Almost Sure

12 October 16

Do Convex and Decreasing Functions Preserve the Semimartingale Property — A Possible Counterexample

f(t,x)

Figure 1: The function f, convex in x and decreasing in t

Here, I attempt to construct a counterexample to the hypotheses of the earlier post, Do convex and decreasing functions preserve the semimartingale property? There, it was asked, for any semimartingale X and function {f\colon{\mathbb R}_+\times{\mathbb R}\rightarrow{\mathbb R}} such that {f(t,x)} is convex in x and right-continuous and decreasing in t, is {f(t,X_t)} necessarily a semimartingale? It was explained how this is equivalent to the hypothesis: for any function {f\colon[0,1]^2\rightarrow{\mathbb R}} such that {f(t,x)} is convex and Lipschitz continuous in x and decreasing in t, does it decompose as {f=g-h} where {g(t,x)} and {h(t,x)} are convex in x and increasing in t. This is the form of the hypothesis which this post will be concerned with, so the example will only involve simple real analysis and no stochastic calculus. I will give some numerical calculations suggesting that the construction below is a counterexample, but do not have any proof of this. So, the hypothesis is still open.

Although the construction given here will be self-contained, it is worth noting that it is connected to the example of a martingale which moves along a deterministic path. If {\{M_t\}_{t\in[0,1]}} is the martingale constructed there, then

\displaystyle  C(t,x)={\mathbb E}[(M_t-x)_+]

defines a function from {[0,1]\times[-1,1]} to {{\mathbb R}} which is convex in x and increasing in t. The question is then whether C can be expressed as the difference of functions which are convex in x and decreasing in t. The example constructed in this post will be the same as C with the time direction reversed, and with a linear function of x added so that it is zero at {x=\pm1}. (more…)

26 September 16

Do Convex and Decreasing Functions Preserve the Semimartingale Property?

Some years ago, I spent considerable effort trying to prove the hypothesis below. After failing at this, I spent time trying to find a counterexample, but also with no success. I did post this as a question on mathoverflow, but it has so far received no conclusive answers. So, as far as I am aware, the following statement remains unproven either way.

Hypothesis H1 Let {f\colon{\mathbb R}_+\times{\mathbb R}\rightarrow{\mathbb R}} be such that {f(t,x)} is convex in x and right-continuous and decreasing in t. Then, for any semimartingale X, {f(t,X_t)} is a semimartingale.

It is well known that convex functions of semimartingales are themselves semimartingales. See, for example, the Ito-Tanaka formula. More generally, if {f(t,x)} was increasing in t rather than decreasing, then it can be shown without much difficulty that {f(t,X_t)} is a semimartingale. Consider decomposing {f(t,X_t)} as

\displaystyle  f(t,X_t)=\int_0^tf_x(s,X_{s-})\,dX_s+V_t, (1)

for some process V. By convexity, the right hand derivative of {f(t,x)} with respect to x always exists, and I am denoting this by {f_x}. In the case where f is twice continuously differentiable then the process V is given by Ito’s formula which, in particular, shows that it is a finite variation process. If {f(t,x)} is convex in x and increasing in t, then the terms in Ito’s formula for V are all increasing and, so, it is an increasing process. By taking limits of smooth functions, it follows that V is increasing even when the differentiability constraints are dropped, so {f(t,X_t)} is a semimartingale. Now, returning to the case where {f(t,x)} is decreasing in t, Ito’s formula is only able to say that V is of finite variation, and is generally not monotonic. As limits of finite variation processes need not be of finite variation themselves, this does not say anything about the case when f is not assumed to be differentiable, and does not help us to determine whether or not {f(t,X_t)} is a semimartingale.

Hypothesis H1 can be weakened by restricting to continuous functions of continuous martingales.

Hypothesis H2 Let {f\colon{\mathbb R}_+\times{\mathbb R}\rightarrow{\mathbb R}} be such that {f(t,x)} is convex in x and continuous and decreasing in t. Then, for any continuous martingale X, {f(t,X_t)} is a semimartingale.

As continuous martingales are special cases of semimartingales, hypothesis H1 implies H2. In fact, the reverse implication also holds so that hypotheses H1 and H2 are equivalent.

Hypotheses H1 and H2 can also be recast as a simple real analysis statement which makes no reference to stochastic processes.

Hypothesis H3 Let {f\colon{\mathbb R}_+\times{\mathbb R}\rightarrow{\mathbb R}} be such that {f(t,x)} is convex in x and decreasing in t. Then, {f=g-h} where {g(t,x)} and {h(t,x)} are convex in x and increasing in t.

(more…)

14 August 16

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.

(more…)

2 July 16

Properties of Quasimartingales

The previous two posts introduced the concept of quasimartingales, and noted that they can be considered as a generalization of submartingales and supermartingales. In this post we prove various basic properties of quasimartingales and of the mean variation, extending results of martingale theory to this situation.

We start with a version of optional stopping which applies for quasimartingales. For now, we just consider simple stopping times, which are stopping times taking values in a finite subset of the nonnegative extended reals {\bar{\mathbb R}_+=[0,\infty]}. Stopping a process can only decrease its mean variation (recall the alternative definitions {{\rm Var}} and {{\rm Var}^*} for the mean variation). For example, a process X is a martingale if and only if {{\rm Var}(X)=0}, so in this case the following result says that stopped martingales are martingales.

Lemma 1 Let X be an adapted process and {\tau} be a simple stopping time. Then

\displaystyle  {\rm Var}^*(X^\tau)\le{\rm Var}^*(X). (1)

Assuming, furthermore, that X is integrable,

\displaystyle  {\rm Var}(X^\tau)\le{\rm Var}(X). (2)

and, more precisely,

\displaystyle  {\rm Var}(X)={\rm Var}(X^\tau)+{\rm Var}(X-X^\tau) (3)

(more…)

22 November 11

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.

(more…)

3 October 11

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. (more…)

18 July 11

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.

(more…)

3 May 11

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. \Box

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.

(more…)

28 March 11

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.

(more…)

11 March 11

The General Theory of Semimartingales

Having completed the series of posts applying the methods of stochastic calculus to various special types of processes, I now return to the development of the theory. The next few posts of these notes will be grouped under the heading `The General Theory of Semimartingales’. Subjects which will be covered include the classification of predictable stopping times, integration with respect to continuous and predictable FV processes, decompositions of special semimartingales, the Bichteler-Dellacherie theorem, the Doob-Meyer decomposition and the theory of quasimartingales.

One of the main results is the Bichteler-Dellacherie theorem describing the class of semimartingales which, in these notes, were defined to be cadlag adapted processes with respect to which the stochastic integral can be defined (that is, they are good integrators). It was shown that these include the sums of local martingales and FV processes. The Bichteler-Dellacherie theorem says that this is the full class of semimartingales. Classically, semimartingales were defined as a sum of a local martingale and an FV process so, an alternative statement of the theorem is that the classical definition agrees with the one used in these notes. Further results, such as the Doob-Meyer decomposition for submartingales and Rao’s decomposition for quasimartingales, will follow quickly from this.

Logically, the structure of these notes will be almost directly opposite to the historical development of the results. Originally, much of the development of the stochastic integral was based on the Doob-Meyer decomposition which, in turn, relied on some advanced ideas such as the predictable and dual predictable projection theorems. However, here, we have already introduced stochastic integration without recourse to such general theory, and can instead make use of this in the theory. The reasons I have taken this approach are as follows. First, stochastic integration is a particularly straightforward and useful technique for many applications, so it is desirable to introduce this early on. Second, although it is possible to use the general theory of processes in the construction of the integral, such an approach seems rather distinct from the intuitive understanding of stochastic integration as well as superfluous to many of its properties. So it seemed more natural from the point of view of these notes to define the integral first, guided by the properties of the (non-stochastic) Lebesgue integral, then show how its elementary properties follow from the definitions, and develop the further theory later. (more…)

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