Modern Stochastics: Theory and Applications

Starting with Dellacherie [4], the following simple model has been studied and intensively used in applications. Given a random variable γ with positive values on a probability space $(\Omega ,\mathcal{F},\mathsf{P})$, one considers the smallest filtration with respect to which γ is a stopping time (or, equivalently, the process ${\mathbb{1}_{\{t\geqslant \gamma \}}}$ is adapted). In particular, Dellacherie gives a formula for the compensator of this single jump process ${\mathbb{1}_{\{t\geqslant \gamma \}}}$. Chou and Meyer [2] describe all local martingales with respect to this filtration and prove a martingale representation theorem. A significant contribution is done in a recent paper by Herdegen and Herrmann [13], where a classification, whether a local martingale in this model is a strict local martingale, or a uniformly integrable martingale, etc., is given. Let us also mention some related papers [1, 15, 16, 3, 8, 21, 12], where, in particular, local martingales with respect to the filtrations generated by jump processes or measures of certain kind are studied.

Let us clarify that in the above model every local martingale has the form or where γ is a random variable with values in, say, $(0,+\infty )$, F, H, and $K=F-H$ are deterministic functions. Denote by G the distribution function of γ, $\overline{G}(t)=1-G(t)$, ${t_{G}}=\sup \{t:G(t)<1\}$ is the right endpoint of the distribution of γ. Assume that $\mathsf{E}|{M_{t}}|<\infty $, then where the corresponding Lebesgue–Stieltjes integral is finite. If $({M_{t}})$ is a martingale, then $\mathsf{E}({M_{t}})=\mathsf{E}({M_{0}})$, and this equality can be written as and can be viewed as a functional equation concerning one of functions in $(F,G,H)$ or $(F,G,K)$, where other two functions are assumed to be given. In fact, this equation takes place for $t<{t_{G}}$ or $t\leqslant {t_{G}}$, the latter in the case where ${t_{G}}<\infty $ and $\mathsf{P}(\gamma ={t_{G}})>0$. Moreover, it turns out that this is not only the necessary condition but also the sufficient one for ${({M_{t}})_{t\in {\mathbb{R}_{+}}}}$ given by (1) to be a local martingale. This consideration allows us to reduce problems to solving this functional equation. For example, to find the compensator $F(t\wedge \gamma )$ of ${\mathbb{1}_{\{t\geqslant \gamma \}}}$ as in [4] one needs to find a solution F given G and $K\equiv 1$. A possible way to explain the idea in [2] is the following: The terminal value ${M_{\infty }}$ of any local martingale M in this model is represented as $H(\gamma )$, and to find a representation (1) for M it is enough to solve the equation for F given G and H; the linear dependence between H and F results in a representation theorem. Contrariwise, in [13] the authors suggest to find H from the equation for given F and G. This allows them to study global properties of M.

In this paper we consider a more general model, where all randomness appears “at time γ” but it may contain much more information than γ does. We start with a random variable γ on a probability space $(\Omega ,\mathcal{F},\mathsf{P})$, and define a single jump filtration $({\mathcal{F}_{t}})$ in such way that nothing happens strictly before γ, γ is a stopping time with respect to it, and the σ-field ${\mathcal{F}_{\gamma }}$ of events that occur before or at time γ coincides with $\mathcal{F}$ (in fact, on the set $\{\gamma <\infty \}$). We prove that every local martingale with respect to this filtration has the representation where now L is a random variable which is not necessarily a function of γ. However, denoting $H(t)=\mathsf{E}[L|\gamma =t]$, we come to the same functional equation of type (2).

Some results of the paper can be deduced from known results for marked point processes, at least if $\mathcal{F}$ is countably generated; this applies, for example, to Theorem 5 about the compensator of a single jump process. Another example is Corollary 1 which says that every local martingale is the sum of a local martingale of form (1) and an “orthogonal” local martingale, the latter being characterised, essentially, by the property $F(t)\equiv 0$. The reader can recognize in this decomposition the representation of a local martingale as the sum of two stochastic integrals with respect to random measures, see [16] and [17]. However, our direct proofs are simpler due to the key feature of our paper. Namely, we obtain a simple necessary and sufficient condition for a process to be a local martingale and later exploit it. A description of all local martingales via a full description of all possible solutions to a functional equation of type (2) is a simple consequence of this necessary and sufficient condition. In particular, an absolute continuity type property of F with respect to G, considered as an assumption in [13], is proved to be a necessary condition. An elementary analysis of a functional equation of type (2) shows that, if γ has no atom at its right endpoint, there are different F satisfying the equation for given H and G. In particular, there is a local martingale M such that ${M_{0}}=1$ and ${M_{\infty }}=0$; M is necessarily a closed supermartingale.

Another important feature of our model, in contrast to Dellacherie’s model, is that it admits σ-martingales which are not local martingales.

Let us also mention some other papers where processes of form (1) or (3) are considered. Processes of form (1) with ${t_{G}}=\infty $ are typical in the modelling of credit risk, see, e.g., [18] and [19, Chapter 7], where usually F is expressed via G and one needs to find H. Since ${t_{G}}=\infty $, such a process is a martingale. For example, in the simplest case $F=1/\overline{G}$ and hence $H=0$. This process is the same that is mentioned in two paragraphs above. Single jump filtrations and processes of form (3) appear in [10] and [11]. It is interesting to note that, in [11], the random “time” γ is, in fact, the global maximum of a random process, say, a convergent continuous local martingale.

Section 2 contains our main results. In Theorem 1 we establish a necessary and sufficient condition for a process of type (3) to be a local martingale. This allows us to obtain a full description of all local martingales through a functional equation of type (2) in Theorem 2. A similar description is available for σ-martingales, see Theorem 3. Finally, Theorem 4 classifies local martingales in accordance with their global behaviour up to ∞. Section 3 contains the proofs of these results. In Section 4 we consider complementary questions. Namely, we find the compensator of a single jump process. We also consider submartingales of class $(\Sigma )$, see [22], and show that their transformation via a change of time leads to processes of type (3). As a consequence, we reprove Theorem 4.1 of [22].

We use the following notation: ${\mathbb{R}_{+}}=[0,+\infty )$, ${\overline{\mathbb{R}}_{+}}=[0,+\infty ]$, $a\wedge b=\min \hspace{0.1667em}\{a,b\}$. The arrows ↑ and ↓ indicate monotone convergence, while ${\lim \nolimits_{s\upuparrows t}}$ stands for ${\lim \nolimits_{s\to t,s<t}}$.

A real-valued function $Z(t)$ defined at least for $t\in [0,s)$ is called càdlàg on $[0,s)$ if it is right-continuous at every $t\in [0,s)$ and has a finite left-hand limit at every $t\in (0,s)$; it is not assumed that it has a limit as $t\upuparrows s$. If, additionally, a finite limit ${\lim \nolimits_{t\upuparrows s}}Z(t)$ exists, then $Z(t)$ is called càdlàg on $[0,s]$. Functions Z of finite variation on compact intervals are understood as usually and are assumed to be càdlàg. The variation at 0 includes $|Z(0)|$ as if Z is extended by 0 on negative axis. The total variation of Z over $[0,t]$ is denoted by $\operatorname{Var}{(Z)_{t}}$. We say that Z has a finite variation over $[0,s)$, $s\leqslant \infty $, if ${\lim \nolimits_{t\upuparrows s}}\operatorname{Var}{(Z)_{t}}<\infty $. We denote $\operatorname{Var}{(Z)_{\infty }}:={\lim \nolimits_{t\to \infty }}\operatorname{Var}{(Z)_{t}}$.

A filtration on a probability space $(\Omega ,\mathcal{F},\mathsf{P})$ is an increasing right-continuous family $\mathbb{F}={({\mathcal{F}_{t}})_{t\in {\mathbb{R}_{+}}}}$ of sub-σ-fields of $\mathcal{F}$. No completeness assumption is made. As usual, we define ${\mathcal{F}_{\infty }}=\sigma \big({\cup _{t\in {\mathbb{R}_{+}}}}{\mathcal{F}_{t}}\big)$ and, for a stopping time τ the σ-field ${\mathcal{F}_{\tau }}$ is defined by A set $B\subset \Omega \times {\mathbb{R}_{+}}$ is evanescent if $B\subseteq A\times {\mathbb{R}_{+}}$, where $A\in \mathcal{F}$ and $\mathsf{P}(A)=0$. We say that two stochastic processes X and Y are indistinguishable if $\{X\ne Y\}$ is an evanescent set.

Since we do not suppose completeness of the filtration $\mathbb{F}$, we cannot expect that processes that we consider have all paths càdlàg. Instead we consider processes whose almost all paths are càdlàg. Obviously, for any càdlàg process X adapted with respect to the completed filtration, there is an a.s. càdlàg $\mathbb{F}$-adapted process indistinguishable from X. Furthermore, any $\mathbb{F}$-adapted process X with a.s. càdlàg paths is indistinguishable from an $\mathbb{F}$-optional process Y whose paths are right-continuous everywhere and have finite left-hand limits for $t<\rho (\omega )$ and $t>\rho (\omega )$, where ρ is a $\mathbb{F}$-stopping time with $\mathsf{P}(\rho <\infty )=0$; let us call such Y regular and ρ a moment of irregularity for Y. Dellacherie and Meyer [6, VI.5 (a), p. 70] prove that, if the filtration is not complete, every supermartingale X (with right-continuous expectation) has a modification Y with the above regularity property. If we are given just an adapted process X with almost all paths càdlàg, we define ρ and Y from values of X on a countable set exactly as is done in [6] in the case where X is a supermartingale. Using [5, Theorem IV.22, p. 94], we obtain that $\rho (\omega )=\infty $ and paths ${X_{\cdot }}(\omega )$ and ${Y_{\cdot }}(\omega )$ coincide for those ω for which ${X_{\cdot }}(\omega )$ is càdlàg everywhere. Moreover, if $\rho (\omega )<\infty $, then ${Y_{t}}(\omega )$ is càdlàg for $t<\rho (\omega )$ and one may put ${Y_{t}}(\omega )=0$ for $t\geqslant \rho (\omega )$.

Processes with finite variation are adapted and not assumed to start from 0. A moment of irregularity for them has additionally the property that their paths have finite variation over $[0,t]$ for all $t<\rho (\omega )$.

It is instructive to mention that, in our model, there is no need to use general results on the existence of (a.s.) càdlàg modifications for martingales since they can be proved directly. For example, if L is an integrable random variable with $\mathsf{E}L=0$, then the process M given by (3) with $F(t)=\mathsf{E}[L|\gamma >t]{\mathbb{1}_{\{t<{t_{G}}\}}}$ satisfies ${M_{t}}=\mathsf{E}[L|{\mathcal{F}_{t}}]$ a.s. for an arbitrary t. It is trivial to check that this function F has finite variation over any $[0,t]$ with $\mathsf{P}(\gamma >t)>0$ (and over $[0,{t_{G}})$ if $\mathsf{P}(\gamma ={t_{G}}<\infty )>0$). Thus M is regular. It may be that, if ${t_{G}}<\infty $ and $\mathsf{P}(\gamma ={t_{G}})=0$, the function F has not a finite limit as $t\upuparrows {t_{G}}$, or, more generally, has unbounded variation over $[0,{t_{G}})$. Then a moment of irregularity is given by It takes a finite value only on the set $\{\gamma \geqslant {t_{G}}\}$ of zero measure. In all other cases we may put $\rho \equiv +\infty $. See Remark 2 in Section 2 for more details.

Let γ be a random variable with values in ${\overline{\mathbb{R}}_{+}}$ on a probability space $(\Omega ,\mathcal{F},\mathsf{P})$. We tacitly assume that $\mathsf{P}(\gamma >0)>0$. $G(t)=\mathsf{P}(\gamma \leqslant t)$, $t\in {\mathbb{R}_{+}}$, stands for the distribution function of γ and $\overline{G}(t)=1-G(t)$. Put also ${t_{G}}=\sup \hspace{0.1667em}\{t\in {\mathbb{R}_{+}}:G(t)<1\}$ and $\mathcal{T}=\{t\in {\mathbb{R}_{+}}:\mathsf{P}(\gamma \geqslant t)>0\}$. Note that $\mathsf{P}(\gamma \notin \mathcal{T})=0$. We will often distinguish between the following two cases: It is clear that $\mathcal{T}=[0,{t_{G}})$ in Case A and $\mathcal{T}=[0,{t_{G}}]$ in Case B.

We define ${\mathcal{F}_{t}}$, $t\in {\mathbb{R}_{+}}$, as the collection of subsets A of Ω such that $A\in \mathcal{F}$ and $A\cap \{t<\gamma \}$ is either ∅ or coincides with $\{t<\gamma \}$.

It is shown in Proposition 1 that ${\mathcal{F}_{t}}$ is a σ-field for every $t\in {\mathbb{R}_{+}}$ and the family $\mathbb{F}={({\mathcal{F}_{t}})_{t\in {\mathbb{R}_{+}}}}$ is a filtration. We call this filtration a single jump filtration. It is determined by generating elements γ and $\mathcal{F}$. In this paper we consider only single jump filtrations and, if necessary to indicate generating elements, we use the notation $\mathbb{F}(\gamma ,\mathcal{F})$ for the single jump filtration generated by γ and $\mathcal{F}$.

In this section a single jump filtration $\mathbb{F}=\mathbb{F}(\gamma ,\mathcal{F})$ is fixed. All notions depending on filtration (stopping times, martingales, local martingales, etc.) refer to this filtration $\mathbb{F}$, unless otherwise specified.

Statement (iv) is not surprising. If the σ-field $\mathcal{F}$ is countably generated, then our filtration is a special case of a filtration generated by a marked point process, and it is known, see [17], that then all martingales are of finite variation. In general, a single jump filtration is a special case of a jumping filtration, see [14], where again all martingales are of finite variation.

In what follows, when we write that the process M has the representation (5), this means that M and the right-hand side of (5) are indistinguishable. Moreover, we tacitly assume that $F(t)$ is right-continuous for $t\geqslant {t_{G}}$ to ensure that the right-hand side of (5) is right-continuous.

Propositions 1 and 2 explain why we call $\mathbb{F}$ a single jump filtration: all randomness appears at time γ. It is not so natural to describe local martingales with respect to $\mathbb{F}$ as single jump processes. As we will see, the function F in (5) need not be continuous, so local martingales may have several jumps.

Our main goal is to provide a complete description of all local martingales. According to Proposition 2 (v), a necessary condition is that it is represented in form (5). Thus, it is enough to study only processes of this form.

In the case where $\mathcal{F}=\sigma \{\gamma \}$, equivalence (i) and (ii) is proved in [2].

Concerning the last statement of the proposition, let us emphasize that if ${t_{G}}<\infty $ and $\mathsf{P}(\gamma ={t_{G}})=0$, a local martingale $M={({M_{t}})_{t\in {\mathbb{R}_{+}}}}$ may not be a martingale on $[0,{t_{G}}]$; obviously, if it is a martingale, then it is uniformly integrable, and necessary and sufficient conditions for this are given in Theorem 4.

If (6) and (7) hold, then and one can define the conditional expectation $H(t)$ of L given that $\gamma =t$ for $t\in \mathcal{T}$: More precisely, $H(t)$ is a Borel function on $\mathcal{T}$ with finite values such that for any $t\in \mathcal{T}$ Note that the function H is $dG$-a.s. unique and is $dG$-integrable over any closed interval in $\mathcal{T}$. It is convenient to introduce a notation for such functions.

Let ${L_{\mathrm{loc}}^{1}}(dG)$ be the set of all Borel functions z on $\mathcal{T}$ such that Given a function $Z:[0,{t_{G}})\to \mathbb{R}$, let us write $Z\stackrel{\mathrm{loc}}{\ll }G$ if there is $z\in {L_{\mathrm{loc}}^{1}}(dG)$ such that $Z(t)=Z(0)+{\textstyle\int _{(0,t]}}z(s)\hspace{0.1667em}dG(s)$ for all $t<{t_{G}}$; in this case we put $\frac{dZ}{dG}(t):=z(t)$ for $0<t<{t_{G}}$. Let us emphasize that in Case B this definition implies that z is $dG$-integrable over $[0,{t_{G}}]$ and, hence, the function Z has a finite variation over $[0,{t_{G}})$ and there is a finite limit ${\lim \nolimits_{t\upuparrows {t_{G}}}}Z(t)=Z(0)+{\textstyle\int _{(0,{t_{G}})}}z(s)\hspace{0.1667em}dG(s)$. Note also that in this definition the value $z(0)$ can be chosen arbitrarily even if $G(0)>0$; the same refers to the value $z({t_{G}})$ in Case B. Correspondingly, $dZ/dG$ is defined only for $0<t<{t_{G}}$.

Let G be a distribution function of a law on $[0,+\infty ]$. We will say that a pair $(F,H)$ satisfies Condition M if and, additionally in Case B,

The statement that the process M given by (20) with ${L^{\prime }}=0$ is a local martingale if $F\stackrel{\mathrm{loc}}{\ll }G$ and H is constructed as in part (b) of Proposition 3, is essentially due to Herdegen and Herrmann [13], though they formulate (17) in an equivalent form: They also prove that, in Case B, if F has infinite variation on $[0,{t_{G}})$ (and hence does not satisfy $F\stackrel{\mathrm{loc}}{\ll }G$), then M given by (6) is not a semimartingale, see [13, Lemma B.6]. (Note that this follows also from our Proposition 2 (iv).) We add that, also in Case B, if H is $dG$-integrable over $(0,{t_{G}})$, F satisfies (15), but $F(0)$ is greater or less than the right-hand side of (16), then M given by (20) with ${L^{\prime }}$ satisfying (19), is a supermartingale or a submartingale, respectively, cf. Theorem 4.

The fact that $H(0)$ can be chosen arbitrarily in Proposition 3 (b) says only that L can be an arbitrary integrable random variable on the set $\{\gamma =0\}$, which is evident ab initio. On the contrary, the fact that $F(0)$ can be chosen arbitrarily in (a) in Case A is an interesting feature of this model. It says that, given the terminal value ${M_{\infty }}$ of M (on $\{\gamma <\infty \}$), one can freely choose the initial value ${M_{0}}$ of M (on $\{\gamma >0\}$) to keep the property of being a local martingale for M.

As the next example shows, the product ${M^{\prime }}{M^{\prime\prime }}$ of local martingales from the above decomposition may not be a local martingale because the first condition in (19) may fail. It will follow from Theorem 3 below that this product is always a σ-martingale.

The previous example shows that our model admits σ-martingales which are not local martingales. In the next theorem we describe all σ-martingales in our model. In particular, it implies that if $\mathcal{F}=\sigma \{\gamma \}$, then all σ-martingales that are integrable at 0 are local martingales.

The next theorem complements the classification of the limit behaviour of local martingales that was considered in Herdegen and Herrmann [13] in the case where $\mathcal{F}=\sigma \{\gamma \}$. Let us say that a local martingale $M={({M_{t}})_{t\in {\mathbb{R}_{+}}}}$ has