Semimartingale Local Times

Figure 1: Brownian motion B with local time L and auxilliary 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”