Compensators of Stopping Times

The previous post introduced the concept of the compensator of a process, which is known to exist for all locally integrable semimartingales. In this post, I’ll just look at the very special case of compensators of processes consisting of a single jump of unit size.

Definition 1 Let {\tau} be a stopping time. The compensator of {\tau} is defined to be the compensator of {1_{[\tau,\infty)}}.

So, the compensator A of {\tau} is the unique predictable FV process such that {A_0=0} and {1_{[\tau,\infty)}-A} is a local martingale. Compensators of stopping times are sufficiently special that we can give an accurate description of how they behave. For example, if {\tau} is predictable, then its compensator is just {1_{\{\tau > 0\}}1_{[\tau,\infty)}}. If, on the other hand, {\tau} is totally inaccessible and almost surely finite then, as we will see below, its compensator, A, continuously increases to a value {A_\infty} which has the exponential distribution.

However, compensators of stopping times are sufficiently general to be able to describe the compensator of any cadlag adapted process X with locally integrable variation. We can break X down into a continuous part plus a sum over its jumps,

\displaystyle  X_t=X_0+X^c_t+\sum_{n=1}^\infty\Delta X_{\tau_n}1_{[\tau_n,\infty)}. (1)

Here, {\tau_n > 0} are disjoint stopping times such that the union {\bigcup_n[\tau_n]} of their graphs contains all the jump times of X. That they are disjoint just means that {\tau_m\not=\tau_n} whenever {\tau_n < \infty}, for any {m\not=n}. As was shown in an earlier post, not only is such a sequence {\tau_n} of the stopping times guaranteed to exist, but each of the times can be chosen to be either predictable or totally inaccessible. As the first term, {X^c_t}, on the right hand side of (1) is a continuous FV process, it is by definition equal to its own compensator. So, the compensator of X is equal to {X^c} plus the sum of the compensators of {\Delta X_{\tau_n}1_{[\tau_n,\infty)}}. The reduces compensators of locally integrable FV processes to those of processes consisting of a single jump at either a predictable or a totally inaccessible time. Continue reading “Compensators of Stopping Times”

Compensators

A very common technique when looking at general stochastic processes is to break them down into separate martingale and drift terms. This is easiest to describe in the discrete time situation. So, suppose that {\{X_n\}_{n=0,1,\ldots}} is a stochastic process adapted to the discrete-time filtered probability space {(\Omega,\mathcal{F},\{\mathcal{F}_n\}_{n=0,1,\ldots},{\mathbb P})}. If X is integrable, then it is possible to decompose it into the sum of a martingale M and a process A, starting from zero, and such that {A_n} is {\mathcal{F}_{n-1}}-measurable for each {n\ge1}. That is, A is a predictable process. The martingale condition on M enforces the identity

\displaystyle  A_n-A_{n-1}={\mathbb E}[A_n-A_{n-1}\vert\mathcal{F}_{n-1}]={\mathbb E}[X_n-X_{n-1}\vert\mathcal{F}_{n-1}].

So, A is uniquely defined by

\displaystyle  A_n=\sum_{k=1}^n{\mathbb E}\left[X_k-X_{k-1}\vert\mathcal{F}_{k-1}\right], (1)

and is referred to as the compensator of X. This is just the predictable term in the Doob decomposition described at the start of the previous post.

In continuous time, where we work with respect to a complete filtered probability space {(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge0},{\mathbb P})}, the situation is much more complicated. There is no simple explicit formula such as (1) for the compensator of a process. Instead, it is defined as follows.

Definition 1 The compensator of a cadlag adapted process X is a predictable FV process A, with {A_0=0}, such that {X-A} is a local martingale.

For an arbitrary process, there is no guarantee that a compensator exists. From the previous post, however, we know exactly when it does. The processes for which a compensator exists are precisely the special semimartingales or, equivalently, the locally integrable semimartingales. Furthermore, if it exists, then the compensator is uniquely defined up to evanescence. Definition 1 is considerably different from equation (1) describing the discrete-time case. However, we will show that, at least for processes with integrable variation, the continuous-time definition does follow from the limit of discrete time compensators calculated along ever finer partitions (see below).

Although we know that compensators exist for all locally integrable semimartingales, the notion is often defined and used specifically for the case of adapted processes with locally integrable variation or, even, just integrable increasing processes. As with all FV processes, these are semimartingales, with stochastic integration for locally bounded integrands coinciding with Lebesgue-Stieltjes integration along the sample paths. As an example, consider a homogeneous Poisson process X with rate {\lambda}. The compensated Poisson process {M_t=X_t-\lambda t} is a martingale. So, X has compensator {\lambda t}.

We start by describing the jumps of the compensator, which can be done simply in terms of the jumps of the original process. Recall that the set of jump times {\{t\colon\Delta X_t\not=0\}} of a cadlag process are contained in the graphs of a sequence of stopping times, each of which is either predictable or totally inaccessible. We, therefore, only need to calculate {\Delta A_\tau} separately for the cases where {\tau} is a predictable stopping time and when it is totally inaccessible.

For the remainder of this post, it is assumed that the underlying filtered probability space is complete. Whenever we refer to the compensator of a process X, it will be understood that X is a special semimartingale. Also, the jump {\Delta X_t} of a process is defined to be zero at time {t=\infty}.

Lemma 2 Let A be the compensator of a process X. Then, for a stopping time {\tau},

  1. {\Delta A_\tau=0} if {\tau} is totally inaccessible.
  2. {\Delta A_\tau={\mathbb E}\left[\Delta X_\tau\vert\mathcal{F}_{\tau-}\right]} if {\tau} is predictable.

Continue reading “Compensators”

Special Semimartingales

For stochastic processes in discrete time, the Doob decomposition uniquely decomposes any integrable process into the sum of a martingale and a predictable process. If {\{X_n\}_{n=0,1,\ldots}} is an integrable process adapted to a filtration {\{\mathcal{F}_n\}_{n=0,1,\ldots}} then we write {X_n=M_n+A_n}. Here, M is a martingale, so that {M_{n-1}={\mathbb E}[M_n\vert\mathcal{F}_{n-1}]}, and A is predictable with {A_0=0}. By saying that A is predictable, we mean that {A_n} is {\mathcal{F}_{n-1}} measurable for each {n\ge1}. It can be seen that this implies that

\displaystyle  A_n-A_{n-1}={\mathbb E}[A_n-A_{n-1}\vert\mathcal{F}_{n-1}]={\mathbb E}[X_n-X_{n-1}\vert\mathcal{F}_{n-1}].

Then it is possible to write A and M as

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle A_n&\displaystyle=\sum_{k=1}^n{\mathbb E}[X_k-X_{k-1}\vert\mathcal{F}_{k-1}],\smallskip\\ \displaystyle M_n&\displaystyle=X_n-A_n. \end{array} (1)

So, the Doob decomposition is unique and, conversely, the processes A and M constructed according to equation (1) can be seen to be respectively, a predictable process starting from zero and a martingale. For many purposes, this allows us to reduce problems concerning processes in discrete time to simpler statements about martingales and separately about predictable processes. In the case where X is a submartingale then things reduce further as, in this case, A will be an increasing process.

The situation is considerably more complicated when looking at processes in continuous time. The extension of the Doob decomposition to continuous time processes, known as the Doob-Meyer decomposition, was an important result historically in the development of stochastic calculus. First, we would usually restrict attention to sufficiently nice modifications of the processes and, in particular, suppose that X is cadlag. When attempting an analogous decomposition to the one above, it is not immediately clear what should be meant by the predictable component. The continuous time predictable processes are defined to be the set of all processes which are measurable with respect to the predictable sigma algebra, which is the sigma algebra generated by the space of processes which are adapted and continuous (or, equivalently, left-continuous). In particular, all continuous and adapted processes are predictable but, due to the existence of continuous martingales such as Brownian motion, this means that decompositions as sums of martingales and predictable processes are not unique. It is therefore necessary to impose further conditions on the term A in the decomposition. It turns out that we obtain unique decompositions if, in addition to being predictable, A is required to be cadlag with locally finite variation (an FV process). The processes which can be decomposed into a local martingale and a predictable FV process are known as special semimartingales. This is precisely the space of locally integrable semimartingales. As usual, we work with respect to a complete filtered probability space {(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge0},{\mathbb P})} and two stochastic processes are considered to be the same if they are equivalent up to evanescence.

Theorem 1 For a process X, the following are equivalent.

  • X is a locally integrable semimartingale.
  • X decomposes as
    \displaystyle  X=M+A (2)

    for a local martingale M and predictable FV process A.

Furthermore, choosing {A_0=0}, decomposition (2) is unique.

Theorem 1 is a general version of the Doob-Meyer decomposition. However, the name `Doob-Meyer decomposition’ is often used to specifically refer to the important special case where X is a submartingale. Historically, the theorem was first stated and proved for that case, and I will look at the decomposition for submartingales in more detail in a later post. Continue reading “Special Semimartingales”

Predictable FV Processes

By definition, an FV process is a cadlag adapted stochastic process which almost surely has finite variation over finite time intervals. These are always semimartingales, because the stochastic integral for bounded integrands can be constructed by taking the Lebesgue-Stieltjes integral along sample paths. Also, from the previous post on continuous semimartingales, we know that the class of continuous FV processes is particularly well behaved under stochastic integration. For one thing, given a continuous FV process X and predictable {\xi}, then {\xi} is X-integrable in the stochastic sense if and only if it is almost surely Lebesgue-Stieltjes integrable along the sample paths of X. In that case the stochastic and Lebesgue-Stieltjes integrals coincide. Furthermore, the stochastic integral preserves the class of continuous FV processes, so that {\int\xi\,dX} is again a continuous FV process. It was also shown that all continuous semimartingales decompose in a unique way as the sum of a local martingale and a continuous FV process, and that the stochastic integral preserves this decomposition.

Moving on to studying non-continuous semimartingales, it would be useful to extend the results just mentioned beyond the class of continuous FV processes. The first thought might be to simply drop the continuity requirement and look at all FV processes. After all, we know that every FV process is a semimartingale and, by the Bichteler-Dellacherie theorem, that every semimartingale decomposes as the sum of a local martingale and an FV process. However, this does not work out very well. The existence of local martingales with finite variation means that the decomposition given by the Bichteler-Dellacherie theorem is not unique, and need not commute with stochastic integration for integrands which are not locally bounded. Also, it is possible for the stochastic integral of a predictable {\xi} with respect to an FV process X to be well-defined even if {\xi} is not Lebesgue-Stieltjes integrable with respect to X along its sample paths. In this case, the integral {\int\xi\,dX} is not itself an FV process. See this post for examples where this happens.

Instead, when we do not want to restrict ourselves to continuous processes, it turns out that the class of predictable FV processes is the correct generalisation to use. By definition, a process is predictable if it is measurable with respect to the set of adapted and left-continuous processes so, in particular, continuous FV processes are predictable. We can show that all predictable FV local martingales are constant (Lemma 2 below), which will imply that decompositions into the sum of local martingales and predictable FV processes are unique (up to constant processes). I do not look at general semimartingales in this post, so will not prove the existence of such decompositions, although they do follow quickly from the results stated here. We can also show that predictable FV processes are very well behaved with respect to stochastic integration. A predictable process {\xi} is integrable with respect to a predictable FV process X in the stochastic sense if and only if it is Lebesgue-Stieltjes integrable along the sample paths, in which case stochastic and Lebesgue-Stieltjes integrals agree. Also, {\int\xi\,dX} will again be a predictable FV process. See Theorem 6 below.

In the previous post on continuous semimartingales, it was also shown that the continuous FV processes can be characterised in terms of their quadratic variations and covariations. They are precisely the semimartingales with zero quadratic variation. Alternatively, they are continuous semimartingales which have zero quadratic covariation with all local martingales. We start by extending this characterisation to the class of predictable FV processes. As always, we work with respect to a complete filtered probability space {(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge0},{\mathbb P})} and two stochastic processes are considered to be equal if they are equivalent up to evanescence. Recall that, in these notes, the notation {[X]^c_t=[X]_t-\sum_{s\le t}(\Delta X_s)^2} is used to denote the continuous part of the quadratic variation of a semimartingale X.

Theorem 1 For a process X, the following are equivalent.

  1. X is a predictable FV process.
  2. X is a predictable semimartingale with {[X]^c=0}.
  3. X is a semimartingale such that {[X,M]} is a local martingale for all local martingales M.
  4. X is a semimartingale such that {[X,M]} is a local martingale for all uniformly bounded cadlag martingales M.

Continue reading “Predictable FV Processes”

Predictable Stopping Times

Although this post is under the heading of `the general theory of semimartingales’ it is not, strictly speaking, about semimartingales at all. Instead, I will be concerned with a characterization of predictable stopping times. The reason for including this now is twofold. First, the results are too advanced to have been proven in the earlier post on predictable stopping times, and reasonably efficient self-contained proofs can only be given now that we have already built up a certain amount of stochastic calculus theory. Secondly, the results stated here are indispensable to the further study of semimartingales. In particular, standard semimartingale decompositions require some knowledge of predictable processes and predictable stopping times.

Recall that a stopping time {\tau} is said to be predictable if there exists a sequence of stopping times {\tau_n\le\tau} increasing to {\tau} and such that {\tau_n < \tau} whenever {\tau > 0}. Also, the predictable sigma-algebra {\mathcal{P}} is defined as the sigma-algebra generated by the left-continuous and adapted processes. Stated like this, these two concepts can appear quite different. However, as was previously shown, stochastic intervals of the form {[\tau,\infty)} for predictable times {\tau} are all in {\mathcal{P}} and, in fact, generate the predictable sigma-algebra.

The main result (Theorem 1) of this post is to show that a converse statement holds, so that {[\tau,\infty)} is in {\mathcal{P}} if and only if the stopping time {\tau} is predictable. This rather simple sounding result does have many far-reaching consequences. We can use it show that all cadlag predictable processes are locally bounded, local martingales are predictable if and only if they are continuous, and also give a characterization of cadlag predictable processes in terms of their jumps. Some very strong statements about stopping times also follow without much difficulty for certain special stochastic processes. For example, if the underlying filtration is generated by a Brownian motion then every stopping time is predictable. Actually, this is true whenever the filtration is generated by a continuous Feller process. It is also possible to give a surprisingly simple characterization of stopping times for filtrations generated by arbitrary non-continuous Feller processes. Precisely, a stopping time {\tau} is predictable if the process is almost surely continuous at time {\tau} and is totally inaccessible if the underlying Feller process is almost surely discontinuous at {\tau}.

As usual, we work with respect to a complete filtered probability space {(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\in{\mathbb R}_+},{\mathbb P})}. I now give a statement and proof of the main result of this post. Note that the equivalence of the four conditions below means that any of them can be used as alternative definitions of predictable stopping times. Often, the first condition below is used instead. Stopping times satisfying the definition used in these notes are sometimes called announceable, with the sequence {\tau_n\uparrow\tau} said to announce {\tau} (this terminology is used by, e.g., Rogers & Williams). Stopping times satisfying property 3 below, which is easily seen to be equivalent to 2, are sometimes called fair. Then, the following theorem says that the sets of predictable, fair and announceable stopping times all coincide.

Theorem 1 Let {\tau} be a stopping time. Then, the following are equivalent.

  1. {[\tau]\in\mathcal{P}}.
  2. {\Delta M_\tau1_{[\tau,\infty)}} is a local martingale for all local martingales M.
  3. {{\mathbb E}[1_{\{\tau < \infty\}}\Delta M_\tau]=0} for all cadlag bounded martingales M.
  4. {\tau} is predictable.

Continue reading “Predictable Stopping Times”

Lévy’s Characterization of Brownian Motion

Standard Brownian motion, {\{B_t\}_{t\ge 0}}, is defined to be a real-valued process satisfying the following properties.

  1. {B_0=0}.
  2. {B_t-B_s} is normally distributed with mean 0 and variance ts independently of {\{B_u\colon u\le s\}}, for any {t>s\ge 0}.
  3. B has continuous sample paths.

As always, it only really matters is that these properties hold almost surely. Now, to apply the techniques of stochastic calculus, it is assumed that there is an underlying filtered probability space {(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},{\mathbb P})}, which necessitates a further definition; a process B is a Brownian motion on a filtered probability space {(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},{\mathbb P})} if in addition to the above properties it is also adapted, so that {B_t} is {\mathcal{F}_t}-measurable, and {B_t-B_s} is independent of {\mathcal{F}_s} for each {t>s\ge 0}. Note that the above condition that {B_t-B_s} is independent of {\{B_u\colon u\le s\}} is not explicitly required, as it also follows from the independence from {\mathcal{F}_s}. According to these definitions, a process is a Brownian motion if and only if it is a Brownian motion with respect to its natural filtration.

The property that {B_t-B_s} has zero mean independently of {\mathcal{F}_s} means that Brownian motion is a martingale. Furthermore, we previously calculated its quadratic variation as {[B]_t=t}. An incredibly useful result is that the converse statement holds. That is, Brownian motion is the only local martingale with this quadratic variation. This is known as Lévy’s characterization, and shows that Brownian motion is a particularly general stochastic process, justifying its ubiquitous influence on the study of continuous-time stochastic processes.

Theorem 1 (Lévy’s Characterization of Brownian Motion) Let X be a local martingale with {X_0=0}. Then, the following are equivalent.

  1. X is standard Brownian motion on the underlying filtered probability space.
  2. X is continuous and {X^2_t-t} is a local martingale.
  3. X has quadratic variation {[X]_t=t}.

Continue reading “Lévy’s Characterization of Brownian Motion”

The Burkholder-Davis-Gundy Inequality

The Burkholder-Davis-Gundy inequality is a remarkable result relating the maximum of a local martingale with its quadratic variation. Recall that [X] denotes the quadratic variation of a process X, and {X^*_t\equiv\sup_{s\le t}\vert X_s\vert} is its maximum process.

Theorem 1 (Burkholder-Davis-Gundy) For any {1\le p<\infty} there exist positive constants {c_p,C_p} such that, for all local martingales X with {X_0=0} and stopping times {\tau}, the following inequality holds.

\displaystyle  c_p{\mathbb E}\left[ [X]^{p/2}_\tau\right]\le{\mathbb E}\left[(X^*_\tau)^p\right]\le C_p{\mathbb E}\left[ [X]^{p/2}_\tau\right]. (1)

Furthermore, for continuous local martingales, this statement holds for all {0<p<\infty}.

A proof of this result is given below. For {p\ge 1}, the theorem can also be stated as follows. The set of all cadlag martingales X starting from zero for which {{\mathbb E}[(X^*_\infty)^p]} is finite is a vector space, and the BDG inequality states that the norms {X\mapsto\Vert X^*_\infty\Vert_p={\mathbb E}[(X^*_\infty)^p]^{1/p}} and {X\mapsto\Vert[X]^{1/2}_\infty\Vert_p} are equivalent.

The special case p=2 is the easiest to handle, and we have previously seen that the BDG inequality does indeed hold in this case with constants {c_2=1}, {C_2=4}. The significance of Theorem 1, then, is that this extends to all {p\ge1}.

One reason why the BDG inequality is useful in the theory of stochastic integration is as follows. Whereas the behaviour of the maximum of a stochastic integral is difficult to describe, the quadratic variation satisfies the simple identity {\left[\int\xi\,dX\right]=\int\xi^2\,d[X]}. Recall, also, that stochastic integration preserves the local martingale property. Stochastic integration does not preserve the martingale property. In general, integration with respect to a martingale only results in a local martingale, even for bounded integrands. In many cases, however, stochastic integrals are indeed proper martingales. The Ito isometry shows that this is true for square integrable martingales, and the BDG inequality allows us to extend the result to all {L^p}-integrable martingales, for {p> 1}.

Theorem 2 Let X be a cadlag {L^p}-integrable martingale for some {1<p<\infty}, so that {{\mathbb E}[\vert X_t\vert^p]<\infty} for each t. Then, for any bounded predictable process {\xi}, {Y\equiv\int\xi\,dX} is also an {L^p}-integrable martingale.

Continue reading “The Burkholder-Davis-Gundy Inequality”

Continuous Local Martingales

Continuous local martingales are a particularly well behaved subset of the class of all local martingales, and the results of the previous two posts become much simpler in this case. First, the continuous local martingale property is always preserved by stochastic integration.

Theorem 1 If X is a continuous local martingale and {\xi} is X-integrable, then {\int\xi\,dX} is a continuous local martingale.

Proof: As X is continuous, {Y\equiv\int\xi\,dX} will also be continuous and, therefore, locally bounded. Then, by preservation of the local martingale property, Y is a local martingale. ⬜

Next, the quadratic variation of a continuous local martingale X provides us with a necessary and sufficient condition for X-integrability.

Theorem 2 Let X be a continuous local martingale. Then, a predictable process {\xi} is X-integrable if and only if

\displaystyle  \int_0^t\xi^2\,d[X]<\infty

for all {t>0}.

Continue reading “Continuous Local Martingales”

Quadratic Variations and the Ito Isometry

As local martingales are semimartingales, they have a well-defined quadratic variation. These satisfy several useful and well known properties, such as the Ito isometry, which are the subject of this post. First, the covariation [X,Y] allows the product XY of local martingales to be decomposed into local martingale and FV terms. Consider, for example, a standard Brownian motion B. This has quadratic variation {[B]_t=t} and it is easily checked that {B^2_t-t} is a martingale.

Lemma 1 If X and Y are local martingales then XY-[X,Y] is a local martingale.

In particular, {X^2-[X]} is a local martingale for all local martingales X.

Proof: Integration by parts gives

\displaystyle  XY-[X,Y] = X_0Y_0+\int X_-\,dY+\int Y_-\,dX

which, by preservation of the local martingale property, is a local martingale. ⬜

Continue reading “Quadratic Variations and the Ito Isometry”

Preservation of the Local Martingale Property

Now that it has been shown that stochastic integration can be performed with respect to any local martingale, we can move on to the following important result. Stochastic integration preserves the local martingale property. At least, this is true under very mild hypotheses. That the martingale property is preserved under integration of bounded elementary processes is straightforward. The generalization to predictable integrands can be achieved using a limiting argument. It is necessary, however, to restrict to locally bounded integrands and, for the sake of generality, I start with local sub and supermartingales.

Theorem 1 Let X be a local submartingale (resp., local supermartingale) and {\xi} be a nonnegative and locally bounded predictable process. Then, {\int\xi\,dX} is a local submartingale (resp., local supermartingale).

Proof: We only need to consider the case where X is a local submartingale, as the result will also follow for supermartingales by applying to -X. By localization, we may suppose that {\xi} is uniformly bounded and that X is a proper submartingale. So, {\vert\xi\vert\le K} for some constant K. Then, as previously shown there exists a sequence of elementary predictable processes {\vert\xi^n\vert\le K} such that {Y^n\equiv\int\xi^n\,dX} converges to {Y\equiv\int\xi\,dX} in the semimartingale topology and, hence, converges ucp. We may replace {\xi_n} by {\xi_n\vee0} if necessary so that, being nonnegative elementary integrals of a submartingale, {Y^n} will be submartingales. Also, {\vert\Delta Y^n\vert=\vert\xi^n\Delta X\vert\le K\vert\Delta X\vert}. Recall that a cadlag adapted process X is locally integrable if and only its jump process {\Delta X} is locally integrable, and all local submartingales are locally integrable. So,

\displaystyle  \sup_n\vert\Delta Y^n_t\vert\le K\vert\Delta X_t\vert

is locally integrable. Then, by ucp convergence for local submartingales, Y will satisfy the local submartingale property. ⬜

For local martingales, applying this result to {\pm X} gives,

Theorem 2 Let X be a local martingale and {\xi} be a locally bounded predictable process. Then, {\int\xi\,dX} is a local martingale.

This result can immediately be extended to the class of local {L^p}-integrable martingales, denoted by {\mathcal{M}^p_{\rm loc}}.

Corollary 3 Let {X\in\mathcal{M}^p_{\rm loc}} for some {0< p\le\infty} and {\xi} be a locally bounded predictable process. Then, {\int\xi\,dX\in\mathcal{M}^p_{\rm loc}}.

Continue reading “Preservation of the Local Martingale Property”