Some Inequalities for the Density of the Area Integral

Abstract

Let X _t be a continuous martingale starting at 0 and set $X^\ast = \text{sup}\{\left\vert X_t \right\vert : t > 0 \}$ and $S(X) = \sqrt{\left\langle X \right\rangle_\infty}$, where 〈X〉_tis the quadratic variation process at time t. We then define a measure μ on ℝ by $\mu(E) = d\left\langle X \right\rangle_t (\{t : X_t \in E\})$, where $d\left\langle X \right\rangle_t$ is the Riemann-Stieltjes measure on 0,∞) associated to the nondecreasing function 〈X〉_t. It is known that μ is absolutely continuous with respect to Lebesgue measure so that there exists a function L(a), (called the local time), so that $\mu(E) = \int_E L(a)da$ for every Borel set E. More generally, for any Borel function f on ℝ:

$$ \int_0^\infty f(X_t)d\left\langle X \right\rangle_t = \int_\mathbb{R} f(a)L(a)da $$

((1.1))

Take f = 1 in (1.1) and we obtain:

$$ S(X)^2 = \int_\mathbb{R}L(a)da = \int_{-X^\ast}^{X^\ast}L(a)da $$

((1.2))

where the last equality follows by noting that $L(a) = 0 \text{ if} a \notin [-X^\ast, X^\ast]$. We now set $L^\ast = \text{sup}\{L(a) : a \in \mathbb{R}\}$; L* is called the maximal local time. Then (1.2) trivially yields $S(X)^2\leq 2L^\ast X^\ast$, and this, the Cauchy-Schwarz inequality, and the Burkholder-Gundy inequality: $\left \Vert S(X) \right \Vert_p\approx \left \Vert X^\ast \right \Vert_p for\ 0< p< \infty$, then gives $\left \Vert S(X) \right \Vert_p\leq C_p\left \Vert L\ast \right \Vert_p \text{ for} 0 < p < \infty$. The reverse inequality is more difficult and was shown by Barlow and Yor [4], [5]. In fact, even more is true. The following theorem is essentially to Bass [6] and independently, Davis [12], however, their statements of these results are different than appears here, but a careful analysis of their methods yields these results.

Partially supported by a National Science Foundation Postdoctoral Fellowship.