Purely Discontinuous Semimartingales

As stated by the Bichteler-Dellacherie theorem, all semimartingales can be decomposed as the sum of a local martingale and an FV process. However, as the terms are only determined up to the addition of an FV local martingale, this decomposition is not unique. In the case of continuous semimartingales, we do obtain uniqueness, by requiring the terms in the decomposition to also be continuous. Furthermore, the decomposition into continuous terms is preserved by stochastic integration. Looking at non-continuous processes, there does exist a unique decomposition into local martingale and predictable FV processes, so long as we impose the slight restriction that the semimartingale is locally integrable.

In this post, I look at another decomposition which holds for all semimartingales and, moreover, is uniquely determined. This is the decomposition into continuous local martingale and purely discontinuous terms which, as we will see, is preserved by the stochastic integral. This is distinct from each of the decompositions mentioned above, except for the case of continuous semimartingales, in which case it coincides with the sum of continuous local martingale and FV components. Before proving the decomposition, I will start by describing the class of purely discontinuous semimartingales which, although they need not have finite variation, do have many of the properties of FV processes. In fact, they comprise precisely of the closure of the set of FV processes under the semimartingale topology. The terminology can be a bit confusing, and it should be noted that purely discontinuous processes need not actually have any discontinuities. For example, all continuous FV processes are purely discontinuous. For this reason, the term quadratic pure jump semimartingale’ is sometimes used instead, referring to the fact that their quadratic variation is a pure jump process. Recall that quadratic variations and covariations can be written as the sum of continuous and pure jump parts,

 $\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle [X]_t&\displaystyle=[X]^c_t+\sum_{s\le t}(\Delta X_s)^2,\smallskip\\ \displaystyle [X,Y]_t&\displaystyle=[X,Y]^c_t+\sum_{s\le t}\Delta X_s\Delta Y_s. \end{array}$ (1)

The statement that the quadratic variation is a pure jump process is equivalent to saying that its continuous part, ${[X]^c}$, is zero. As the only difference between the generalized Ito formula for semimartingales and for FV processes is in the terms involving continuous parts of the quadratic variations and covariations, purely discontinuous semimartingales behave much like FV processes under changes of variables and integration by parts. Yet another characterisation of purely discontinuous semimartingales is as sums of purely discontinuous local martingales — which were studied in the previous post — and of FV processes.

Rather than starting by choosing one specific property to use as the definition, I prove the equivalence of various statements, any of which can be taken to define the purely discontinuous semimartingales.

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

1. ${[X]^c=0}$.
2. ${[X,Y]^c=0}$ for all semimartingales Y.
3. ${[X,Y]=0}$ for all continuous semimartingales Y.
4. ${[X,M]=0}$ for all continuous local martingales M.
5. ${X=M+V}$ for a purely discontinuous local martingale M and FV process V.
6. there exists a sequence ${\{X^n\}_{n=1,2,\ldots}}$ of FV processes such that ${X^n\rightarrow X}$ in the semimartingale topology.

Purely Discontinuous Local Martingales

The previous post introduced the idea of a purely discontinuous local martingale. In the context of that post, such processes were used to construct local martingales with prescribed jumps, and enabled us to obtain uniqueness in the constructions given there. However, purely discontinuous local martingales are a very useful concept more generally in martingale and semimartingale theory, so I will go into more detail about such processes now. To start, we restate the definition from the previous post.

Definition 1 A local martingale X is said to be purely discontinuous iff XM is a local martingale for all continuous local martingales M.

We can show that every local martingale decomposes uniquely into continuous and purely discontinuous parts. Continuous local martingales are well understood — for instance, they can always be realized as time-changed Brownian motions. On the other hand, as we will see in a moment, purely discontinuous local martingales can be realized as limits of FV processes, and arguments involving FV local martingales can often to be extended to the purely discontinuous case. So, decomposition (1) below is useful as it allows arguments involving continuous-time local martingales to be broken down into different approaches involving their continuous and purely discontinuous parts. As always, two processes are considered to be equal if they are equivalent up to evanescence.

Theorem 2 Every local martingale X decomposes uniquely as

 $\displaystyle X = X^{\rm c} + X^{\rm d}$ (1)

where ${X^{\rm c}}$ is a continuous local martingale with ${X^{\rm c}_0=0}$ and ${X^{\rm d}}$ is a purely discontinuous local martingale.

Proof: As the process ${H=\Delta X}$ is, by definition, equal to the jump process of a local martingale then it satisfies the hypothesis of Theorem 5 of the previous post. So, there exists a purely discontinuous local martingale ${X^{\rm d}}$ with ${\Delta X^{\rm d}=H=\Delta X}$. We can take ${X^{\rm d}_0=X_0}$ so that ${X^{\rm c}=X-X^{\rm d}}$ is a continuous local martingale starting from 0.

If ${X=\tilde X^{\rm c}+\tilde X^{\rm d}}$ is another such decomposition, then ${\tilde X^{\rm d}}$ and ${X^{\rm d}}$ have the same jumps and initial value so, by Lemma 3 of the previous post, ${\tilde X^{\rm d}=X^{\rm d}}$. ⬜

Throughout the remainder of this post, the notation ${X^{\rm c}}$ and ${X^{\rm d}}$ will be used to denote the continuous and purely discontinuous parts of a local martingale X, as given by decomposition (1). Using the notation ${\mathcal{M}_{\rm loc}}$, ${\mathcal{M}_{{\rm loc},0}^{\rm c}}$ and ${\mathcal{M}_{\rm loc}^{\rm d} }$ respectively for the spaces of local martingales, continuous local martingales starting from zero and the purely discontinuous local martingales, Theorem 2 can be expressed succinctly as

 $\displaystyle \mathcal{M}_{\rm loc} = \mathcal{M}_{{\rm loc},0}^{\rm c} \oplus \mathcal{M}_{\rm loc}^{\rm d}.$ (2)

That is, ${\mathcal{M}_{\rm loc}}$ is the direct sum of ${\mathcal{M}_{{\rm loc},0}^{\rm c}}$ and ${\mathcal{M}_{\rm loc}^{\rm d}}$. Definition 2 identifies the purely discontinuous local martingales to be, in a sense, orthogonal to the continuous local martingales. Then, (2) can be understood as the decomposition of ${\mathcal{M}_{\rm loc}}$ into the direct sum of the closed subspace ${\mathcal{M}_{{\rm loc},0}^{\rm c}}$ and its orthogonal complement. This does in fact give an alternative, elementary, and commonly used, method of proving decomposition (1). As we have already shown the rather strong result of Theorem 5 from the previous post, the quickest way of proving the decomposition was to simply apply this result. I’ll give more details on the more elementary approach further below.

Definition 1 used above for the class of purely discontinuous local martingales was very convenient for our purposes, as it leads immediately to the proof of Theorem 2. However, there are many alternative characterizations of such processes. For example, they are precisely the processes which are limits of FV local martingales in a strong enough sense. They can also be characterized in terms of their quadratic variations and covariations. Recall that the quadratic variation and covariation are FV processes with jumps ${\Delta[X]=(\Delta X)^2}$ and ${\Delta[X,Y]=\Delta X\Delta Y}$, so that they can be decomposed into continuous and pure jump components,

 $\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle [X]_t &\displaystyle=[X]^c_t+\sum_{s\le t}(\Delta X_s)^2,\smallskip\\ \displaystyle [X,Y]_t &\displaystyle=[X,Y]^c_t+\sum_{s\le t}\Delta X_s\Delta Y_s. \end{array}$ (3)

The following theorem gives several alternative characterizations of the class of purely discontinuous local martingales.

Theorem 3 For a local martingale X, the following are equivalent.

1. X is purely discontinuous.
2. ${[X,Y]=0}$ for all continuous local martingales Y.
3. ${[X,Y]^c=0}$ for all local martingales Y.
4. ${[X]^c=0}$.
5. there exists a sequence ${\{X^n\}_{n=1,2,\ldots}}$ of FV local martingales such that

$\displaystyle {\mathbb E}\left[\sup_{t\ge0}(X^n_t-X_t)^2\right]\rightarrow0.$

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.

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.

Continuous Semimartingales

A stochastic process is a semimartingale if and only if it can be decomposed as the sum of a local martingale and an FV process. This is stated by the Bichteler-Dellacherie theorem or, alternatively, is often taken as the definition of a semimartingale. For continuous semimartingales, which are the subject of this post, things simplify considerably. The terms in the decomposition can be taken to be continuous, in which case they are also unique. As usual, we work with respect to a complete filtered probability space ${(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge0},{\mathbb P})}$, all processes are real-valued, and two processes are considered to be the same if they are indistinguishable.

Theorem 1 A continuous stochastic process X is a semimartingale if and only if it decomposes as

 $\displaystyle X=M+A$ (1)

for a continuous local martingale M and continuous FV process A. Furthermore, assuming that ${A_0=0}$, decomposition (1) is unique.

Proof: As sums of local martingales and FV processes are semimartingales, X is a semimartingale whenever it satisfies the decomposition (1). Furthermore, if ${X=M+A=M^\prime+A^\prime}$ were two such decompositions with ${A_0=A^\prime_0=0}$ then ${M-M^\prime=A^\prime-A}$ is both a local martingale and a continuous FV process. Therefore, ${A^\prime-A}$ is constant, so ${A=A^\prime}$ and ${M=M^\prime}$.

It just remains to prove the existence of decomposition (1). However, X is continuous and, hence, is locally square integrable. So, Lemmas 4 and 5 of the previous post say that we can decompose ${X=M+A}$ where M is a local martingale, A is an FV process and the quadratic covariation ${[M,A]}$ is a local martingale. As X is continuous we have ${\Delta M=-\Delta A}$ so that, by the properties of covariations,

 $\displaystyle -[M,A]_t=-\sum_{s\le t}\Delta M_s\Delta A_s=\sum_{s\le t}(\Delta A_s)^2.$ (2)

We have shown that ${-[M,A]}$ is a nonnegative local martingale so, in particular, it is a supermartingale. This gives ${\mathbb{E}[-[M,A]_t]\le\mathbb{E}[-[M,A]_0]=0}$. Then (2) implies that ${\Delta A}$ is zero and, hence, A and ${M=X-A}$ are continuous. ⬜

Using decomposition (1), it can be shown that a predictable process ${\xi}$ is X-integrable if and only if it is both M-integrable and A-integrable. Then, the integral with respect to X breaks down into the sum of the integrals with respect to M and A. This greatly simplifies the construction of the stochastic integral for continuous semimartingales. The integral with respect to the continuous FV process A is equivalent to Lebesgue-Stieltjes integration along sample paths, and it is possible to construct the integral with respect to the continuous local martingale M for the full set of M-integrable integrands using the Ito isometry. Many introductions to stochastic calculus focus on integration with respect to continuous semimartingales, which is made much easier because of these results.

Theorem 2 Let ${X=M+A}$ be the decomposition of the continuous semimartingale X into a continuous local martingale M and continuous FV process A. Then, a predictable process ${\xi}$ is X-integrable if and only if

 $\displaystyle \int_0^t\xi^2\,d[M]+\int_0^t\vert\xi\vert\,\vert dA\vert < \infty$ (3)

almost surely, for each time ${t\ge0}$. In that case, ${\xi}$ is both M-integrable and A-integrable and,

 $\displaystyle \int\xi\,dX=\int\xi\,dM+\int\xi\,dA$ (4)

gives the decomposition of ${\int\xi\,dX}$ into its local martingale and FV terms.

The Bichteler-Dellacherie Theorem

In this post, I will give a statement and proof of the Bichteler-Dellacherie theorem describing the space of semimartingales. A semimartingale, as defined in these notes, is a cadlag adapted stochastic process X such that the stochastic integral ${\int\xi\,dX}$ is well-defined for all bounded predictable integrands ${\xi}$. More precisely, an integral should exist which agrees with the explicit expression for elementary integrands, and satisfies bounded convergence in the following sense. If ${\{\xi^n\}_{n=1,2,\ldots}}$ is a uniformly bounded sequence of predictable processes tending to a limit ${\xi}$, then ${\int_0^t\xi^n\,dX\rightarrow\int_0^t\xi\,dX}$ in probability as n goes to infinity. If such an integral exists, then it is uniquely defined up to zero probability sets.

An immediate consequence of bounded convergence is that the set of integrals ${\int_0^t\xi\,dX}$ for a fixed time t and bounded elementary integrands ${\vert\xi\vert\le1}$ is bounded in probability. That is,

 $\displaystyle \left\{\int_0^t\xi\,dX\colon\xi{\rm\ is\ elementary},\ \vert\xi\vert\le1\right\}$ (1)

is bounded in probability, for each ${t\ge0}$. For cadlag adapted processes, it was shown in a previous post that this is both a necessary and sufficient condition to be a semimartingale. Some authors use the property that (1) is bounded in probability as the definition of semimartingales (e.g., Protter, Stochastic Calculus and Differential Equations). The existence of the stochastic integral for arbitrary predictable integrands does not follow particularly easily from this definition, at least, not without using results on extensions of vector valued measures. On the other hand, if you are content to restrict to integrands which are left-continuous with right limits, the integral can be constructed very efficiently and, furthermore, such integrands are sufficient for many uses (integration by parts, Ito’s formula, a large class of stochastic differential equations, etc).

It was previously shown in these notes that, if X can be decomposed as ${X=M+V}$ for a local martingale M and FV process V then it is possible to construct the stochastic integral, so X is a semimartingale. The importance of the Bichteler-Dellacherie theorem is that it tells us that a process is a semimartingale if and only if it is the sum of a local martingale and an FV process. In fact this was the historical definition used of semimartingales, and is still probably the most common definition.

Throughout, we work with respect to a complete filtered probability space ${(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge0},{\mathbb P})}$, and all processes are real-valued.

Theorem 1 (Bichteler-Dellacherie) For a cadlag adapted process X, the following are equivalent.

1. X is a semimartingale.
2. For each ${t\ge0}$, the set given by (1) is bounded in probability.
3. X is the sum of a local martingale and an FV process.

Furthermore, the local martingale term in 3 can be taken to be locally bounded.

Poisson Processes

A Poisson process is a continuous-time stochastic process which counts the arrival of randomly occurring events. Commonly cited examples which can be modeled by a Poisson process include radioactive decay of atoms and telephone calls arriving at an exchange, in which the number of events occurring in each consecutive time interval are assumed to be independent. Being piecewise constant, Poisson processes have very simple pathwise properties. However, they are very important to the study of stochastic calculus and, together with Brownian motion, forms one of the building blocks for the much more general class of Lévy processes. I will describe some of their properties in this post.

A random variable N has the Poisson distribution with parameter ${\lambda}$, denoted by ${N\sim{\rm Po}(\lambda)}$, if it takes values in the set of nonnegative integers and

 $\displaystyle {\mathbb P}(N=n)=\frac{\lambda^n}{n!}e^{-\lambda}$ (1)

for each ${n\in{\mathbb Z}_+}$. The mean and variance of N are both equal to ${\lambda}$, and the moment generating function can be calculated,

$\displaystyle {\mathbb E}\left[e^{aN}\right] = \exp\left(\lambda(e^a-1)\right),$

which is valid for all ${a\in{\mathbb C}}$. From this, it can be seen that the sum of independent Poisson random variables with parameters ${\lambda}$ and ${\mu}$ is again Poisson with parameter ${\lambda+\mu}$. The Poisson distribution occurs as a limit of binomial distributions. The binomial distribution with success probability p and m trials, denoted by ${{\rm Bin}(m,p)}$, is the sum of m independent ${\{0,1\}}$-valued random variables each with probability p of being 1. Explicitly, if ${N\sim{\rm Bin}(m,p)}$ then

$\displaystyle {\mathbb P}(N=n)=\frac{m!}{n!(m-n)!}p^n(1-p)^{m-n}.$

In the limit as ${m\rightarrow\infty}$ and ${p\rightarrow 0}$ such that ${mp\rightarrow\lambda}$, it can be verified that this tends to the Poisson distribution (1) with parameter ${\lambda}$.

Poisson processes are then defined as processes with independent increments and Poisson distributed marginals, as follows.

Definition 1 A Poisson process X of rate ${\lambda\ge0}$ is a cadlag process with ${X_0=0}$ and ${X_t-X_s\sim{\rm Po}(\lambda(t-s))}$ independently of ${\{X_u\colon u\le s\}}$ for all ${s\le t}$.

An immediate consequence of this definition is that, if X and Y are independent Poisson processes of rates ${\lambda}$ and ${\mu}$ respectively, then their sum ${X+Y}$ is also Poisson with rate ${\lambda+\mu}$. Continue reading “Poisson Processes”

Being able to handle quadratic variations and covariations of processes is very important in stochastic calculus. Apart from appearing in the integration by parts formula, they are required for the stochastic change of variables formula, known as Ito’s lemma, which will be the subject of the next post. Quadratic covariations satisfy several simple relations which make them easy to handle, especially in conjunction with the stochastic integral.

Recall from the previous post that the covariation ${[X,Y]}$ is a cadlag adapted process, so that its jumps ${\Delta [X,Y]_t\equiv [X,Y]_t-[X,Y]_{t-}}$ are well defined.

Lemma 1 If ${X,Y}$ are semimartingales then

 $\displaystyle \Delta [X,Y]=\Delta X\Delta Y.$ (1)

In particular, ${\Delta [X]=\Delta X^2}$.

Proof: Taking the jumps of the integration by parts formula for ${XY}$ gives

 $\displaystyle \Delta XY = X_{-}\Delta Y + Y_{-}\Delta X + \Delta [X,Y],$

and rearranging this gives the result. ⬜

An immediate consequence is that quadratic variations and covariations involving continuous processes are continuous. Another consequence is that the sum of the squares of the jumps of a semimartingale over any bounded interval must be finite.

Corollary 2 Every semimartingale ${X}$ satisfies

 $\displaystyle \sum_{s\le t}\Delta X^2_s\le [X]_t<\infty.$

Proof: As ${[X]}$ is increasing, the inequality ${[X]_t\ge \sum_{s\le t}\Delta [X]_s}$ holds. Substituting in ${\Delta[X]=\Delta X^2}$ gives the result. ⬜

Next, the following result shows that covariations involving continuous finite variation processes are zero. As Lebesgue-Stieltjes integration is only defined for finite variation processes, this shows why quadratic variations do not play an important role in standard calculus. For noncontinuous finite variation processes, the covariation must have jumps satisfying (1), so will generally be nonzero. In this case, the covariation is just given by the sum over these jumps. Integration with respect to any FV process ${V}$ can be defined as the Lebesgue-Stieltjes integral on the sample paths, which is well defined for locally bounded measurable integrands and, when the integrand is predictable, agrees with the stochastic integral.

Lemma 3 Let ${X}$ be a semimartingale and ${V}$ be an FV process. Their covariation is

 $\displaystyle [X,V]_t = \int_0^t \Delta X\,dV = \sum_{s\le t}\Delta X_s\Delta V_s.$ (2)

In particular, if either of ${X}$ or ${V}$ is continuous then ${[X,V]=0}$.

Properties of the Stochastic Integral

In the previous two posts I gave a definition of stochastic integration. This was achieved via an explicit expression for elementary integrands, and extended to all bounded predictable integrands by bounded convergence in probability. The extension to unbounded integrands was done using dominated convergence in probability. Similarly, semimartingales were defined as those cadlag adapted processes for which such an integral exists.

The current post will show how the basic properties of stochastic integration follow from this definition. First, if ${V}$ is a cadlag process whose sample paths are almost surely of finite variation over an interval ${[0,t]}$, then ${\int_0^t\xi\,dV}$ can be interpreted as a Lebesgue-Stieltjes integral on the sample paths. If the process is also adapted, then it will be a semimartingale and the stochastic integral can be used. Fortunately, these two definitions of integration do agree with each other. The term FV process is used to refer to such cadlag adapted processes which are almost surely of finite variation over all bounded time intervals. The notation ${\int_0^t\vert\xi\vert\,\vert dV\vert}$ represents the Lebesgue-Stieltjes integral of ${\vert\xi\vert}$ with respect to the variation of ${V}$. Then, the condition for ${\xi}$ to be ${V}$-integrable in the Lebesgue-Stieltjes sense is precisely that this integral is finite.

Lemma 1 Every FV process ${V}$ is a semimartingale. Furthermore, let ${\xi}$ be a predictable process satisfying

 $\displaystyle \int_0^t\vert\xi\vert\,\vert dV\vert<\infty$ (1)

almost surely, for each ${t\ge 0}$. Then, ${\xi\in L^1(V)}$ and the stochastic integral ${\int\xi\,dV}$ agrees with the Lebesgue-Stieltjes integral, with probability one.