{align*}\mathbb{E}[D_t] = \mathbb{E}[D_t | \mathscr{F}_0] = D_0 < \infty,\end{align*}

and hence $D_t \in \mathscr{L}^1$ , proving that D is a martingale. Analogously, we show that the processes $S^1 Z, \ldots, S^d Z$ are martingales, proving that Z is an EMD for $\mathbb{S}$ .

(iii) $\Rightarrow$ (ii): Note that D is an EMD for $\mathbb{S} B^{-1}$ . Let $\mathbb{Q} \approx \mathbb{P}$ be the equivalent probability measure on $(\Omega,\mathscr{F}_T)$ with Radon–Nikodym derivative $\frac{d \mathbb{Q}}{d \mathbb{P}} = D_T / D_0$ . Then $\mathbb{Q}$ is an EMM for $\mathbb{S} B^{-1}$ .

(ii) $\Rightarrow$ (i): Let $\xi \in \mathscr{I}_0(\mathbb{S} B^{-1}) \cap L_+^0$ be arbitrary. Then there exists a strategy $\delta \in \Delta(\mathbb{S} B^{-1})$ such that $(\delta \bullet (S B^{-1}))_T = \xi$ . Since $\mathbb{Q}$ is an EMM for $\mathbb{S} B^{-1}$ , the process $M \,:\!=\, \delta \bullet (S B^{-1})$ is a d-martingale transform under $\mathbb{Q}$ . Therefore, by [Reference Jacod and Shiryaev38, Theorem 1] the process M is a generalized $\mathbb{Q}$ -martingale. Hence, we have $\mathbb{Q}$ -almost surely

\begin{align*}\mathbb{E}_{\mathbb{Q}}[\xi] = \mathbb{E}_{\mathbb{Q}}[M_T] = \mathbb{E}_{\mathbb{Q}}[M_T | \mathscr{F}_0] = M_0 = 0.\end{align*}

Since $\xi \geq 0$ and $\mathbb{Q} \approx \mathbb{P}$ , we deduce that $\xi = 0$ . Hence $\mathscr{I}_0(\mathbb{S} B^{-1})$ satisfies NA, and by Proposition B.1 it follows that $\mathscr{P}_{{\text{sf}},0}(\mathbb{S} \cup \{ B \})$ satisfies NA.

In the previous result, it is important to note that NA is equivalent to the existence of an EMM $\mathbb{Q} \approx \mathbb{P}$ , which connects to the well-known no-arbitrage result in discrete time; see, for example, [Reference Föllmer and Schied25]. This is due to the fact that, in the present discrete-time setting, for every deflator as in Theorem 4.2 which is a multiplicative special semimartingale, the local martingale part is a true martingale, which gives rise to the aforementioned measure change. This finding indicates that without moving to continuous-time modeling, one would be unable to exploit less expensive real-world pricing that would work in practice when the existing market had an ELMD that was a strict local martingale.