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 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”

Semimartingale Local Times

Figure 1: Brownian motion B with local time L and auxiliary Brownian motion W

For a stochastic process X taking values in a state space E, its local time at a point {x\in E} is a measure of the time spent at x. For a continuous time stochastic process, we could try and simply compute the Lebesgue measure of the time at the level,

\displaystyle  L^x_t=\int_0^t1_{\{X_s=x\}}ds. (1)

For processes which hit the level {x} and stick there for some time, this makes some sense. However, if X is a standard Brownian motion, it will always give zero, so is not helpful. Even though X will hit every real value infinitely often, continuity of the normal distribution gives {{\mathbb P}(X_s=x)=0} at each positive time, so that that {L^x_t} defined by (1) will have zero expectation.

Rather than the indicator function of {\{X=x\}} as in (1), an alternative is to use the Dirac delta function,

\displaystyle  L^x_t=\int_0^t\delta(X_s-x)\,ds. (2)

Unfortunately, the Dirac delta is not a true function, it is a distribution, so (2) is not a well-defined expression. However, if it can be made rigorous, then it does seem to have some of the properties we would want. For example, the expectation {{\mathbb E}[\delta(X_s-x)]} can be interpreted as the probability density of {X_s} evaluated at {x}, which has a positive and finite value, so it should lead to positive and finite local times. Equation (2) still relies on the Lebesgue measure over the time index, so will not behave as we may expect under time changes, and will not make sense for processes without a continuous probability density. A better approach is to integrate with respect to the quadratic variation,

\displaystyle  L^x_t=\int_0^t\delta(X_s-x)d[X]_s (3)

which, for Brownian motion, amounts to the same thing. Although (3) is still not a well-defined expression, since it still involves the Dirac delta, the idea is to come up with a definition which amounts to the same thing in spirit. Important properties that it should satisfy are that it is an adapted, continuous and increasing process with increments supported on the set {\{X=x\}},

\displaystyle  L^x_t=\int_0^t1_{\{X_s=x\}}dL^x_s.

Local times are a very useful and interesting part of stochastic calculus, and finds important applications to excursion theory, stochastic integration and stochastic differential equations. However, I have not covered this subject in my notes, so do this now. Recalling Ito’s lemma for a function {f(X)} of a semimartingale X, this involves a term of the form {\int f^{\prime\prime}(X)d[X]} and, hence, requires {f} to be twice differentiable. If we were to try to apply the Ito formula for functions which are not twice differentiable, then {f^{\prime\prime}} can be understood in terms of distributions, and delta functions can appear, which brings local times into the picture. In the opposite direction, which I take in this post, we can try to generalise Ito’s formula and invert this to give a meaning to (3). Continue reading “Semimartingale Local Times”