On the tail distributions of the supremum and the quadratic variation of a càdlàg local martingale

Abstract

We study a tail property of the distribution of the supremum and the quadratic variation of a local martingale. In the case when the local martingale is continuous, there are works by Azéma, Gundy, and Yor [1], Novikov [9], Elworthy, Li, and Yor [2], Madan and Yor [8], Takaoka [10] etc. Recently, Liptser and Novikov [7] extended these studies to the case of a local martingale with uniformly bounded jumps; here is their main result:Theorem 1.1 Let M = $M = \{ M_t \} _{t \in R_ + }$ be a locally square integrable càdlàg martingale defined on a filtered probability space $(\Omega ,\mathcal{F},\{ \mathcal{F}_t \} _{t \in R_ + } ,P)$ with standard general conditions. Assume that 〈M〉_∞ = lim _t→∞ 〈M〉 _t < ∞ a.s and $\{ M_\tau ^ + \} _{\tau \in \tau }$ is uniformly integrable, where τ is the set of stopping times τ. Then

• (i) 0 ≤ E[M _∞] ≤ E[M ⁺_∞ ] < ∞

  Furthermore,

• (ii) if $\{ \Delta M_\tau \} _{\tau \in \tau }$ is uniformly integrable, then

  $$\mathop {\lim }\limits_{\lambda \to \infty } \lambda P(\mathop {\sup }\limits_{t \in R_ + } (M_t^ - ) > \lambda ) = E[M_\infty ];$$

• (iii) if |ΔM| ≤ K and $E[e^{ \in M_\infty } ] < \infty$ and for K > 0 and ∈ > 0, then

  $$\mathop {\lim }\limits_{\lambda \to \infty } \lambda P\left( {\sqrt {\left\langle M \right\rangle _\infty } > \lambda } \right) = \mathop {\lim }\limits_{\lambda \to \infty } \lambda P\left( {\sqrt {\left[ M \right]_\infty } > \lambda } \right) = \sqrt {\frac{2}{\pi }} E[M_\infty ].$$