Time-Changed Brownian Motion

From the definition of standard Brownian motion B, given any positive constant c, ${B_{ct}-B_{cs}}$ will be normal with mean zero and variance c(t–s) for times ${t>s\ge 0}$ . So, scaling the time axis of Brownian motion B to get the new process ${B_{ct}}$ just results in another Brownian motion scaled by the factor ${\sqrt{c}}$ .

This idea is easily generalized. Consider a measurable function ${\xi\colon{\mathbb R}_+\rightarrow{\mathbb R}_+}$ and Brownian motion B on the filtered probability space ${(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},{\mathbb P})}$ . So, ${\xi}$ is a deterministic process, not depending on the underlying probability space ${\Omega}$ . If ${\theta(t)\equiv\int_0^t\xi^2_s\,ds}$ is finite for each ${t>0}$ then the stochastic integral ${X=\int\xi\,dB}$ exists. Furthermore, X will be a Gaussian process with independent increments. For piecewise constant integrands, this results from the fact that linear combinations of joint normal variables are themselves normal. The case for arbitrary deterministic integrands follows by taking limits. Also, the Ito isometry says that ${X_t-X_s}$ has variance

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle{\mathbb E}\left[\left(\int_s^t\xi\,dB\right)^2\right]&\displaystyle={\mathbb E}\left[\int_s^t\xi^2_u\,du\right]\smallskip\\ &\displaystyle=\theta(t)-\theta(s)\smallskip\\ &\displaystyle={\mathbb E}\left[(B_{\theta(t)}-B_{\theta(s)})^2\right]. \end{array}$

So, ${\int\xi\,dB=\int\sqrt{\theta^\prime(t)}\,dB_t}$ has the same distribution as the time-changed Brownian motion ${B_{\theta(t)}}$ .

With the help of Lévy’s characterization, these ideas can be extended to more general, non-deterministic, integrands and to stochastic time-changes. In fact, doing this leads to the startling result that all continuous local martingales are just time-changed Brownian motion. Continue reading “Time-Changed Brownian Motion” →

Lévy’s Characterization of Brownian Motion

Standard Brownian motion, ${\{B_t\}_{t\ge 0}}$ , is defined to be a real-valued process satisfying the following properties.

${B_0=0}$ .
${B_t-B_s}$ is normally distributed with mean 0 and variance t–s independently of ${\{B_u\colon u\le s\}}$ , for any ${t>s\ge 0}$ .
B has continuous sample paths.

As always, it only really matters is that these properties hold almost surely. Now, to apply the techniques of stochastic calculus, it is assumed that there is an underlying filtered probability space ${(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},{\mathbb P})}$ , which necessitates a further definition; a process B is a Brownian motion on a filtered probability space ${(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},{\mathbb P})}$ if in addition to the above properties it is also adapted, so that ${B_t}$ is ${\mathcal{F}_t}$ -measurable, and ${B_t-B_s}$ is independent of ${\mathcal{F}_s}$ for each ${t>s\ge 0}$ . Note that the above condition that ${B_t-B_s}$ is independent of ${\{B_u\colon u\le s\}}$ is not explicitly required, as it also follows from the independence from ${\mathcal{F}_s}$ . According to these definitions, a process is a Brownian motion if and only if it is a Brownian motion with respect to its natural filtration.

The property that ${B_t-B_s}$ has zero mean independently of ${\mathcal{F}_s}$ means that Brownian motion is a martingale. Furthermore, we previously calculated its quadratic variation as ${[B]_t=t}$ . An incredibly useful result is that the converse statement holds. That is, Brownian motion is the only local martingale with this quadratic variation. This is known as Lévy’s characterization, and shows that Brownian motion is a particularly general stochastic process, justifying its ubiquitous influence on the study of continuous-time stochastic processes.

Theorem 1 (Lévy’s Characterization of Brownian Motion) Let X be a local martingale with ${X_0=0}$ . Then, the following are equivalent.

X is standard Brownian motion on the underlying filtered probability space.

X is continuous and ${X^2_t-t}$ is a local martingale.

X has quadratic variation ${[X]_t=t}$ .

Continue reading “Lévy’s Characterization of Brownian Motion” →

Special Processes

The point in these stochastic calculus notes has been reached where the theory of stochastic integration is sufficiently well developed to apply in a wide range of situations.

Results such as Ito’s lemma, properties of quadratic variations and existence and uniqueness of solutions to stochastic differential equations followed quite directly from the definition of stochastic integration. Then, once it was shown that integration with respect to martingales is well-defined, results such as preservation of the local martingale property and Ito’s isometry also followed without too much effort.

Over the next few posts, I will take a break from further development of the general theory. Instead, I look at certain special processes, applying the calculus developed so far and gaining a few examples to motivate further development of the theory.

This will include properties of Brownian motion, such as Lévy’s characterization, Girsanov transforms, stochastic time changes and martingale representation. Other important processes which I take a brief look at include Bessel processes, the Poisson process and the Cauchy process. We will also derive a general description of processes with independent increments, including the Lévy-Khintchine formula characterizing Lévy processes.

The Burkholder-Davis-Gundy Inequality

The Burkholder-Davis-Gundy inequality is a remarkable result relating the maximum of a local martingale with its quadratic variation. Recall that [X] denotes the quadratic variation of a process X, and ${X^*_t\equiv\sup_{s\le t}\vert X_s\vert}$ is its maximum process.

Theorem 1 (Burkholder-Davis-Gundy) For any ${1\le p<\infty}$ there exist positive constants ${c_p,C_p}$ such that, for all local martingales X with ${X_0=0}$ and stopping times ${\tau}$ , the following inequality holds.

$\displaystyle c_p{\mathbb E}\left[ [X]^{p/2}_\tau\right]\le{\mathbb E}\left[(X^*_\tau)^p\right]\le C_p{\mathbb E}\left[ [X]^{p/2}_\tau\right].$ (1)

Furthermore, for continuous local martingales, this statement holds for all ${0<p<\infty}$ .

A proof of this result is given below. For ${p\ge 1}$ , the theorem can also be stated as follows. The set of all cadlag martingales X starting from zero for which ${{\mathbb E}[(X^*_\infty)^p]}$ is finite is a vector space, and the BDG inequality states that the norms ${X\mapsto\Vert X^*_\infty\Vert_p={\mathbb E}[(X^*_\infty)^p]^{1/p}}$ and ${X\mapsto\Vert[X]^{1/2}_\infty\Vert_p}$ are equivalent.

The special case p=2 is the easiest to handle, and we have previously seen that the BDG inequality does indeed hold in this case with constants ${c_2=1}$ , ${C_2=4}$ . The significance of Theorem 1, then, is that this extends to all ${p\ge1}$ .

One reason why the BDG inequality is useful in the theory of stochastic integration is as follows. Whereas the behaviour of the maximum of a stochastic integral is difficult to describe, the quadratic variation satisfies the simple identity ${\left[\int\xi\,dX\right]=\int\xi^2\,d[X]}$ . Recall, also, that stochastic integration preserves the local martingale property. Stochastic integration does not preserve the martingale property. In general, integration with respect to a martingale only results in a local martingale, even for bounded integrands. In many cases, however, stochastic integrals are indeed proper martingales. The Ito isometry shows that this is true for square integrable martingales, and the BDG inequality allows us to extend the result to all ${L^p}$ -integrable martingales, for ${p> 1}$ .

Theorem 2 Let X be a cadlag ${L^p}$ -integrable martingale for some ${1<p<\infty}$ , so that ${{\mathbb E}[\vert X_t\vert^p]<\infty}$ for each t. Then, for any bounded predictable process ${\xi}$ , ${Y\equiv\int\xi\,dX}$ is also an ${L^p}$ -integrable martingale.

Continue reading “The Burkholder-Davis-Gundy Inequality” →

Continuous Local Martingales

Continuous local martingales are a particularly well behaved subset of the class of all local martingales, and the results of the previous two posts become much simpler in this case. First, the continuous local martingale property is always preserved by stochastic integration.

Theorem 1 If X is a continuous local martingale and ${\xi}$ is X-integrable, then ${\int\xi\,dX}$ is a continuous local martingale.

Proof: As X is continuous, ${Y\equiv\int\xi\,dX}$ will also be continuous and, therefore, locally bounded. Then, by preservation of the local martingale property, Y is a local martingale. ⬜

Next, the quadratic variation of a continuous local martingale X provides us with a necessary and sufficient condition for X-integrability.

Theorem 2 Let X be a continuous local martingale. Then, a predictable process ${\xi}$ is X-integrable if and only if

$\displaystyle \int_0^t\xi^2\,d[X]<\infty$

for all ${t>0}$ .

Continue reading “Continuous Local Martingales” →

Quadratic Variations and the Ito Isometry

As local martingales are semimartingales, they have a well-defined quadratic variation. These satisfy several useful and well known properties, such as the Ito isometry, which are the subject of this post. First, the covariation [X,Y] allows the product XY of local martingales to be decomposed into local martingale and FV terms. Consider, for example, a standard Brownian motion B. This has quadratic variation ${[B]_t=t}$ and it is easily checked that ${B^2_t-t}$ is a martingale.

Lemma 1 If X and Y are local martingales then XY-[X,Y] is a local martingale.

In particular, ${X^2-[X]}$ is a local martingale for all local martingales X.

Proof: Integration by parts gives

$\displaystyle XY-[X,Y] = X_0Y_0+\int X_-\,dY+\int Y_-\,dX$

which, by preservation of the local martingale property, is a local martingale. ⬜

Continue reading “Quadratic Variations and the Ito Isometry” →

Preservation of the Local Martingale Property

Now that it has been shown that stochastic integration can be performed with respect to any local martingale, we can move on to the following important result. Stochastic integration preserves the local martingale property. At least, this is true under very mild hypotheses. That the martingale property is preserved under integration of bounded elementary processes is straightforward. The generalization to predictable integrands can be achieved using a limiting argument. It is necessary, however, to restrict to locally bounded integrands and, for the sake of generality, I start with local sub and supermartingales.

Theorem 1 Let X be a local submartingale (resp., local supermartingale) and ${\xi}$ be a nonnegative and locally bounded predictable process. Then, ${\int\xi\,dX}$ is a local submartingale (resp., local supermartingale).

Proof: We only need to consider the case where X is a local submartingale, as the result will also follow for supermartingales by applying to -X. By localization, we may suppose that ${\xi}$ is uniformly bounded and that X is a proper submartingale. So, ${\vert\xi\vert\le K}$ for some constant K. Then, as previously shown there exists a sequence of elementary predictable processes ${\vert\xi^n\vert\le K}$ such that ${Y^n\equiv\int\xi^n\,dX}$ converges to ${Y\equiv\int\xi\,dX}$ in the semimartingale topology and, hence, converges ucp. We may replace ${\xi_n}$ by ${\xi_n\vee0}$ if necessary so that, being nonnegative elementary integrals of a submartingale, ${Y^n}$ will be submartingales. Also, ${\vert\Delta Y^n\vert=\vert\xi^n\Delta X\vert\le K\vert\Delta X\vert}$ . Recall that a cadlag adapted process X is locally integrable if and only its jump process ${\Delta X}$ is locally integrable, and all local submartingales are locally integrable. So,

$\displaystyle \sup_n\vert\Delta Y^n_t\vert\le K\vert\Delta X_t\vert$

is locally integrable. Then, by ucp convergence for local submartingales, Y will satisfy the local submartingale property. ⬜

For local martingales, applying this result to ${\pm X}$ gives,

Theorem 2 Let X be a local martingale and ${\xi}$ be a locally bounded predictable process. Then, ${\int\xi\,dX}$ is a local martingale.

This result can immediately be extended to the class of local ${L^p}$ -integrable martingales, denoted by ${\mathcal{M}^p_{\rm loc}}$ .

Corollary 3 Let ${X\in\mathcal{M}^p_{\rm loc}}$ for some ${0< p\le\infty}$ and ${\xi}$ be a locally bounded predictable process. Then, ${\int\xi\,dX\in\mathcal{M}^p_{\rm loc}}$ .

Continue reading “Preservation of the Local Martingale Property” →

Martingales are Integrators

A major foundational result in stochastic calculus is that integration can be performed with respect to any local martingale. In these notes, a semimartingale was defined to be a cadlag adapted process with respect to which a stochastic integral exists satisfying some simple desired properties. Namely, the integral must agree with the explicit formula for elementary integrands and satisfy bounded convergence in probability. Then, the existence of integrals with respect to local martingales can be stated as follows.

Theorem 1 Every local martingale is a semimartingale.

This result can be combined directly with the fact that FV processes are semimartingales.

Corollary 2 Every process of the form X=M+V for a local martingale M and FV process V is a semimartingale.

Working from the classical definition of semimartingales as sums of local martingales and FV processes, the statements of Theorem 1 and Corollary 2 would be tautologies. Then, the aim of this post is to show that stochastic integration is well defined for all classical semimartingales. Put in another way, Corollary 2 is equivalent to the statement that classical semimartingales satisfy the semimartingale definition used in these notes. The converse statement will be proven in a later post on the Bichteler-Dellacherie theorem, so the two semimartingale definitions do indeed agree.

Continue reading “Martingales are Integrators” →

Semimartingale Completeness

A sequence of stochastic processes, ${X^n}$ , is said to converge to a process X under the semimartingale topology, as n goes to infinity, if the following conditions are met. First, ${X^n_0}$ should tend to ${X_0}$ in probability. Also, for every sequence ${\xi^n}$ of elementary predictable processes with ${\vert\xi^n\vert\le 1}$ ,

$\displaystyle \int_0^t\xi^n\,dX^n-\int_0^t\xi^n\,dX\rightarrow 0$

in probability for all times t. For short, this will be denoted by ${X^n\xrightarrow{\rm sm}X}$ .

The semimartingale topology is particularly well suited to the class of semimartingales, and to stochastic integration. Previously, it was shown that the cadlag and adapted processes are complete under semimartingale convergence. In this post, it will be shown that the set of semimartingales is also complete. That is, if a sequence ${X^n}$ of semimartingales converge to a limit X under the semimartingale topology, then X is also a semimartingale.

Theorem 1 The space of semimartingales is complete under the semimartingale topology.

The same is true of the space of stochastic integrals defined with respect to any given semimartingale. In fact, for a semimartingale X, the set of all processes which can be expressed as a stochastic integral ${\int\xi\,dX}$ can be characterized as follows; it is precisely the closure, under the semimartingale topology, of the set of elementary integrals of X. This result was originally due to Memin, using a rather different proof to the one given here. The method used in this post only relies on the elementary properties of stochastic integrals, such as the dominated convergence theorem.

Theorem 2 Let X be a semimartingale. Then, a process Y is of the form ${Y=\int\xi\,dX}$ for some ${\xi\in L^1(X)}$ if and only if there is a sequence ${\xi^n}$ of bounded elementary processes with ${\int\xi^n\,dX\xrightarrow{\rm sm}Y}$ .

Writing S for the set of processes of the form ${\int\xi\,dX}$ for bounded elementary ${\xi}$ , and ${\bar S}$ for its closure under the semimartingale topology, the statement of the theorem is equivalent to

$\displaystyle \bar S=\left\{\int\xi\,dX\colon \xi\in L^1(X)\right\}.$

(1)

Continue reading “Semimartingale Completeness” →

Further Properties of the Stochastic Integral

We move on to properties of stochastic integration which, while being fairly elementary, are rather difficult to prove directly from the definitions.

First, recall that for a semimartingale X, the X-integrable processes ${L^1(X)}$ were defined to be predictable processes ${\xi}$ which are ‘good dominators’. That is, if ${\xi^n}$ are bounded predictable processes with ${\vert\xi^n\vert\le\vert\xi\vert}$ and ${\xi^n\rightarrow 0}$ pointwise, then ${\int_0^t\xi^n\,dX}$ tends to zero in probability. This definition is a bit messy. Fortunately, the following result gives a much cleaner characterization of X-integrability.

Theorem 1 Let X be a semimartingale. Then, a predictable process ${\xi}$ is X-integrable if and only if the set

$\displaystyle \left\{\int_0^t\zeta\,dX\colon\zeta\in{\rm b}\mathcal{P},\vert\zeta\vert\le\vert\xi\vert\right\}$ (1)

is bounded in probability for each ${t\ge 0}$ .

Continue reading “Further Properties of the Stochastic Integral” →

	Anonymous on Spitzer’s Formula
	Anonymous on Spitzer’s Formula
	Anonymous on Brownian Bridges
	Anonymous on Spitzer’s Formula
	Anonymous on Spitzer’s Formula
	Anonymous on Spitzer’s Formula
	Anonymous on Spitzer’s Formula
	Anonymous on Spitzer’s Formula
	Anonymous on Spitzer’s Formula
	Anonymous on Spitzer’s Formula