Almost Sure

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.

Proof of 1 ⇒ 2: If {[X]^c=0} then the Cauchy-Schwarz inequality gives

\displaystyle  \lvert [X,Y]^c\rvert\le\sqrt{[X]^c[Y]^c}=0.

Proof of 2 ⇒ 3: If Y is a continuous semimartingale then {[X,Y]} is continuous, so

\displaystyle  [X,Y]=[X,Y]^c=0.

Proof of 3 ⇒ 4: As every continuous local martingale M is also a semimartingale, we have {[X,M]=0}. ⬜

Proof of 4 ⇒ 5: Since X is a semimartingale, it decomposes as {X=M+V} for a local martingale M and FV process V. For any continuous local martingale N, the fact that V is FV gives {[V,N]=0} and, hence,

\displaystyle  [M,N]=[X,N]-[V,N]=[X,N]=0

So M is a purely discontinuous local martingale. ⬜

Proof of 5 ⇒ 6: By Theorem 3 of the post on purely discontinuous local martingales, there exists a sequence {M^n} of FV local martingales such that {\sup_{t\ge0}\lvert M^n_t-M_t\rvert} tends to zero in {L^2}. This implies that it converges to zero in the semimartingale topology, and the sequence of FV processes {X^n=M^n+V} tends to X in the semimartingale topology. ⬜

Proof of 6 ⇒ 1: As FV processes do not contribute to the continuous parts of quadratic variations, {[X]^c} is equal to {[X-X^n]^c}. So, continuity of quadratic covariations with respect to the semimartingale topology gives,

\displaystyle  [X]^c=[X-X^n]^c\le[X-X^n]\rightarrow0.

Definition 2 A semimartingale X is purely discontinuous iff any (and then, all) of the equivalent statements of Theorem 1 are satisfied.

Before going any further, we should check that the terminology does not contradict that used for local martingales.

Lemma 3 A local martingale is a purely discontinuous local martingale (Definition 1 of the previous post) if and only if it is a purely discontinuous semimartingale (Definition 2 above).

Proof: For a local martingale X, by Theorem 1 above and Theorem 3 of the previous post, both definitions are equivalent to {[X,M]=0} for all local martingales M. ⬜

The simplest examples of semimartingales which are purely discontinuous are those with finite variation.

Lemma 4 Every FV process is a purely discontinuous semimartingale.

Proof: This follows immediately from statement 5 of Theorem 1. ⬜

More specifically, we can identify the set of purely discontinuous semimartingales in terms of the set of FV processes using the semimartingale topology.

Lemma 5 The set of purely discontinuous semimartingales is the closure of the FV processes in the semimartingale topology.

Proof: This follows immediately from statement 6 of Theorem 1. ⬜

Not all purely discontinuous semimartingales are FV, so the converse to Lemma 4 does not hold. A simple necessary and sufficient condition can be given in terms of the jumps of the process though.

Lemma 6 A purely discontinuous semimartingale X is an FV process if and only if {\sum_{s\le t}\lvert\Delta X_s\rvert} is almost surely finite for each {t\in{\mathbb R}_+}.

Proof: As {\sum_{s\le t}\lvert\Delta X_s\rvert} is bounded by the variation of X over the interval {[0,t]}, it is finite whenever X is an FV process.

Conversely, decompose X as {M+V} for a purely discontinuous local martingale M and FV process V. If {\sum_{s\le t}\lvert\Delta X_s\rvert} is almost surely finite, then so is {\sum_{s\le t}\lvert\Delta M_s\rvert} and, by Lemma 4 of the previous post, M is an FV process. ⬜

It follows immediately from this that, for continuous processes, the purely discontinuous semimartingales coincide with the FV processes.

Corollary 7 A continuous semimartingale is purely discontinuous if and only if it is an FV process.

Since continuous adapted processes are predictable, the following is a generalization of Corollary 7.

Lemma 8 A predictable semimartingale is purely discontinuous if and only if it is an FV process.

Proof: If X is an FV process then it is purely discontinuous by Lemma 4. Conversely, suppose that X is a predictable purely discontinuous semimartingale. Then, it is locally bounded, and so decomposes as {X=M+A} for a local martingale M and predictable FV process A. As M is a predictable local martingale it is continuous and, as V is FV, we have {[M,V]=0}. Therefore,

\displaystyle  [M]=[X,M]=0.

So M is almost surely constant. Hence, {X=M_0+V} is an FV process. ⬜

An important property of the class of purely discontinuous semimartingales is that it is preserved by the stochastic integral.

Lemma 9 If X is a purely discontinuous semimartingale and {\xi} is a predictable X-integrable process, then {\int\xi\,dX} is purely discontinuous.

Proof: As quadratic covariations commute with stochastic integration,

\displaystyle  \left[\int\xi\,dX,M\right]=\int\xi\,d[X,M]=0.

for all continuous local martingales M. ⬜

The terminology `purely discontinuous’ was previously applied to Lévy processes and to more general processes with independent increments. As stated in the following lemma, this agrees with the terminology applied to semimartingales in this post.

Lemma 10 Let X be a real-valued semimartingale with the independent increments property and which is continuous in probability, and has characteristics {(\Sigma_t,b_t,\mu)}. Then, X is a purely discontinuous semimartingale iff {\Sigma_t=0}.

Proof: As the quadratic variation satisfies {[X]^c_t=\Sigma_t}, the result follows from the first condition of Theorem 1 above. ⬜

In particular, a Lévy process with characterics {(\Sigma,b,\nu)} is a semimartingale, and is purely discontinuous if and only if {\Sigma=0}. It is FV if and only if

\displaystyle  \int 1\wedge \lvert x\rvert\,d\nu(x) < \infty.

This gives plenty of examples of purely discontinuous semimartingales which are not FV processes. For instance, we have the (symmetric and asymmetric) stable processes with exponents {1\le\alpha < 2}, which have Lévy measure

\displaystyle  d\nu(x)=(a1_{\{x > 0\}}+b1_{\{x < 0\}})\lvert x\rvert^{-\alpha}dx

for nonnegative constants a and b. In particular, for {\alpha=1}, this includes the Cauchy process.

The Semimartingale Decomposition

I now give the decomposition theorem for semimartingales.

Theorem 11 Every semimartingale X decomposes uniquely as

\displaystyle  X = X^{\rm c} + X^{\rm d} (2)

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

Proof: By the Bichteler-Dellacherie theorem, X decomposes as {M+V} for a local martingale M and FV process V. Then, decomposes} into its continuous local martingale {M^{\rm c}} and purely discontinuous part {M^{\rm d}}. Writing

\displaystyle  X=M^{\rm c}+(M^{\rm d}+V)

gives the required decomposition with {X^{\rm c}=M^{\rm c}} and {X^{\rm d}=M^{\rm d}+V}.

If {X=\tilde X^{\rm c}+\tilde X^{\rm d}} was any other such decomposition then

\displaystyle  N\equiv X^{\rm c}-\tilde X^{\rm d}=\tilde X^{\rm d}-X^{\rm d}

is both a continuous local martingale starting from zero and is purely discontinuous, so is zero. Hence, the decomposition is unique. ⬜

Throughout the remainder of this post, for any semimartingale X, the notation {X^{\rm c}} and {X^{\rm d}} will be used to denote its continuous local martingale and purely discontinuous parts. The continuous part of the quadratic covariation of semimartingales is equal to the quadratic variation of their continuous parts.

Lemma 12 If X and Y are semimartingales then

\displaystyle  [X,Y]^c=[X^{\rm c},Y^{\rm c}]. (3)

Proof: This follows from the second statement of Theorem 1 and bilinearity of quadratic covariations.

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle [X,Y]^c &\displaystyle=[X^{\rm c},Y^{\rm c}]^c+[X^{\rm c},Y^{\rm d}]^c+[X^{\rm d},Y]^c\smallskip\\ &\displaystyle=[X^{\rm c},Y^{\rm c}]^c=[X^{\rm c},Y^{\rm c}]. \end{array}

Combining equation (3) with (1) gives an expression for the quadratic covariation in terms of their continuous parts and jumps

\displaystyle  [X,Y]_t=[X^{\rm c},Y^{\rm c}]_t+\sum_{s\le t}\Delta X_s\Delta Y_s. (4)

This is sometimes used as the definition of the quadratic covariation, by first defining {[M,N]} for continuous local martingales M, N as the unique continuous FV process starting from zero such that {MN-[M,N]} is a local martingale, and extending to all semimartingales using (4).

Decomposition (2) is preserved by stochastic integration.

Theorem 13 Let {X = X^{\rm c} + X^{\rm d}} be decomposition 2. Then, a predictable process {\xi} is X-integrable if and only if it is both {X^{\rm c}} and {X^{\rm d}}-integrable and,

\displaystyle  \int\xi\,dX = \int\xi\,dX^{\rm c}+\int\xi\,dX^{\rm d} (5)

is the unique decomposition of the integral into its continuous martingale and purely discontinuous parts.

Proof: By the elementary properties of the stochastic integral, if {\xi} is both {X^{\rm c}} and {X^{\rm d}}-integrable then it is X-integrable.

Conversely, suppose that {\xi} is X-integrable. Then,

\displaystyle  \int\xi^2\,d[X^{\rm c}]=\int\xi^2\,d[X]^c\le\int\xi^2\,d[X] < \infty,

so, {\xi} is {X^{\rm c}}-integrable and, hence, is integrable with respect to {X^{\rm d}=X-X^{\rm c}}.

Next, stochastic integration preserves both continuous local martingales and, by Lemma 9, purely discontinuous semimartingales. So, the terms on the right of (5) are, respectively, a continuous local martingale and a purely discontinuous semimartingale. ⬜

For Lévy processes, we have seen decomposition (2) before.

Lemma 14 Let X be a cadlag Lévy process and

\displaystyle  X = W+Y

be its decomposition as the sum of an initially zero continuous centered Gaussian process with independent increments and a purely discontinuous Lévy process.

Then, {W=X^{\rm c}} and {Y=X^{\rm d}}.

Proof: As W has the independent increments property and is centered, it is a martingale. Also, by Lemma 10, Y is a purely discontinuous semimartingale. ⬜

Finally, I show that decomposition (2) is continuous in the semimartingale topology. For brevity, I am denoting semimartingale convergence by {X^n\xrightarrow{\rm sm}X}.

Lemma 15 Let X and {\{X^n\}_{n=1,2,\ldots}} be semimartingales. Then, {X^n\xrightarrow{\rm sm}X} if and only if {(X^n)^{\rm c}\xrightarrow{\rm sm}X^{\rm c}} and {(X^n)^{\rm d}\xrightarrow{\rm sm}X^{\rm d}}.

In particular, the maps on the space of semimartingales

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{c} \displaystyle \mathcal{S}\rightarrow\mathcal{S}\smallskip\\ \displaystyle X\mapsto X^{\rm c},\ X\mapsto X^{\rm d} \end{array}

are continuous with respect to the semimartingale topology.

Proof: As addition is clearly continuous under the semimartingale topology, if {(X^n)^{\rm c}\xrightarrow{\rm sm}X^{\rm c}} and {(X^n)^{\rm d}\xrightarrow{\rm sm}X^{\rm d}} then {X^n\xrightarrow{\rm sm}X}.

Conversely, if {X^n\xrightarrow{\rm sm}X} then, by continuity of quadratic variations under the semimartingale topology,

\displaystyle  [(X^n)^{\rm c}-X^{\rm c}]=[X^n-X]^{\rm c}\le[X^n-X]\xrightarrow{\rm sm}0.

So, {(X^n)^{\rm c}} converges in the semimartingale topology to {X^{\rm c}}. Hence,

\displaystyle  (X^n)^{\rm d} = X^n-(X^n)^{\rm c}\xrightarrow{\rm sm}X-X^{\rm c}=X^{\rm d}

as required. This also shows that the maps {X\mapsto X^{\rm c}} and {X\mapsto X^{\rm d}} are continuous in the semimartingale topology. ⬜

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