Local Time-Space Calculus and Extensions of Itô’s Formula

Abstract

Let X = (X _t ) _t≥0 be a continuous semimartingale and let F : ℝ₊ × ℝ → × be a C ¹ function. Then the change-of-variable formula is valid:

$$\begin{array}{*{20}{c}} {F(t,{{X}_{t}}) = F(0,{{X}_{0}}) + \int_{0}^{t} {{{F}_{t}}(s,{{X}_{s}})ds + \int_{0}^{t} {{{F}_{x}}(s,{{X}_{s}})d{{X}_{s}}} } } \\ { - \frac{1}{2}\int_{0}^{t} {\int_{\mathbb{R}} {{{F}_{x}}(s,x)d\ell _{s}^{x}} } } \\ \end{array}$$

where ℓ ^x _s is the local time of X defined by:

$$\ell _{s}^{x} = \mathbb{P} - \mathop{{\lim }}\limits_{{\varepsilon \downarrow 0}} \frac{1}{\varepsilon }\int_{0}^{s} {I(x \leqslant {{X}_{r}} < x + \varepsilon )d{{{\langle X,X\rangle }}_{r}}}$$

and dℓ ^x _s to an area integration with respect to (s,x) ↦ ℓ ^x _s Further extensions of this formula for non-smooth functions F are also briefly examined. The approach leads to a formal dℓ ^x _s calculus which appears useful in guessing a candidate formula for before a rigorous proof is known or given.