The Jacod-Yor Theorem for Sigma Martingales and the Second Fundamental Theorem of Asset Pricing

1. Introduction

The question of determining criteria for a class of processes $\mathcal{A}$ to be representable as a stochastic integral with respect to a given semimartingale X is fundamental to many applications, particularly in modeling, risk management, and decision-making frameworks. For example, martingale representation theorems enable the expression of portfolio wealth in self-financing trading strategies, directly linking the hedgeability of a claim to its representation as a stochastic integral. Beyond finance, these theorems are indispensable in filtering theory for constructing optimal filters for stochastic signals and in stochastic control for characterizing solutions to stochastic optimal control problems.

Most commonly, the martingale representation theorem is studied in the context of Brownian motion. Originally developed by Itô [16], the theory was extended to square-integrable martingales by Kunita and Watanabe [23] and by Davis and Varaiya [5]. Occasionally, representation theorems are also developed for processes other than Brownian motion such as Possion processes ([22] or Lévy processes ([2,34]).

For local martingales, representation theory is notably sparse, with only a few significant contributions, such as [4,9,24]. Many of these works depend on the Jacod–Yor Theorem, introduced by Jacod and Yor [18], which provides necessary and sufficient conditions for the existence of a martingale representation. While this theorem has been extensively studied for $L^2$-martingales and, in some cases, local martingales, it has not been extended to sigma martingales.

No general martingale representation theorem exists for sigma martingales. In particular, the Jacod–Yor Theorem has not been established for this class of processes. Karandikar and Rao [20] explicitly acknowledge this shortfall:

"To the best of our knowledge, the martingale representation property in the framework of sigma martingales is not available in the literature. Indeed, most treatments deal with square-integrable martingales where the notion of orthogonality of martingales is available, which simplifies the treatment."

Sigma martingales generalize local martingales and are particularly important in financial mathematics. For example, the First Fundamental Theorem of Asset Pricing (FTAP) asserts the existence of an equivalent sigma martingale measure under the condition of No Lunch with Vanishing Risk (NFLVR), as shown by Delbaen and Schachermayer [7]. This result states that, under certain no-arbitrage conditions, discounted price processes are sigma martingales.

The lack of a martingale representation theory for sigma martingales also affects the Second Fundamental Theorem of Asset Pricing (Second FTAP), which concerns market completeness. The classical Second FTAP, attributed to Harrison et al. [12,13,14], is limited to cases where discounted price processes are martingales or local martingales. While versions of the Second FTAP for sigma martingales are presented in [37] and [20], these works deviate from the economically justified approaches commonly associated with the First FTAP.

This paper aims to fill these gaps by providing a version of the Jacod–Yor Theorem for sigma martingales. As an application, we prove the Second Fundamental Theorem of Asset Pricing in a general form, thereby laying the foundation for a unified theory that bridges sigma martingales, their representation properties, and their role in financial mathematics.

2. Definitions and Main Theorems

Sigma martingales were first introduced as processes "de la classe (${\Sigma }_m$)" in [3] and were later formalized in [10]. They arise naturally in financial mathematics, as any process that can be represented as a stochastic integral with respect to a local martingale is necessarily a sigma martingale.

There are several equivalent definitions of sigma martingales. Our definition aligns with that of Goll and Kallsen [11]. Different definitions are used by Jacod and Shiryaev [19], Protter [35], and Delbaen and Schachermayer [7]. Nevertheless, they are all equivalent, as is shown in Theorem 1.

 Definition 1.

Let $\mathcal{P}$ denote the predictable σ-algebra.

• A one-dimensional semimartingale S is called a sigma martingale if there exists a sequence of predictable sets $D_n\subset \mathcal{P}$, satisfying  $D_n\subset D_{n+1}$ for all n, and ${\bigcup }_{n=1}^{\infty }{D}_n=\Omega \times {\mathbb{R}}_{+}$. Additionally, for any $n\ge 1$, the process  $1_{D_n}•$ S  $is a uniformly integrable martingale.$
• A d$-dimensional semimartingale is called a\mathit{sigma\; martingale}if each of its components is a one-dimensional sigma martingale.$
• The vector space of all d$-dimensional sigma martingales is denoted by$ ${\mathcal{M}}_d^v$.

We recall that ${\mathcal{H}}^p$ denotes the space of ${\mathcal{H}}^p$-martingales (see Appendix A for details).

The following theorem provides multiple equivalent characterizations of sigma martingales.

Theorem 1 (Characterization of Sigma Martingales) Let $X=(X^1,\dots ,X^d)$ be a d-dimensional semimartingale. The following are equivalent:

• The process X is a sigma martingale.
• There exists a strictly positive predictable process H and an ${\mathcal{H}}^1$-martingale $M=(M^1,\dots ,M^d)$such that  

  $$X^i=H•M^i\mathit{for}\mathit{all}i\in \{1,\dots ,d\}.$$
• There exists a local martingale M = (M¹, …, M^d) $and a predictable process$ H = (H¹, …, H^d)  $with$ 

  $$H^i\in L\left(M^i\right)and{X}^i=H^i•M^i\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathit{for}\mathit{all}i\in \{1,\dots ,d\}.$$
• There exists a sequence D_n ⊂ $\mathcal{P}$ $such that$ D_n ⊂ D_n+1, ${\cup }_{n=1}^{\infty }$ D_n = Ω × $\mathbb{R}$₊ $,and for any$ n ≥ 1, $1_{D_n}$ • X $is a local martingale.$
• There exists a sequenceD_n ⊂ $\mathcal{P}$ $such that$ D_n ⊂ $\mathcal{P}$ such that D_n⊂ D_n+1, ${\cup }_{n=1}^{\infty }$ D_n = Ω × $\mathbb{R}$₊, $and for any$ n ≥ 1, $1_{D_n}$•X $is a sigma martingale.$

The predictable representation property (PRP) is usually defined for $L^2$-martingales; see, for example, [35]. Generalizations exist for local martingales as in [15,37]. We extend this concept further to sigma martingales.

Definition 2. For $X\in {\mathcal{M}}_{\sigma ,0}^d$, define:

$$\mathcal{I}\left(X\right):=\{H•X;H\in L(X)$$

We say that Xhas the predictable representation property (PRP) if 

$$\mathcal{I}\left(X\right)={\mathcal{M}}_{\sigma ,0}^1.$$

$Furthermore, define$

$${\mathcal{I}}^1\left(X\right):=\mathcal{I}\left(X\right)\cap {\mathcal{H}}^1.$$

The PRP is fundamental in applications like mathematical finance, where it enables us to express contingent claims or wealth processes in terms of integrals with respect to a sigma martingale. This property is also central to the Jacod-Yor Theorem for sigma martingales.

Remark 1. The set $\mathcal{I}\left(X\right)$ depends on the probability measure  $\mathbf{P}$. However, for the sake of simplicity, we only write  $\mathcal{I}\left(X\right)$ instead of  $\mathcal{I}(X,\mathbf{P})$.

The importance of orthogonality in stochastic analysis traces back to foundational work by P. A. Meyer in the 1960s. In his seminal papers [30,31], Meyer laid the groundwork for understanding martingales using Hilbert-space methods. This perspective was further developed by Itô, Motoo and Kunita [17,23,32], who demonstrated that the space of $L^2$-martingales could be treated as a Hilbert space. These techniques and tools allowed for significant advancements in the theory of stochastic analysis.

Inspired by this classical notion of orthogonality, we extend the concept to sigma martingales.

Definition 3. Let $X\in {\mathcal{M}}_{\sigma }^d$. We say a process $Z\in {\mathcal{M}}_{\sigma }^1$ is orthogonal  to X if  $XZ\in {\mathcal{M}}_{\sigma }^d$, and we write  $X\perp Z$. Furthermore, we define the set of all processes orthogonal to X as

$$X^{\perp }=\{Z\in {\mathcal{M}}_{\sigma }^1;X\perp Z\}.$$

Remark 2. Clearly,  $X^{\perp }$ contains all constant processes and for$d=1$, by partial integration, two sigma martingales  $X,Y$ are orthogonal if and only if $[X,Y]\in {\mathcal{M}}_{\sigma }^1$.

Definition 4. Let $X\in {\mathcal{M}}_{\sigma }^d$. Then  ${\mathfrak{P}}_{\sigma }\left(X\right)$ denotes the sets of all probability measures  $\mathbf{Q}$ on $(\Omega ,\mathcal{F})$ such that X is a sigma martingale under $\mathbf{Q}$. A measure $\mathbf{P}\in {\mathfrak{P}}_{\sigma }\left(X\right)$ is extremal if, for any $\mathbf{Q},\mathbf{R}\in {\mathfrak{P}}_{\sigma }\left(X\right)$ and $\lambda \in (0,1)$ such that $\mathbf{P}=\lambda \mathbf{Q}+(1-\lambda )\mathbf{R}$, we have $\mathbf{P}=\mathbf{Q}=\mathbf{R}$.

The main result of this paper is a generalization of the classical Jacod-Yor Theorem to sigma martingales.

Theorem 2   (Jacod-Yor Theorem for Sigma Martingales).   Let  $X\in {\mathcal{M}}_{\sigma }^d$. The following are equivalent:

• The process X has the predictable representation property.
• The set $X^{\perp }$ only contains the constant processes.
• For any $\mathbf{Q}\in {\mathfrak{P}}_{\sigma }\left(X\right)$ with $\mathbf{Q}\ll \mathbf{P}$,  we have $\mathbf{P}=\mathbf{Q}$.
• The probability measure $\mathbf{P}$ is an extremal point of ${\mathfrak{P}}_{\sigma }\left(X\right)$.

As a classical martingale representation theorem, the Jacod-Yor Theorem has multiple applications. As probably the most popular one, we are going to prove the Second Fundamental Theorem of Asset Pricing for unbounded stochastic processes for a market that satisfies NFLVR. By the First Fundamental Theorem of Asset Pricing [7], we can assume that the discounted price processes are sigma martingales with respect to to some equivalent probability measure.

Our definition of the financial market is essentially the same as in [8] and [7]. For simplicity, we assume discounted processes under an equivalent sigma martingale measure. These simplifications can be justified via the First Fundamental Theorem of Asset Pricing and basic results about general markets. For a general derivation of that setting, starting from a non-discounted setup with a real-world probability measure, we refer to [38].

We consider a financial market consisting of $d+1$ tradeable securities whose price processes are represented by the $d+1$-dimensional process $S_t={(S_t^0,S_t^1,\dots ,S_t^d)}_{t\in {\mathbb{R}}_{+}}$ adapted to the filtration $\mathbb{F}={\left({\mathcal{F}}_t\right)}_{t\in {\mathbb{R}}_{+}}$. We make the following assumptions:

• The processes $S_t^i$ are sigma martingales for all $i=0,\dots ,d$.
• The filtration $\mathbb{F}$ satisfies the usual conditions, and the σ-algebra ${\mathcal{F}}_0$ is trivial; that is, $A\in {\mathcal{F}}_0$ implies $\mathbb{P}\left(A\right)=0$ or $\mathbb{P}\left(A\right)=1$.

Definition 5. 

• We call a $d+1$-dimensional process $\phi =({\phi }^0,\dots ,{\phi }^d)\in L\left(S\right)$ a trading strategy.
• The wealth process of the investor is defined as

  $$V_t:=\sum _{i=0}^d{\phi }_t^i{S}_t^i,t\in {\mathbb{R}}_{+},$$
  where ${\phi }_t^i$ represents the number of the i-th security that the investor holds in their portfolio at t.
• A trading strategy $\phi =({\phi }^0,{\phi }^1,\dots ,{\phi }^d)$ is called self-financing if the wealth process $V_t\left(\phi \right)$ satisfies

  $$V_t=\phi ·S_t, t\in {ℝ}_{+}.$$

  (1)
• A self-financing strategy  $\phi =({\phi }^0,{\phi }^1,\dots ,{\phi }^d)\in L\left(S\right)$ is called admissible if there exists an $\alpha \in {\mathbb{R}}_{+}$ such that 

  $$V_t\ge -\alpha \forall t\in [0,\infty )$$

   for the corresponding wealth process V.
• A contingent claim X settled at time T is a non-negative ${\mathcal{F}}_T$ -measurable, integrable random variable.

We now turn to the Second Fundamental Theorem of Asset Pricing. In the case of finitely many assets, a version of this theorem was first proved for the discrete-time setting in [12]. The proof for the continuous-time case followed in [13,14], but Sigrid Müller showed in [33] that the statement in a higher-dimensional case is not valid for the componentwise stochastic integral without additional assumptions. However, one can bypass the additional assumptions by not restricting the class $L\left(S\right)$ to the componentwise-integrable integrands, as shown in [27] and [28]. In all these publications, however, a probability measure is assumed under which the process $S^i$ is a local martingale and not just a sigma martingale.

Theorem 3  (Second Fundamental Theorem of Asset Pricing). Let ${\left(S_t\right)}_{t\in [0,T]}$ (with  $T<\infty$) be a $d+1$-dimensional sigma martingale under $\mathbf{P}$. The following are equivalent:

• The market is complete, that means for each contingent claim X, there exists an admissible self-financing $\phi \in L\left(S\right)$ such that $\phi •$ S $is a true martingale and$ (•S)T = X
• $For each uniformly integrable martingale$ M $,there is a$ φ ∈ L(S) $with$

  $$M=\phi •S.$$
• $For each$ X ∈ ${ℳ}_{\mathrm{loc}}^1$ $,there is a$ φ ∈ L(S) $with$

  $$M=\phi •S.$$
• $For each$ X ∈ ${ℳ}_{\sigma }^1$ $,there is a$ φ ∈ L(S) $with$

  $$X=\phi •S.$$
• $For any other measure$  Q $equivalent to$ $\mathbf{P}$ $under which$ S $is a sigma martingale, we must have$ $\mathbf{Q}$ = $\mathbf{P}$.

Proof. (i) ⇒ (ii). Assume the market is complete. Let M be a uniformly integrable martingale. Then $\left|M\right|$ is a submartingale with ${sup}_{0\le t\le T}\mathbf{E}\left[\right|M_t\left|\right]\le \mathbf{E}\left[\right|M_T\left|\right]<\infty$. By the Krickeberg (or Doob–Meyer) decomposition, M can be assumed w.l.o.g. nonnegative. Then $M_T\ge 0$ is a claim. Completeness implies there is a self-financing φ such that  $(\phi •$S)  $\mathrm{i}\mathrm{s} \mathrm{a} \mathrm{t}\mathrm{r}\mathrm{u}\mathrm{e}$ L¹  $-\mathrm{m}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{a}\mathrm{l}\mathrm{e} \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} \mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l} \mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}$ M_T  $.\mathrm{T}\mathrm{w}\mathrm{o} \mathrm{U}\mathrm{I} \mathrm{m}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{s} \mathrm{a}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{a}\mathrm{t} \mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}$ T  $\mathrm{m}\mathrm{u}\mathrm{s}\mathrm{t} \mathrm{c}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{i}\mathrm{d}\mathrm{e} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{l}\mathrm{l}$ t ≤ T $.\mathrm{T}\mathrm{h}\mathrm{u}\mathrm{s}$ M = φ •S.

• (ii) ⇒ (iv). Take $X\in {\mathcal{M}}_{\sigma }^1$. By definition (see [3,10,11] or Theorem 1), there is a strictly positive predictable process H such that $H·X$ is a true (UI) martingale. Hence (ii) says $H·X=\psi •S$ $\mathrm{f}\mathrm{o}\mathrm{r} \mathrm{s}\mathrm{o}\mathrm{m}\mathrm{e}$ Ψ ∈ L(S)  $.\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{e}$   H > 0   $,\mathrm{w}\mathrm{e} \mathrm{r}\mathrm{e}-\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{e} \mathrm{o}\mathrm{r} \mathrm{r}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e} \mathrm{t}\mathrm{o} \mathrm{g}\mathrm{e}\mathrm{t}$ X = Φ •S$. \mathrm{H}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e},$   X   $\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{o} \mathrm{a}\mathrm{d}\mathrm{m}\mathrm{i}\mathrm{t}\mathrm{s} \mathrm{a} \mathrm{m}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{a}\mathrm{l}\mathrm{e} \mathrm{r}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{o}\mathrm{f}$ S.
• (iv) ⇔ (v). This follows directly from the Jacod-Yor Theorem (Theorem 2).
• (iv) ⇒ (iii) ⇒ (ii). These inclusions are obvious.
• (ii) ⇒ (i). Finally, if every UI martingale is a true martingale stochastic integral in S, then the model is complete. Indeed, let $X\ge 0$ be any claim. Define $M_t:=\mathbf{E}\left[X\right|{\mathcal{F}}_t]$, a UI martingale. By (ii), $M=\phi •$S $\mathrm{for} \mathrm{some} \mathrm{predictable}$ φ $.\mathrm{Since} \mathrm{we} \mathrm{start} \mathrm{the} \mathrm{trading} \mathrm{strategy} \mathrm{at} \mathrm{time}$ 0 $\mathrm{with} \mathrm{wealth}$ M₀ = $\mathbf{E}$[X] $, \mathrm{this}$ φ •S $\mathrm{must} \mathrm{be} \mathrm{a} \mathrm{self}-\mathrm{financing} \mathrm{wealth} \mathrm{process} \mathrm{and}$ (φ•S)_T = X$. \mathrm{Thus},$ X $\mathrm{is} \mathrm{replicated}, \mathrm{proving} \mathrm{completeness}.$ □

Remark 3. In the statement of completeness, we require that any replicating strategy $\phi •$ S $be a true($ L^1- $) martingale rather than just a local martingale (or a super martingale). Local martingale strategies do not necessarily pin down the terminal value at time$ T $due to possible unbounded variations or jumps, and uniform integrability is essential to guarantee the desired payoff$ X $is matched exactly at$ T .

Some authors ensure the true martingale property by requiring that all replicating strategies be bounded (or that φ be bounded), which trivially forces $(\phi •$S) $to be a UI martingale. However, this is clearly a stronger restriction than ours, as we only require$  (φ •S) $itself to be a true martingale.$ 

For an example of a market that admits a hedging strategy replicating a claim but lacks a martingale hedging strategy and, consequently, market completeness, refer to Example 7.3 in [37].