Month: October 2020

Local Time Continuity

Local time surface
Figure 1: Brownian motion and its local time surface

The local time of a semimartingale at a level x is a continuous increasing process, giving a measure of the amount of time that the process spends at the given level. As the definition involves stochastic integrals, it was only defined up to probability one. This can cause issues if we want to simultaneously consider local times at all levels. As x can be any real number, it can take uncountably many values and, as a union of uncountably many zero probability sets can have positive measure or, even, be unmeasurable, this is not sufficient to determine the entire local time ‘surface’

\displaystyle (t,x)\mapsto L^x_t(\omega)

for almost all {\omega\in\Omega}. This is the common issue of choosing good versions of processes. In this case, we already have a continuous version in the time index but, as yet, have not constructed a good version jointly in the time and level. This issue arose in the post on the Ito–Tanaka–Meyer formula, for which we needed to choose a version which is jointly measurable. Although that was sufficient there, joint measurability is still not enough to uniquely determine the full set of local times, up to probability one. The ideal situation is when a version exists which is jointly continuous in both time and level, in which case we should work with this choice. This is always possible for continuous local martingales.

Theorem 1 Let X be a continuous local martingale. Then, the local times

\displaystyle (t,x)\mapsto L^x_t

have a modification which is jointly continuous in x and t. Furthermore, this is almost surely {\gamma}-Hölder continuous w.r.t. x, for all {\gamma < 1/2} and over all bounded regions for t.

Continue reading “Local Time Continuity” →

The Kolmogorov Continuity Theorem

Fractional BM
Figure 1: Fractional Brownian motion with H = 1/4, 1/2, 3/4

One of the common themes throughout the theory of continuous-time stochastic processes, is the importance of choosing good versions of processes. Specifying the finite distributions of a process is not sufficient to determine its sample paths so, if a continuous modification exists, then it makes sense to work with that. A relatively straightforward criterion ensuring the existence of a continuous version is provided by Kolmogorov’s continuity theorem.

For any positive real number {\gamma}, a map {f\colon E\rightarrow F} between metric spaces E and F is said to be {\gamma}-Hölder continuous if there exists a positive constant C satisfying

\displaystyle d(f(x),f(y))\le Cd(x,y)^\gamma

for all {x,y\in E}. The smallest value of C satisfying this inequality is known as the {\gamma}-Hölder coefficient of {f}. Hölder continuous functions are always continuous and, at least on bounded spaces, is a stronger property for larger values of the coefficient {\gamma}. So, if E is a bounded metric space and {\alpha\le\beta}, then every {\beta}-Hölder continuous map from E is also {\alpha}-Hölder continuous. In particular, 1-Hölder and Lipschitz continuity are equivalent.

Kolmogorov’s theorem gives simple conditions on the pairwise distributions of a process which guarantee the existence of a continuous modification but, also, states that the sample paths {t\mapsto X_t} are almost surely locally Hölder continuous. That is, they are almost surely Hölder continuous on every bounded interval. To start with, we look at real-valued processes. Throughout this post, we work with repect to a probability space {(\Omega,\mathcal F, {\mathbb P})}. There is no need to assume the existence of any filtration, since they play no part in the results here

Theorem 1 (Kolmogorov) Let {\{X_t\}_{t\ge0}} be a real-valued stochastic process such that there exists positive constants {\alpha,\beta,C} satisfying

\displaystyle {\mathbb E}\left[\lvert X_t-X_s\rvert^\alpha\right]\le C\lvert t-s\vert^{1+\beta},

for all {s,t\ge0}. Then, X has a continuous modification which, with probability one, is locally {\gamma}-Hölder continuous for all {0 < \gamma < \beta/\alpha}.

Continue reading “The Kolmogorov Continuity Theorem” →

Quantum Entanglement States

In an earlier post, I described four simple thought experiments, involving some black boxes and two or more participants. As described there, the results of these experiments were inconsistent with any classical description, assuming that the boxes cannot communicate. However, I also stated that all of these experiments are consistent with quantum probability, and that I would give the mathematical details in a further post. I will do this now. Continue reading “Quantum Entanglement States” →

The Ito-Tanaka-Meyer Formula

Ito’s lemma is one of the most important and useful results in the theory of stochastic calculus. This is a stochastic generalization of the chain rule, or change of variables formula, and differs from the classical deterministic formulas by the presence of a quadratic variation term. One drawback which can limit the applicability of Ito’s lemma in some situations, is that it only applies for twice continuously differentiable functions. However, the quadratic variation term can alternatively be expressed using local times, which relaxes the differentiability requirement. This generalization of Ito’s lemma was derived by Tanaka and Meyer, and applies to one dimensional semimartingales.

The local time of a stochastic process X at a fixed level x can be written, very informally, as an integral of a Dirac delta function with respect to the continuous part of the quadratic variation {[X]^{c}},

\displaystyle L^x_t=\int_0^t\delta(X-x)d[X]^c. (1)

This was explained in an earlier post. As the Dirac delta is only a distribution, and not a true function, equation (1) is not really a well-defined mathematical expression. However, as we saw, with some manipulation a valid expression can be obtained which defines the local time whenever X is a semimartingale.

Going in a slightly different direction, we can try multiplying (1) by a bounded measurable function {f(x)} and integrating over x. Commuting the order of integration on the right hand side, and applying the defining property of the delta function, that {\int f(X-x)\delta(x)dx} is equal to {f(X)}, gives

\displaystyle \int_{-\infty}^{\infty} L^x_t f(x)dx=\int_0^tf(X)d[X]^c. (2)

By eliminating the delta function, the right hand side has been transformed into a well-defined expression. In fact, it is now the left side of the identity that is a problem, since the local time was only defined up to probability one at each level x. Ignoring this issue for the moment, recall the version of Ito’s lemma for general non-continuous semimartingales,

\displaystyle \begin{aligned} f(X_t)=& f(X_0)+\int_0^t f^{\prime}(X_-)dX+\frac12A_t\\ &\quad+\sum_{s\le t}\left(\Delta f(X_s)-f^\prime(X_{s-})\Delta X_s\right). \end{aligned} (3)

where {A_t=\int_0^t f^{\prime\prime}(X)d[X]^c}. Equation (2) allows us to express this quadratic variation term using local times,

\displaystyle A_t=\int_{-\infty}^{\infty} L^x_t f^{\prime\prime}(x)dx.

The benefit of this form is that, even though it still uses the second derivative of {f}, it is only really necessary for this to exist in a weaker, measure theoretic, sense. Suppose that {f} is convex, or a linear combination of convex functions. Then, its right-hand derivative {f^\prime(x+)} exists, and is itself of locally finite variation. Hence, the Stieltjes integral {\int L^xdf^\prime(x+)} exists. The infinitesimal {df^\prime(x+)} is alternatively written {f^{\prime\prime}(dx)} and, in the twice continuously differentiable case, equals {f^{\prime\prime}(x)dx}. Then,

\displaystyle A_t=\int _{-\infty}^{\infty} L^x_t f^{\prime\prime}(dx). (4)

Using this expression in (3) gives the Ito-Tanaka-Meyer formula. Continue reading “The Ito-Tanaka-Meyer Formula” →

The Stochastic Fubini Theorem

Fubini’s theorem states that, subject to precise conditions, it is possible to switch the order of integration when computing double integrals. In the theory of stochastic calculus, we also encounter double integrals and would like to be able to commute their order. However, since these can involve stochastic integration rather than the usual deterministic case, the classical results are not always applicable. To help with such cases, we could do with a new stochastic version of Fubini’s theorem. Here, I will consider the situation where one integral is of the standard kind with respect to a finite measure, and the other is stochastic. To start, recall the classical Fubini theorem.

Theorem 1 (Fubini) Let {(E,\mathcal E,\mu)} and {(F,\mathcal F,\nu)} be finite measure spaces, and {f\colon E\times F\rightarrow{\mathbb R}} be a bounded {\mathcal E\otimes\mathcal F}-measurable function. Then,

\displaystyle y\mapsto\int f(x,y)d\mu(x)

is {\mathcal F}-measurable,

\displaystyle x\mapsto\int f(x,y)d\nu(y)

is {\mathcal E}-measurable, and,

\displaystyle \int\int f(x,y)d\mu(x)d\nu(y)=\int\int f(x,y)d\nu(x)d\mu(y). (1)

Continue reading “The Stochastic Fubini Theorem” →

WordPress LaTeX Issues

This is just a brief notice. There appears to be a new problem with WordPress.com displaying LaTeX maths equations. The issue appears to be sporadic, with some of the formulas on both old and new posts now showing “Formula does not parse”. It is affecting other blogs hosted on WordPress too. Thanks to Timothy Gowers on twitter for pointing this out.
The WordPress team are looking into it. Hopefully this will be fixed soon. Update: This issue has now been fixed.

For now, the following method can be used to display the equations using MathJax instead. Click here, then drag the displayed link to the address bar of your browser. Then click on it while viewing any page on this blog, and all formulas will be displayed with MathJax. Magic!

Some tests follow. At the time of writing, some of the following formulas show an error:
Test: {X^n\xrightarrow{\rm ucp}X}.

Another test: {X^n\xrightarrow{\rm ucp}X}

Yet another test: {X^n\rightarrow X}

And another: {X^n X}

And another: X^n X

And another: {[S]_\infty}

yet more: {[S]_\infty}

and more: [S]_\infty

and more: [S]

more: S_\infty