Almost Sure

27 December 11

Compensators of Counting Processes

A counting process, X, is defined to be an adapted stochastic process starting from zero which is piecewise constant and right-continuous with jumps of size 1. That is, letting {\tau_n} be the first time at which {X_t=n}, then

\displaystyle  X_t=\sum_{n=1}^\infty 1_{\{\tau_n\le t\}}.

By the debut theorem, {\tau_n} are stopping times. So, X is an increasing integer valued process counting the arrivals of the stopping times {\tau_n}. A basic example of a counting process is the Poisson process, for which {X_t-X_s} has a Poisson distribution independently of {\mathcal{F}_s}, for all times {t > s}, and for which the gaps {\tau_n-\tau_{n-1}} between the stopping times are independent exponentially distributed random variables. As we will see, although Poisson processes are just one specific example, every quasi-left-continuous counting process can actually be reduced to the case of a Poisson process by a time change. As always, we work with respect to a complete filtered probability space {(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge0},{\mathbb P})}.

Note that, as a counting process X has jumps bounded by 1, it is locally integrable and, hence, the compensator A of X exists. This is the unique right-continuous predictable and increasing process with {A_0=0} such that {X-A} is a local martingale. For example, if X is a Poisson process of rate {\lambda}, then the compensated Poisson process {X_t-\lambda t} is a martingale. So, the compensator of X is the continuous process {A_t=\lambda t}. More generally, X is said to be quasi-left-continuous if {{\mathbb P}(\Delta X_\tau=0)=1} for all predictable stopping times {\tau}, which is equivalent to the compensator of X being almost surely continuous. Another simple example of a counting process is {X=1_{[\tau,\infty)}} for a stopping time {\tau > 0}, in which case the compensator of X is just the same thing as the compensator of {\tau}.

As I will show in this post, compensators of quasi-left-continuous counting processes have many parallels with the quadratic variation of continuous local martingales. For example, Lévy’s characterization states that a local martingale X starting from zero is standard Brownian motion if and only if its quadratic variation is {[X]_t=t}. Similarly, as we show below, a counting process is a homogeneous Poisson process of rate {\lambda} if and only if its compensator is {A_t=\lambda t}. It was also shown previously in these notes that a continuous local martingale X has a finite limit {X_\infty=\lim_{t\rightarrow\infty}X_t} if and only if {[X]_\infty} is finite. Similarly, a counting process X has finite value {X_\infty} at infinity if and only if the same is true of its compensator. Another property of a continuous local martingale X is that it is constant over all intervals on which its quadratic variation is constant. Similarly, a counting process X is constant over any interval on which its compensator is constant. Finally, it is known that every continuous local martingale is simply a continuous time change of standard Brownian motion. In the main result of this post (Theorem 5), we show that a similar statement holds for counting processes. That is, every quasi-left-continuous counting process is a continuous time change of a Poisson process of rate 1. (more…)

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25 February 11

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). (more…)

23 November 10

Lévy Processes

A Poisson process sample path

Figure 1: A Cauchy process sample path

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<t}.
  • 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).

(more…)

15 September 10

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).

(more…)

24 June 10

Poisson Processes

A Poisson process sample path

Figure 1: A Poisson process sample path

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 {X\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}. (more…)

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