I am trying to apply a simple example of Meyer-Itô's formula, which generalizes the classical Itô's formula over convex functions (instead of twice continuously differentiable functions). Applied to continuous semimartingales, Meyer-Itô's formula can be summarized as (see Theorem 70 in Protter's book):
Let $X$ be a continuous semimartingale and let $f$ be a convex function. The following equality holds: \begin{equation} f(X_t) = f(X_0) + \int_{0^+}^t f_-^{\prime}(X_s) \mathrm{d}X_s + \frac12 \int_{\mathbb{R}} L_t^x f''(x) \mathrm{d}x, \label{meyer-ito} \end{equation}
where the signed measure $f''$ is the second derivative of $f$ in the sense of distributions, and where $L_t^x$ is the family of local times of $X$.
Suppose for simplicity I want to apply this equation with a standard Brownian motion $W = \{W_t : t \geq 0\}$ and with the convex function \begin{equation} f(x) = { \left\{ \begin{array}{ll} -x & \mbox{if } x < 0, \\ 0 & \mbox{if } x \geq 0. \end{array} \right. } \end{equation}
How to deal with the last term of Meyer-Itô's formula? How do I proceed once I have computed the second derivative of $f$ in the sense of distributions?
I don't really know where to begin to get an explicit integral for the last term. Any help would be appreciated. Thank you!
First to be clear, there is even a generalization for discontinuous martingales see The Ito-Tanaka-Meyer Formula involving jumps
Theorem 2 (Ito-Tanaka-Meyer) Let $X$ be a semimartingale, $f\colon{\mathbb R}\rightarrow{\mathbb R}$ be convex, and $L^x_t$ be a jointly measurable version of the local times. Then,
\begin{aligned} f(X_t)=& f(X_0)+\int_0^t f^\prime(X_-)dX+\frac12\int_{-\infty}^\infty L^x_t\,f^{\prime\prime}(dx)\\ &\quad+\sum_{s\le t}\left(\Delta f(X_s)-f^\prime(X_{s-})\Delta X_s\right), \end{aligned} almost surely, for each $t\ge0$.
We first compute the distribution second derivative for $\phi=-x1_{x<0}$ similar to Delta Function and Heaviside Function. Note that we have the formulas
$$-x1_{x<0}=\frac{|x|-x}{2}=\int_{-\infty}^{-x}y\delta_{y}dy.$$
Therefore, we have the distributional derivative in terms of the Heaviside $\phi'(x)=sgn(x)1_{x<0}=-(1-H(x))$. So the second distributional derivative will be $\phi''(x)=\delta_{x}$.
For the case of continuous local martingale (eg.1d-Brownian motion) we have that $L^x_t$ is bi-continuous (eg. (1.7) Theorem in Revuz-Yor) and so we can directly evaluate the last term.