Concepts of Statistical Causality and Strong and Weak Properties of Predictable Representation

Abstract

The paper considers the statistical concept of causality in continuous time, which is based on Granger’s definition of causality. We give necessary and sufficient conditions, in terms of statistical causality, for the preservation of the strong property of predictable representation for stopped martingales when filtration is decreased. This concept of causality is also connected to the preservation of the strong property of predictable representation under a change in measure. In addition, we give conditions, in terms of statistical causality, for martingales to have strong and weak properties of predictable representation. The results are applied to the problem of pricing claims in incomplete financial markets.

1. Introduction

In recent decades, causality has been a topic of great interest to philosophers and scientists from many different areas.

The problem of finding the causal relationships between two appearances is to answer the question “Where do the relationships among the phenomena come from?” The causal relations can be detected if controlled experiments or an observation under specified conditions can be conducted. In that case, the hypotheses concerning the causal relationships can be verified much easier. But if such experiments are not available, hypotheses about causal relationships can be verified by establishing the correct theory based on facts that are known, which permits correct predictions when the theory is applied to new phenomena. This procedure provides us with a better understanding of the causal relationships between objects.

This kind of prediction is commonly used in many areas of science. For example, chemists have concluded that, in the case of a certain class of hydrocarbons, the boiling point decreases as the number of carbon atoms in a molecule increases. Therefore, it can be predicted that a new type of molecule with more carbon atoms than any of those yet created will probably have a lower boiling point. In physics, it was discovered that there exist isotopes of each element, which are different kinds of atoms that have the same chemical properties but different atomic weights. From physical theory, it is known that different isotopes diffuse at different rates when there are differences in concentration. On the basis of this predicted difference, a method was developed that made the separation of the two isotopes of uranium possible. This method of separation was essential for creating a nuclear reactor. Based on the known theory, it was predicted that uranium exposed to neutrons should be transformed into a new element, plutonium, which was not produced before. Many physical and chemical properties of this new element were predicted, too (see [1,2]).

In this paper, we investigate a concept of statistical causality that links the Granger causality with some associated concepts (see [3,4,5]). But in many situations, it is very difficult to investigate relations of causality in a discrete-time model. The continuous time framework is fruitful for the statistical analysis of stochastic processes that quickly evolve (see [6]). Therefore, in applications, continuous time models are increasingly used (see, for example, [4,5,7]). This concept of causality is shown to be closely connected to orthogonal martingales ([8]), extremal measures (see [9]), and optional and predictable projections (see [10]). Let us mention that these connections exist even when, instead of an infinite horizon, we consider stopping times and suitable stopped (truncated) filtrations (see [8,9,10]). It is also known that the martingale property is preserved if the filtration decreases, but if the filtration increases, the preservation of the martingale property crucially depends on the concept of causality (see [11]).

When we consider the strong and weak properties of predictable representation, it is important to know whether this representation property is relative to filtration $\left\{{\mathcal{F}}_t\right\}$ or $\left\{{\mathcal{G}}_t\right\}$ ($\left\{{\mathcal{G}}_t\right\}\subseteq \left\{{\mathcal{F}}_t\right\}$). There are a lot of examples where a process has a strong property of predictable representation with respect to $\left\{{\mathcal{G}}_t\right\}$ but not with respect to $\left\{{\mathcal{F}}_t\right\}$. Thus, it is natural to ask whether the converse statement holds, i.e., if a process has a strong property of predictable representation with respect to $\left\{{\mathcal{F}}_t\right\}$ under which conditions this will also remain true for filtration $\left\{{\mathcal{G}}_t\right\}$. The conditions for the stability of the strong property of predictable representation under enlargement of filtration for a semimartingale X are discussed in [12], and for the stability of the weak property of predictable representation for semimartingales, they are discussed in [13]. Let us mention that under initial enlargement or when enlargement starts from a trivial $\sigma$-algebra, this property, in general, is not preserved (see [12]). The authors of [14] proved that the preservation of this property crucially depends on the concept of causality. Here, we investigate the conditions in terms of stopped causality for the preservation of the property of predictable representation for the stopped processes.

The definitions of causality, stopped causality, and some properties of the statistical concept of causality between flows of information are represented in Section 2. Also, this section presents some known results about stopping times and stopped filtrations.

Section 3 contains the main results. This section establishes the equivalence between the concept of stopped causality and the preservation of the strong property of predictable representation when the stopped filtration is decreasing. Also, the invariance of the strong property of predictable representation under the change in measure is strongly connected to the concept of stopped causality for stopped filtrations. The necessary and sufficient conditions are established for a process to have a strong property of predictable representation. The relationship between the concept of causality and the weak property of predictable representation is also considered.

Section 4 gives an application of the results of the problem of pricing claims in incomplete financial markets. Some concluding remarks are presented at the end of the article.

2. Preliminaries and Notation

Let $(\mathsf{\Omega },\mathcal{A},P)$ be a probability space and $\mathbf{F}=\{{\mathcal{F}}_t,t\in I,I\subseteq R^{+}\}$ be a filtration that satisfies the usual conditions of right continuity and completeness. A stochastic basis for a time-dependent system is introduced by $(\mathsf{\Omega },\mathcal{A},{\mathcal{F}}_t,P)$. The filtration can be described as a flow of information. The smallest $\sigma$-algebra containing all $\left({\mathcal{F}}_t\right)$ (even if $supI<+\infty$) is given with ${\mathcal{F}}_{\infty }={\bigvee }_{t\in I}{\mathcal{F}}_t$. For filtrations $\mathbf{J}=\left\{{\mathcal{J}}_{\mathbf{t}}\right\}$, $\mathbf{G}=\left\{{\mathcal{G}}_{\mathbf{t}}\right\}$ and $\mathbf{E}=\left\{{\mathcal{E}}_{\mathbf{t}}\right\}$, an analogous notation is used. Natural filtration of the process $X=\{X_t,t\in I\}$ is the smallest $\sigma$-algebra with respect to which the random variables $X_u,u\le t$ are measurable, and it is given by ${\mathbf{F}}^X=\{{\mathcal{F}}_t^X,t\in I\}$, where ${\mathcal{F}}_t^X=\sigma \{X_u,u\in I,u\le t\}$.

The intuitive notion of causality in continuous time is given in [15]. Here, the same notion of causality is used for filtrations that use the concept of conditional independence between $\sigma$-algebras.

Definition 1

(see [3,15]). It is said that $\mathbf{J}$ is a cause of $\mathbf{E}$ within $\mathbf{F}$ relative to P (and written as $\mathbf{E}|<\mathbf{J};\mathbf{F}$;P) if ${\mathcal{E}}_{\infty }\subseteq {\mathcal{F}}_{\infty }$, $\mathbf{J}\subseteq \mathbf{F}$ and if ${\mathcal{E}}_{\infty }$ is conditionally independent of $\left\{{\mathcal{F}}_t\right\}$ given $\left\{{\mathcal{J}}_t\right\}$ for each t i.e., ${\mathcal{E}}_{\infty }\perp {\mathcal{F}}_t|{\mathcal{J}}_t$ (i.e., ${\mathcal{E}}_u\perp {\mathcal{F}}_t|{\mathcal{J}}_t$ holds for each t and each u) or

$$(\forall A\in {\mathcal{E}}_{\infty })P\left(A\right|{\mathcal{F}}_t)=P\left(A\right|{\mathcal{J}}_t).$$

Intuitively, $\mathbf{E}|<\mathbf{J};\mathbf{F};P$ indicates that, for arbitrary t, $\left\{{\mathcal{J}}_t\right\}$ carries all the information from the $\left\{{\mathcal{F}}_t\right\}$ necessary for forecasting the ${\mathcal{E}}_{\infty }$.

If $\mathbf{J}$ and $\mathbf{F}$ are such that $\mathbf{J}|<\mathbf{J};\mathbf{F};P$, we shall say that $\mathbf{J}$ is self-caused within $\mathbf{F}$ (in [5], the term “its own cause” is used). Let us point out that “$\mathbf{J}$ is self-caused” can be usefully applied in theory of martingales and projections (see [11]).

It is possible to apply these definitions to stochastic processes if we take into account their natural filtrations. For example, the $\left\{{\mathcal{F}}_t\right\}$-adapted stochastic process $X_t$ is self caused if $\left\{{\mathcal{F}}_t^X\right\}$ is self-caused within $\left\{{\mathcal{F}}_t\right\}$ i.e., if ${\mathbf{F}}^X|<{\mathbf{F}}^X;\mathbf{F};P$ holds.

The self-caused process X is entirely determined by its way of behaving with respect to its natural filtration ${\mathbf{F}}^X$ (see [7]). For example, process $X=\{X_t,t\in I\}$ is a Markov process with respect to ${\mathbf{F}}^X$, and it is self-caused within $\mathbf{F}$ relative to P if and only if X is a Markov process with respect to the filtration $\mathbf{F}=\{{\mathcal{F}}_t,t\in I\}$. Obviously, the same holds for Brownian motion $W=\{W_t,t\in I\}$ (see [7]).

It is of interest to mention that in many applications, we consider systems up to some random time. The extension of the Definition 1 from fixed times to a random variable, i.e., the definition of causality using the stopped filtrations is introduced in [16].

Definition 2

([17]). A ${\mathbb{R}}_{+}$ random variable T defined on $(\mathsf{\Omega },\mathcal{A})$ is called a $\left\{{\mathcal{F}}_t\right\}$ stopping time if for each $t\ge 0$, $\{T\le t\}\in \left\{{\mathcal{F}}_t\right\}$.

The stopping time $\sigma$-algebra is defined by ${\mathcal{F}}_T=\{A\in \mathcal{F}:A\cap \{T\le t\}\in {\mathcal{F}}_t\}$ where T is $\left\{{\mathcal{F}}_t\right\}$ stopping time. Intuitively, ${\mathcal{F}}_T$ is the information available at time T. For a process X, $X_T\left(\omega \right)=X_{T\left(\omega \right)}\left(\omega \right)$ whenever $T\left(\omega \right)<+\infty$. $X^T=\{X_{t\wedge T},t\in I\}$ with

$$X_t^T\left(\omega \right)=X_{t\wedge T\left(\omega \right)}\left(\omega \right)=X_t{\chi }_{\{t<T\}}+X_T{\chi }_{\{t\ge T\}}$$

is the stopped process. If T is a stopping time and X is adapted and cadlag, then the stopped process $X^T=X_{t\wedge T}$ is adapted, too. Also, a martingale that stopps at a stopping time T is still a martingale (Theorem I.18 in [18]). The stopped filtration $\left\{{\mathcal{F}}_{t\wedge T}\right\}$ is defined as

$${\mathcal{F}}_{t\wedge T}={\mathcal{F}}_t\cap {\mathcal{F}}_T=\left\{\begin{array}{c}{\mathcal{F}}_t,t<T,\hfill \\ {\mathcal{F}}_T,t\ge T,\hfill \end{array}\right.$$

For the stopped martingale $X_{t\wedge T}$, the natural filtration is ${\mathbf{F}}^{X^T}=\left\{{\mathcal{F}}_{t\wedge T}^X\right\}$, with respect to which the process $X_{t\wedge T}$ is entirely described. So we can use the definition of causality in continuous time, which involves the stopping times.

The generalization of the Definition 1 from fixed to stopping time is introduced in [16].

Definition 3

([16]). Let $\mathbf{F}=\left\{{\mathcal{F}}_t\right\}$, $\mathbf{J}=\left\{{\mathcal{J}}_t\right\}$ and $\mathbf{E}=\left\{{\mathcal{E}}_t\right\},t\in I$ be given filtrations on the probability space $(\mathsf{\Omega },\mathcal{A},P)$, and let T be a stopping time with respect to filtration $\mathbf{E}$. The filtration ${\mathbf{J}}^T$ entirely causes ${\mathbf{E}}^T$ within ${\mathbf{F}}^T$ relative to P (and written as ${\mathbf{E}}^T|<{\mathbf{J}}^T;{\mathbf{F}}^T;P$) if ${\mathbf{E}}^T\subseteq {\mathbf{F}}^T$, ${\mathbf{J}}^T\subseteq {\mathbf{F}}^T$ and if $\left\{{\mathcal{E}}_T\right\}$ is conditionally independent on $\left\{{\mathcal{F}}_{t\wedge T}\right\}$ given $\left\{{\mathcal{J}}_{t\wedge T}\right\}$ for each t, i.e., $(\forall t){\mathcal{E}}_T\perp {\mathcal{F}}_{t\wedge T}\mid {\mathcal{J}}_{t\wedge T},$ or

$$(\forall t\in I)(\forall A\in {\mathcal{E}}_T)P(A\mid {\mathcal{F}}_{t\wedge T})=P(A\mid {\mathcal{J}}_{t\wedge T}).$$

(1)

Remark 1.

Florens and Fougères investigated a related concept of causality with stopping times in [19], and they gave characterizations of causality associated with stopping times (see [19] Theorem 2.1).

Some elementary properties of this concept are introduced in [8,16].

The strong and weak predictable representation theory is important in applications in stochastic control, filtering theory, and finance.

Definition 4

([17]). Let $M_t$ be a local martingale with $M_0=0$. Then, $M_t$ has the strong property of predictable representation if, for any $\left\{{\mathcal{F}}_t^M\right\}$-local martingale X, there is an $\left\{{\mathcal{F}}_t^M\right\}$-predictable process $K_t$ such that $X=K·M$, i.e.,

$$X_t=b+{\int }_0^t{K}_{u}d{M}_u.$$

Lemma 1

([17]). Assume that the local martingale $M_t$ has the strong property of predictable representation. Then, for any stopping time T, $M^T=M_{t\wedge T}$ has a strong property of predictable representation with respect to $\left\{{\mathcal{F}}_{t\wedge T}\right\}$.

Definition 5

([17]). Let $Z_t$ be a semimartingale with μ, $Z^c$, and let $(A,B,\nu )$ be its jump measure, continuous martingale part, and predictable characteristics, respectively. Process Z has the weak property of predictable representation if

$${\mathcal{M}}_{loc,0}=\mathcal{L}\left(Z^c\right)+\mathcal{K}\left(\mu \right)$$

where ${\mathcal{M}}_{loc,0}$ is a set of local martingales, vanishing at 0, $\mathcal{L}\left(Z^c\right)$ is the set of stochastic integrals

$$b+\int K_{u}d{Z}^c$$

and $\mathcal{K}\left(\mu \right)$ is a set of stochastic integrals

$$\int Y(s,z)(\mu (ds,dz)-\nu (ds,dz))$$

in the sense of Definition 11.16 in [17], where Y is a predictable process.

The next definition that deals with the notion of extremal measure has a great application. For example, the extremal point of some convex set of probability measures can only have a trivial decomposition. If some set of measures consists of a single element P (singleton), this measure is extremal, and this triviality is very important (see [20,21]).

Definition 6

([22]). A probability measure P of $\mathcal{M}$ is called extremal if, whenever $P=\alpha Q+(1-\alpha )R$ with $0<\alpha <1$ and $Q,R\in \mathcal{M}$, then $P=Q=R$.

3. Causality and Strong and Weak Properties of Predictable Representation

The strong property of predictable representation has been a constantly interesting theme in stochastic analysis ever since the results of K. Ito for Brownian filtration.

The connection between causality and martingale representation property is established in [14]. This paper investigates the concepts of the strong property of predictable representation (in the sense of Definition 4), the weak property of predictable representation (in the sense of Definition 5), and their connections to the stopped filtrations and processes in terms of statistical causality concepts.

Theorem 1.

Let $\mathbf{G}\subseteq \mathbf{F}$ be given filtrations on the probability space $(\mathsf{\Omega },\mathcal{A},{\mathcal{F}}_t,P)$, where T is a $\left\{{\mathcal{G}}_t\right\}$—stopping time and $\left\{{\mathcal{F}}_{t\wedge T}\right\}$—local martingale $M_{t\wedge T}$, which is ${\mathcal{G}}_T$-measurable and has a strong property of predictable representation with respect to $\left\{{\mathcal{F}}_{t\wedge T}\right\}$. Then. $M_{t\wedge T}$ has a strong property of predictable representation with respect to $\left\{{\mathcal{G}}_{t\wedge T}\right\}$ if and only if $\left\{{\mathcal{G}}_{t\wedge T}\right\}$ is self-caused within $\left\{{\mathcal{F}}_{t\wedge T}\right\}$, i.e., ${\mathbf{G}}^T|<{\mathbf{G}}^T;{\mathbf{F}}^T;P$ holds.

Proof. 

Let $(\mathsf{\Omega },\mathcal{A},{\mathcal{F}}_t,P)$ be a probability space and $\mathbf{G}\subseteq \mathbf{F}$. Suppose that for a $\left\{{\mathcal{G}}_t\right\}$-stopping time T, process $M^T=M_{t\wedge T}$ has a strong property of predictable representation with respect to $\left\{{\mathcal{G}}_{t\wedge T}\right\}$, i.e., there is a $\left\{{\mathcal{G}}_{t\wedge T}\right\}$-stopped local martingale $X^T=X_{t\wedge T}$ for which we have

$$X_{t\wedge T}=E(X_T\mid {\mathcal{G}}_{t\wedge T})=b+{\int }_0^{t\wedge T}{K}_{u}d{M}_u$$

where $H_{t\wedge T}$ is a $\left\{{\mathcal{G}}_{t\wedge T}\right\}$-predictable process. Also, there is a sequence $\left\{T_n\right\}$ of $\left\{{\mathcal{G}}_t\right\}$-stopping times such that $\left\{X_{t\wedge T\wedge T_n}\right\}$ is a sequence of $\left\{{\mathcal{G}}_{t\wedge T}\right\}$-stopped martingales, because $\mathbf{G}\subseteq \mathbf{F}$, $\left\{T_n\right\}$ is a sequence of $\left\{{\mathcal{F}}_t\right\}$—stopping times as well. By assumption, $M_{t\wedge T}$ has a strong property of predictable representation, so $\left\{X_{t\wedge T\wedge T_n}\right\}$ is a sequence of $\left\{{\mathcal{F}}_{t\wedge T}\right\}$-stopped martingales. According to Definition 3, from

$$E(X_T\mid {\mathcal{F}}_{t\wedge T})=E(X_T\mid {\mathcal{G}}_{t\wedge T})=X_{t\wedge T}$$

where $X_T$ is ${\mathcal{G}}_T$—measurable process, follows ${\mathbf{G}}^T|<{\mathbf{G}}^T;{\mathbf{F}}^T;P$.

Conversely, let $M_{t\wedge T}$ have strong property of predictable representation with respect to $\left\{{\mathcal{F}}_{t\wedge T}\right\}$

$$X_{t\wedge T}=E(X_T\mid {\mathcal{F}}_{t\wedge T})=b+{\int }_0^{t\wedge T}{K}_{u}d{M}_u$$

(2)

where $K_{t\wedge T}$ is a $\left\{{\mathcal{F}}_{t\wedge T}\right\}$-predictable process and $X_{t\wedge T}$ is a $\left\{{\mathcal{F}}_{t\wedge T}\right\}$-stopped local martingale. By Theorem 2 in [10], $K_{t\wedge T}$ is a $\left\{{\mathcal{G}}_{t\wedge T}\right\}$-predictable process. Process $M_{t\wedge T}$ is ${\mathcal{G}}_T$-measurable, and because of causality ${\mathbf{G}}^T|<{\mathbf{G}}^T;{\mathbf{F}}^T;P$, for every ${\mathcal{G}}_T$-measurable process $M_T$, we have

$$E(M_T\mid {\mathcal{F}}_{t\wedge T})=E(M_T\mid {\mathcal{G}}_{t\wedge T})=M_{t\wedge T}$$

so process $M_{t\wedge T}$ is a $\left\{{\mathcal{G}}_{t\wedge T}\right\}$-stopped martingale. According to this, $M_{t\wedge T}$ can be represented as

$$L_{t\wedge T}=E(L_T\mid {\mathcal{G}}_{t\wedge T})=b+{\int }_0^{t\wedge T}{K}_{u}d{M}_u$$

(3)

where $L_{t\wedge T}$ is a $\left\{{\mathcal{G}}_{t\wedge T}\right\}$-stopped local martingale.

In order to prove the equality of (2) and (3), assume the opposite, that $L_{t\wedge T}\ne X_{t\wedge T}$. For a sequence $\left\{T_n\right\}$ of $\left\{{\mathcal{G}}_t\right\}$-stopping times, sequence $\left\{L_{t\wedge T\wedge T_n}\right\}$ is a sequence of $\left\{{\mathcal{G}}_{t\wedge T}\right\}$-stopped martingales. Because of $\mathbf{G}\subseteq \mathbf{F}$, sequence $\left\{T_n\right\}$ is a sequence of $\left\{{\mathcal{F}}_t\right\}$-stopping times, too. Thus, because of causality, sequence $\left\{L_{t\wedge T\wedge T_n}\right\}$ is a sequence of $\left\{{\mathcal{F}}_{t\wedge T}\right\}$-stopped martingales. Because the right-hand sides of (2) and (3) are equal, we have $L_{t\wedge T}=X_{t\wedge T}$, and that is a contradiction. Therefore, $L_{t\wedge T}=X_{t\wedge T}$ and the process $X_{t\wedge T}$ is a $\left\{{\mathcal{G}}_{t\wedge T}\right\}$-stopped local martingale and can be represented as

$$X_{t\wedge T}=E(X_T\mid {\mathcal{G}}_{t\wedge T})=b+{\int }_0^{t\wedge T}{K}_{u}d{M}_u.$$

Therefore, $M_{t\wedge T}$ has a strong property of predictable representation with respect to $\left\{{\mathcal{G}}_{t\wedge T}\right\}$, too. □

Suppose that Q is a probability measure on $(\mathsf{\Omega },\mathcal{A})$ that is absolutely continuous with respect to P and $N_{\infty }=\frac{dQ}{dP}$ denotes the Radon–Nikodym derivative. The next theorem establishes conditions in terms of causality for the preservation of the strong property of predictable representation under an absolute, continuous change in measure.

Theorem 2.

Let $\mathbf{G}\subseteq \mathbf{F}$ be given filtrations on $(\mathsf{\Omega },\mathcal{A})$, where T is a $\left\{{\mathcal{G}}_t\right\}$-stopping time, P and Q probability measures on $\mathcal{A}$ satisfying $Q\ll P$ with $\frac{dQ}{dP}$ as ${\mathcal{G}}_T$-measurable. Let ${\mathbf{G}}^T|<{\mathbf{G}}^T;{\mathbf{F}}^T;P$ hold; then, $M^T=M_{t\wedge T}$ has a strong property of predictable representation with respect to $({\mathcal{F}}_{t\wedge T},P)$ if and only if $M^T=M_{t\wedge T}$ has a strong property of predictable representation with respect to $({\mathcal{F}}_{t\wedge T},Q)$.

Proof. 

Let T be a $\left\{{\mathcal{G}}_t\right\}$-stopping time, $Q\ll P$, and let $E_P$ and $E_Q$ represent a conditional expectation of the measures P and Q, respectively. The Radon–Nykodim derivative can be represented as $N_{\infty }=\frac{dQ}{dP}$. Then, the right regular version of the density process $N_{t\wedge T}=N_t{I}_{\{t<T\}}+N_T{I}_{\{t\ge T\}}$ is the càdlàg modification of the $E_P(N_{\infty }^T\mid {\mathcal{F}}_{t\wedge T})$. Since the causality relation holds and $N_{\infty }^T$ is $\left\{{\mathcal{G}}_{t\wedge T}\right\}$—measurable, we have

$$N_{t\wedge T}=E_P(N_{\infty }^T\mid {\mathcal{F}}_{t\wedge T})=E_P(N_{\infty }^T\mid {\mathcal{G}}_{t\wedge T}).$$

Let $M_{t\wedge T}$ have a strong property of predictable representation with respect to $({\mathcal{F}}_{t\wedge T},P)$, i.e.,

$$X_{t\wedge T}=E_P(X_T\mid {\mathcal{F}}_{t\wedge T})=b+{\int }_0^{t\wedge T}{K}_{u}dP\left(M_u\right)$$

where $X_{t\wedge T}$ is a $({\mathcal{F}}_{t\wedge T},P)$-stopped local martingale. Because of causality and using the same procedure as in the proof of the Theorem 3.1 in [8], the process ${\left(XN\right)}^T$ is a $({\mathcal{F}}_{t\wedge T},P)$-martingale. For $s<t<T$ and $F\in \left\{{\mathcal{F}}_{s\wedge T}\right\}$, we have

$$\begin{array}{cc}{\int }_F{X}_{t\wedge T}{N}_{t\wedge T}dP& ={\int }_F{X}_{t\wedge T}\frac{d{Q}_t}{dP}dP={\int }_F{X}_{t\wedge T}d{Q}_t={\int }_F{X}_{s\wedge T}d{Q}_s={\int }_F{X}_{s\wedge T}\frac{d{Q}_s}{dP}dP\\ & ={\int }_F{X}_{s\wedge T}{N}_{s\wedge T}dP,\end{array}$$

(4)

and for $T\le t\le r$, by (4), and due to the Optional Sampling Theorem for $F\in \left\{{\mathcal{F}}_{t\wedge T}\right\}$, we have

$$\begin{array}{ccc}\hfill {\int }_F{X}_{t\wedge T}{N}_{t\wedge T}dP& =& {\int }_F{X}_{t\wedge T}{N}_{T}dP={\int }_F{X}_{t\wedge T}{N}_{t}dP={\int }_F{X}_{t\wedge T}\frac{d{Q}_t}{dP}dP={\int }_F{X}_{t\wedge T}dQ\hfill \\ & =& {\int }_F{X}_{r\wedge T}dQ={\int }_F{X}_{r\wedge T}{N}_{r\wedge T}dP.\hfill \end{array}$$

Therefore,

$$X_{t\wedge T}{N}_{t\wedge T}=E_P(X_T{N}_T\mid {\mathcal{F}}_{t\wedge T}).$$

(5)

So for $N_{t\wedge T}\ne 0$, from the last relation, we have

$$X_{t\wedge T}=\frac{1}{N_{t\wedge T}}{E}_P\left(X_T\frac{dQ}{dP}\mid {\mathcal{F}}_{t\wedge T}\right)=E_Q(X_T\mid {\mathcal{F}}_{t\wedge T})=b+{\int }_0^{t\wedge T}{K}_{u}dQ\left(M_u\right).$$

Therefore, $M_{t\wedge T}$ has a strong property of predictable representation with respect to $({\mathcal{F}}_{t\wedge T},Q)$.

Conversely, suppose that ${\mathbf{G}}^T|<{\mathbf{G}}^T;{\mathbf{F}}^T;P$ holds and $M^T=M_{t\wedge T}$ has a strong property of predictable representation with respect to $({\mathcal{F}}_{t\wedge T},Q)$, i.e.,

$$X_{t\wedge T}=E_Q(X_T\mid {\mathcal{F}}_{t\wedge T})=b+{\int }_0^{t\wedge T}{K}_{u}dQ\left(M_u\right).$$

Because of causality and for $N_{t\wedge T}\ne 0$, we have

$$E_Q(X_T\mid {\mathcal{F}}_{t\wedge T})=\frac{1}{N_{t\wedge T}}{E}_P\left(X_T\frac{dQ}{dP}\mid {\mathcal{F}}_{t\wedge T}\right)=\frac{1}{N_{t\wedge T}}{E}_P\left(X_T\frac{dQ}{dP}\mid {\mathcal{G}}_{t\wedge T}\right).$$

Due to the ${\mathcal{G}}_T$-measurability of $N_{\infty }^T$, from the previous equality, we have

$$\begin{array}{ccc}\hfill X_{t\wedge T}& =& \frac{1}{N_{t\wedge T}}{E}_P(X_T\mid {\mathcal{G}}_{t\wedge T})E_P(N_{\infty }^T\mid {\mathcal{G}}_{t\wedge T})=E_P(X_T\mid {\mathcal{F}}_{t\wedge T})\hfill \\ & =& b+{\int }_0^{t\wedge T}{K}_{u}dP\left(M_u\right).\hfill \end{array}$$

Therefore, $X_{t\wedge T}$ has a strong property of predictable representation with respect to $({\mathcal{F}}_{t\wedge T},P)$. □

Scientists have been focused on extremal measure problems for a very long time. The connection between the extremality of a measure and the strong property of predictable representation was introduced by Dellacherie (see [23]) and further investigated by Jacod and Yor (see [22,24]).

Let us consider a set of right continuous modifications of processes vanishing at 0 ($M_0=0$)

$$G=\{M_t\mid M_t=P(A\mid {\mathcal{G}}_t),A\in {\mathcal{G}}_{\infty }\}.$$

(6)

Let G be a set of the form (6), and let $\mathcal{M}$ be a set of measures Q on $\mathcal{A}$ for which $Q=P$ on ${\mathcal{F}}_0$ and elements of G are $({\mathcal{F}}_t,Q)$-martingales. ${\mathcal{F}}_0$ is a trivial $\sigma$-field generated by all P null sets. Then, we have the following results to hold.

Theorem 3.

Let $\mathbf{G}\subseteq \mathbf{F}$ be filtrations on $(\mathsf{\Omega },\mathcal{A},P)$. The martingale $M_t\in G$ vanishing at 0, with trivial${\mathcal{F}}_0$, has a strong property of predictable representation if and only if $\left\{{\mathcal{G}}_t\right\}$ is self-caused within $\left\{{\mathcal{F}}_t\right\}$, i.e., if and only if $\mathbf{G}|<\mathbf{G};\mathbf{F};P$ holds.

Proof. 

Let $\mathbf{G}\subseteq \mathbf{F}$ on $(\mathsf{\Omega },\mathcal{A},P)$, and let martingale $M_t\in G$ have a strong property of predictable representation with respect to $\left\{{\mathcal{G}}_t\right\}$. Due to Theorem 13.11 in [17], measure P is an extremal point in the set of measures $\mathcal{M}$. Thus, according to Theorem 3.2 in [9], we have $\mathbf{G}|<\mathbf{G};\mathbf{F};P$.

Conversely, suppose the causality relation holds, i.e., $\mathbf{G}|<\mathbf{G};\mathbf{F};P$. According to Theorem 3.2 in [9], the measure $P\in \mathcal{M}$ is an extremal point. Thus, by Theorem 13.11 in [17], martingale $M_t$ has a strong property of predictable representation with respect to $\left\{{\mathcal{G}}_t\right\}$. □

Theorem 4.

Let $\mathbf{G}\subseteq \mathbf{F}$ be filtrations on $(\mathsf{\Omega },\mathcal{A},P)$. The martingale $M_t\in G$ vanishes at 0, where G is of the form (6) with trivial ${\mathcal{F}}_0$ having a weak property of predictable representation if $\left\{{\mathcal{G}}_t\right\}$ is self-caused within $\left\{{\mathcal{F}}_t\right\}$, i.e., if $\mathbf{G}|<\mathbf{G};\mathbf{F};P$ holds.

Proof. 

Let $\mathbf{G}\subseteq \mathbf{F}$ be filtrations on $(\mathsf{\Omega },\mathcal{A},P)$, and let $\mathbf{G}|<\mathbf{G};\mathbf{F};P$ hold. By Theorem 3, $M_t$ has a strong property of predictable representation. Due to Theorem 13.14 in [17], martingale $M_t$ has a weak property of predictable representation as well. □

Let now consider a set

$$L=\{L_t\mid L_t=P(A\mid {\mathcal{F}}_t^M),A\in {\mathcal{F}}_{\infty }^M\}$$

(7)

where $\mathcal{L}$ is a set of measures Q for which $P=Q$ on ${\mathcal{F}}_0$ and elements of the set L are $({\mathcal{F}}_t,P)$-martingales. The next Lemma follows directly from Theorem 3.

Lemma 2.

The martingale $L_t\in L$ has a strong property of predictable representation if and only if $\left\{{\mathcal{F}}_t^M\right\}$ is self-caused within $\left\{{\mathcal{F}}_t\right\}$, i.e., if and only if ${\mathbf{F}}^M|<{\mathbf{F}}^M;\mathbf{F};P$ holds.

Proof. 

The proof follows directly by Theorem 3, Lemma 3.3 in [9] and Theorem 13.11 in [17]. □

Example 1.

The process of Brownian motion $\{B_t,t\in I\}$ is a self-caused process (by Proposition 2.1 in [9]) with respect to $\left\{{\mathcal{F}}_t\right\}$. Due to Lemma 2, this process has a strong property of predictable representation with respect to its natural filtration $\left\{{\mathcal{F}}_t^B\right\}$ (see also, Excercise 4.15 in [24]).

The next Lemma is a consequence of Theorem 4.

Lemma 3.

The martingale $L_t\in L$ has a weak property of predictable representation if $\left\{{\mathcal{F}}_t^M\right\}$ is self-caused within $\left\{{\mathcal{F}}_t\right\}$, i.e., if ${\mathbf{F}}^M|<{\mathbf{F}}^M;\mathbf{F};P$ holds.

Example 2.

Suppose that there is a signal process X that describes the state of a system but that cannot be observed directly. Instead, only process Y, with dynamics dependent on the value of X, can be observed. The goal is to find the “best estimate” of $X_t$ by observing the σ-algebra ${\mathcal{Y}}_t=\sigma \{Y_s,s\le t\}$ up to time t. In other words, by observations of Y, the state of hidden signal X is filtered. An innovative approach introduces a process $V_t$ defined by Definition 22.1.6 in [25], called the innovation process. Formally, $V_{t+h}-V_t$ represents the new information about X obtained from observations between t and $t+h$. The innovation process $V_t$ is a self-caused process. This statement follows directly from Theorem 22.1.8 in [25] and Lemma 2 (see also, Lemma 22.1.7 in [25], Corollary 2.2.1 in [7]).

Example 3.

Suppose the running cost is determined by a real, bounded, measurable function c, and the terminal cost is given by a real, bounded, measurable function g. Then, if control $u\in \mathcal{U}$ (the set of admissible controls $\mathcal{U}$ is the set of $\left\{{\mathcal{F}}_t\right\}$—predictable processes) is used, the total expected cost is

$$J\left(u\right)=E_u\left({\int }_0^{t}c(t,x_t,u_t)dt+g\left(x_t\right)\right)$$

where $E_u$ denotes the expectation with respect to $P_u$. The question is how the control u should be selected so that the total expected cost is minimized. The cost process is defined by $M_t^u={\int }_0^t{c}_t^{u}ds+V_t$, where $V_t$ is the value function (Definitions 16.8 and 16.16 in [26]). Both the running cost c and terminal cost g are bounded, so from Remark 16.21 in [26], the cost process $M_t^u$ can be represented as $M_t^u=W_0+A_t^u+N_t^u$, where $A_t^u$ is a unique, predictable, increasing process and $N_t^u$ is a uniformly integrable martingale. The sufficient condition for a control u to be optimal is that the process $N_t^u$ is self-caused. This follows directly from Lemma 2, Remark 16.26, and Corollary 16.23 in [26].

4. Application

The problem in the representation theory is to find a martingale of underlying filtration, which can be uniquely represented as the stochastic integral with respect to the local martingale from the given collection $\mathcal{X}$ for a suitable predictable process. The parallel question in finance theory is which contingent claims could be written as a stochastic integral with respect to the discounted security prices.

Harison and Pliska in [27] proved that every contingent claim in a financial market is attainable (hedgeable) if and only if the set of martingale measures has only one element. Such a market is called complete. The question is which claims are then attainable in an incomplete financial market.

Basically any dynamic hedging strategy is consistent with a (vector-valued) predictable process ${\varphi }_t$, and the (discounted) value of the considering portfolio suites to the stochastic integral of ${\varphi }_t$ relative to the (discounted) security prices. The question of attainability is the same as a question of the representation property.

We assume that $\mathcal{X}$ is a finite set of non-negative discounted security prices (given as the vector Z), stopped at some finite horizon $\tau ,P\in \mathcal{M}\left(\mathcal{X}\right)$, where $\mathcal{M}\left(\mathcal{X}\right)$ is a set of all martingale measures P for which $X\in \mathcal{X}$ is a $({\mathcal{F}}_t,P)$-local martingale.

Following [27] X is a contingent claim if $X\ge 0P$-a.s. holds. Let ${\beta }_t$ be the discounted factor at time t. The next definition holds.

Definition 7

([27,28]). The contingent claim X is said to be P-integrable if $E_P\left({\beta }_{\tau }X\right)<\infty$ and is said to be P-attainable if there exists a V such that

(i) 

${\displaystyle V_t=V_0+{\int }_0^t{\varphi }_s·d{Z}_s}$, where ϕ is predictable,

(ii) 

V is a P-martingale,

(iii) 

$V_{\tau }={\beta }_{\tau }X$.

Finally, X is said to be bounded if $\left|\right|{\beta }_{\tau }X{\left|\right|}_{\infty }<\infty .$

The price of a claim under P in [27] is defined as ${\pi }_P\left(X\right)=E_P\left({\beta }_{\tau }X\right),$ and the price of a P-attainable claim X under P is ${\pi }_P\left(X\right)=V_0.$

In the following theorem, we consider the connection between the concept of causality and P-attainability of a contingent claim.

Theorem 5.

Let X be a contingent claim. Then $X_t$ is a P-attainable if and only if the vector of the non-negative discounted security prices $Z_t$ is self-caused within $\left\{{\mathcal{F}}_t\right\}$ and the process $V_{\tau }$ stopped at some finite horizon is $V_{\tau }={\beta }_{\tau }X$.

Proof. 

Let $X_t$ be a contingent claim. According to Definition 7, process $X_t$ is P-attainable if there exists a process $V_t$ such that

$$V_t=b+{\int }_0^t{\varphi }_{s}d{Z}_s.$$

Due to Definition 4, process $Z_t$ has the strong property of predictable representation with respect to $\left\{{\mathcal{F}}_t^Z\right\}$. According to Theorem 3, $Z_t$ is a self-caused process, i.e., ${\mathbf{F}}^Z|<{\mathbf{F}}^Z;\mathbf{F};P.$

Conversely, suppose that ${\mathbf{F}}^Z|<{\mathbf{F}}^Z;\mathbf{F};P$ holds, and the process $V_{\tau }$ that stopped at some finite horizon is $V_{\tau }={\beta }_{\tau }X.$ According to Theorem 3, $Z_t$ has the strong property of predictable representation, and the process $V_t$ is of the form

$$V_t=E(V_{\infty }\mid {\mathcal{F}}_t^Z)=b+{\int }_0^t{\varphi }_{s}d{Z}_s$$

where ${\varphi }_s$ is a predictable process. All conditions of the Definition 7 are satisfied, because $V_t=E(V_{\infty }\mid {\mathcal{F}}_t^Z)$ is a martingale. So X is P-attainable. □

The next results follow as a consequence of Theorem 5. First, we give conditions, in terms of causality, for the same price of contingent claim under all martingale measures.

Theorem 6.

The contingent claim X has the same price under all measures $Q\in \mathcal{M}\left(\mathcal{X}\right)$ for which $Q\sim P$ and $\left|\right|\frac{dQ}{dP}{\left|\right|}_{\infty }\wedge \left|\right|\frac{dP}{dQ}{\left|\right|}_{\infty }<\infty$ if and only if the vector of the non-negative discounted security prices $Z_t$ is self-caused within $\left\{{\mathcal{F}}_t\right\}$ and the process $V_{\tau }$ stopped at some finite horizon is $V_{\tau }={\beta }_{\tau }X$.

Proof. 

This follows directly from Theorem 5 and Theorem 8 in [28]. □

Theorem 7.

Let X be a bounded claim. Then X is Q-attainable for all $Q\in \mathcal{M}\left(\mathcal{X}\right)$ and $Q\sim P$ if and only if the vector of the non-negative discounted security prices $Z_t$ is self-caused within $\left\{{\mathcal{F}}_t\right\}$ and the process $V_{\tau }$ that stopped at some finite horizon is $V_{\tau }={\beta }_{\tau }X$.

Proof. 

It follows directly from Theorem 5 and Theorem 9 in [28]. □

Let us mention that this application does not limit the generalizability of these findings to broader financial market settings.

5. Discussion and Conclusions

This paper investigates the connection between the concept of causality and the strong property of predictable representation.

The predictable representation property for semi-martingales with respect to enlarged filtrations is considered in [12,13]. The authors of [14] considered the connection between the concept of statistical causality and the martingale representation property. Similar results are obtained here for stopped filtrations. Theorem 1 characterizes the decreased stopped subfiltration $\left\{{\mathcal{G}}_{t\wedge T}\right\}$ relative to which the strong property of predictable representation is preserved, although, in general, the preservation does not hold. Theorem 2 shows that the invariance of the strong predictable representation property relative to the absolute continuous change in measure strongly depends on the concept of stopped causality. The conditions for a process to have strong and weak properties of predictable representation are given in Theorems 3 and 4.

An open question is when a stopped process has a strong or weak property of predictable representation. Although the equivalence between the stopped causality and extremal measures is proved in [9], it remains to connect it to the strong property of predictable representation. Also, the question of enlargement of filtration in the sense of application in mathematical finance is very interesting, together with the preservation of the strong property of predictable representation in terms of causality, and it should be treated more carefully.