# 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.

# Properties of Lévy Processes

Lévy processes, which are defined as having stationary and independent increments, were introduced in the previous post. It was seen that the distribution of a d-dimensional Lévy process X is determined by the characteristics ${(\Sigma,b,\nu)}$ via the Lévy-Khintchine formula,

 $\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle{\mathbb E}\left[e^{ia\cdot (X_t-X_0)}\right] = \exp(t\psi(a)),\smallskip\\ &\displaystyle\psi(a)=ia\cdot b-\frac12a^{\rm T}\Sigma a+\int_{{\mathbb R}^d}\left(e^{ia\cdot x}-1-\frac{ia\cdot x}{1+\Vert x\Vert}\right)\,d\nu(x). \end{array}$ (1)

The positive semidefinite matrix ${\Sigma}$ describes the Brownian motion component of X, b is a drift term, and ${\nu}$ is a measure on ${{\mathbb R}^d}$ such that ${\nu(A)}$ is the rate at which jumps ${\Delta X\in A}$ of X occur. Then, equation (1) gives us the characteristic function of the increments of the process.

In the current post, I will investigate some of the properties of such processes, and how they are related to the characteristics. In particular, we will be concerned with pathwise properties of X. It is known that Brownian motion and Cauchy processes have infinite variation in every nonempty time interval, whereas other Lévy processes — such as the Poisson process — are piecewise constant, only jumping at a discrete set of times. There are also purely discontinuous Lévy processes which have infinitely many discontinuities, yet are of finite variation, on every interval (e.g., the gamma process). Continue reading “Properties of Lévy Processes”

# Lévy Processes

Continuous-time stochastic processes with stationary independent increments are known as Lévy processes. In the previous post, it was seen that processes with independent increments are described by three terms — the covariance structure of the Brownian motion component, a drift term, and a measure describing the rate at which jumps occur. Being a special case of independent increments processes, the situation with Lévy processes is similar. However, stationarity of the increments does simplify things a bit. We start with the definition.

Definition 1 (Lévy process) A d-dimensional Lévy process X is a stochastic process taking values in ${{\mathbb R}^d}$ such that

• independent increments: ${X_t-X_s}$ is independent of ${\{X_u\colon u\le s\}}$ for any ${s.
• stationary increments: ${X_{s+t}-X_s}$ has the same distribution as ${X_t-X_0}$ for any ${s,t>0}$.
• continuity in probability: ${X_s\rightarrow X_t}$ in probability as s tends to t.

More generally, it is possible to define the notion of a Lévy process with respect to a given filtered probability space ${(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge0},{\mathbb P})}$. In that case, we also require that X is adapted to the filtration and that ${X_t-X_s}$ is independent of ${\mathcal{F}_s}$ for all ${s < t}$. In particular, if X is a Lévy process according to definition 1 then it is also a Lévy process with respect to its natural filtration ${\mathcal{F}_t=\sigma(X_s\colon s\le t)}$. Note that slightly different definitions are sometimes used by different authors. It is often required that ${X_0}$ is zero and that X has cadlag sample paths. These are minor points and, as will be shown, any process satisfying the definition above will admit a cadlag modification.

The most common example of a Lévy process is Brownian motion, where ${X_t-X_s}$ is normally distributed with zero mean and variance ${t-s}$ independently of ${\mathcal{F}_s}$. Other examples include Poisson processes, compound Poisson processes, the Cauchy process, gamma processes and the variance gamma process.

For example, the symmetric Cauchy distribution on the real numbers with scale parameter ${\gamma > 0}$ has probability density function p and characteristic function ${\phi}$ given by,

 $\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle p(x)=\frac{\gamma}{\pi(\gamma^2+x^2)},\smallskip\\ &\displaystyle\phi(a)\equiv{\mathbb E}\left[e^{iaX}\right]=e^{-\gamma\vert a\vert}. \end{array}$ (1)

From the characteristic function it can be seen that if X and Y are independent Cauchy random variables with scale parameters ${\gamma_1}$ and ${\gamma_2}$ respectively then ${X+Y}$ is Cauchy with parameter ${\gamma_1+\gamma_2}$. We can therefore consistently define a stochastic process ${X_t}$ such that ${X_t-X_s}$ has the symmetric Cauchy distribution with parameter ${t-s}$ independent of ${\{X_u\colon u\le t\}}$, for any ${s < t}$. This is called a Cauchy process, which is a purely discontinuous Lévy process. See Figure 1.

Lévy processes are determined by the triple ${(\Sigma,b,\nu)}$, where ${\Sigma}$ describes the covariance structure of the Brownian motion component, b is the drift component, and ${\nu}$ describes the rate at which jumps occur. The distribution of the process is given by the Lévy-Khintchine formula, equation (3) below.

Theorem 2 (Lévy-Khintchine) Let X be a d-dimensional Lévy process. Then, there is a unique function ${\psi\colon{\mathbb R}\rightarrow{\mathbb C}}$ such that

 $\displaystyle {\mathbb E}\left[e^{ia\cdot (X_t-X_0)}\right]=e^{t\psi(a)}$ (2)

for all ${a\in{\mathbb R}^d}$ and ${t\ge0}$. Also, ${\psi(a)}$ can be written as

 $\displaystyle \psi(a)=ia\cdot b-\frac{1}{2}a^{\rm T}\Sigma a+\int _{{\mathbb R}^d}\left(e^{ia\cdot x}-1-\frac{ia\cdot x}{1+\Vert x\Vert}\right)\,d\nu(x)$ (3)

where ${\Sigma}$, b and ${\nu}$ are uniquely determined and satisfy the following,

1. ${\Sigma\in{\mathbb R}^{d^2}}$ is a positive semidefinite matrix.
2. ${b\in{\mathbb R}^d}$.
3. ${\nu}$ is a Borel measure on ${{\mathbb R}^d}$ with ${\nu(\{0\})=0}$ and,
 $\displaystyle \int_{{\mathbb R}^d}\Vert x\Vert^2\wedge 1\,d\nu(x)<\infty.$ (4)

Furthermore, ${(\Sigma,b,\nu)}$ uniquely determine all finite distributions of the process ${X-X_0}$.

Conversely, if ${(\Sigma,b,\nu)}$ is any triple satisfying the three conditions above, then there exists a Lévy process satisfying (2,3).

# Processes with Independent Increments

In a previous post, it was seen that all continuous processes with independent increments are Gaussian. We move on now to look at a much more general class of independent increments processes which need not have continuous sample paths. Such processes can be completely described by their jump intensities, a Brownian term, and a deterministic drift component. However, this class of processes is large enough to capture the kinds of behaviour that occur for more general jump-diffusion processes. An important subclass is that of Lévy processes, which have independent and stationary increments. Lévy processes will be looked at in more detail in the following post, and includes as special cases, the Cauchy process, gamma processes, the variance gamma process, Poisson processes, compound Poisson processes and Brownian motion.

Recall that a process ${\{X_t\}_{t\ge0}}$ has the independent increments property if ${X_t-X_s}$ is independent of ${\{X_u\colon u\le s\}}$ for all times ${0\le s\le t}$. More generally, we say that X has the independent increments property with respect to an underlying filtered probability space ${(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge0},{\mathbb P})}$ if it is adapted and ${X_t-X_s}$ is independent of ${\mathcal{F}_s}$ for all ${s < t}$. In particular, every process with independent increments also satisfies the independent increments property with respect to its natural filtration. Throughout this post, I will assume the existence of such a filtered probability space, and the independent increments property will be understood to be with regard to this space.

The process X is said to be continuous in probability if ${X_s\rightarrow X_t}$ in probability as s tends to t. As we now state, a d-dimensional independent increments process X is uniquely specified by a triple ${(\Sigma,b,\mu)}$ where ${\mu}$ is a measure describing the jumps of X, ${\Sigma}$ determines the covariance structure of the Brownian motion component of X, and b is an additional deterministic drift term.

Theorem 1 Let X be an ${{\mathbb R}^d}$-valued process with independent increments and continuous in probability. Then, there is a unique continuous function ${{\mathbb R}^d\times{\mathbb R}_+\rightarrow{\mathbb C}}$, ${(a,t)\mapsto\psi_t(a)}$ such that ${\psi_0(a)=0}$ and

 $\displaystyle {\mathbb E}\left[e^{ia\cdot (X_t-X_0)}\right]=e^{i\psi_t(a)}$ (1)

for all ${a\in{\mathbb R}^d}$ and ${t\ge0}$. Also, ${\psi_t(a)}$ can be written as

 $\displaystyle \psi_t(a)=ia\cdot b_t-\frac{1}{2}a^{\rm T}\Sigma_t a+\int _{{\mathbb R}^d\times[0,t]}\left(e^{ia\cdot x}-1-\frac{ia\cdot x}{1+\Vert x\Vert}\right)\,d\mu(x,s)$ (2)

where ${\Sigma_t}$, ${b_t}$ and ${\mu}$ are uniquely determined and satisfy the following,

1. ${t\mapsto\Sigma_t}$ is a continuous function from ${{\mathbb R}_+}$ to ${{\mathbb R}^{d^2}}$ such that ${\Sigma_0=0}$ and ${\Sigma_t-\Sigma_s}$ is positive semidefinite for all ${t\ge s}$.
2. ${t\mapsto b_t}$ is a continuous function from ${{\mathbb R}_+}$ to ${{\mathbb R}^d}$, with ${b_0=0}$.
3. ${\mu}$ is a Borel measure on ${{\mathbb R}^d\times{\mathbb R}_+}$ with ${\mu(\{0\}\times{\mathbb R}_+)=0}$, ${\mu({\mathbb R}^d\times\{t\})=0}$ for all ${t\ge 0}$ and,
 $\displaystyle \int_{{\mathbb R}^d\times[0,t]}\Vert x\Vert^2\wedge 1\,d\mu(x,s)<\infty.$ (3)

Furthermore, ${(\Sigma,b,\mu)}$ uniquely determine all finite distributions of the process ${X-X_0}$.

Conversely, if ${(\Sigma,b,\mu)}$ is any triple satisfying the three conditions above, then there exists a process with independent increments satisfying (1,2).