Generalized It$\hat {o}$ Formulae and Space-Time Lebesgue-Stieltjes Integrals of Local Times
K.D. Elworthy\inst{1}
A. Truman\inst{2} and H.Z. Zhao\inst{3}
Generalised It${\hat{o}}$ formulae are proved for time dependent functions of continuous real valued semi-martingales.The conditions involve left space and time first derivatives, with the left space derivative required to have locally bounded 2-dimensional variation. In particular a class of functions with discontinuous first derivative is included. An estimate of Krylov allows further weakening of these conditions when the semi-martingale is a diffusion. \vskip5pt Keywords: Local time, continuous semi-martingale, generalized It$\hat {o}$'s formula, two-dimensional Lebesgue-Stieltjes integral. \vskip5pt AMS 2000 subject classifications: 60H05, 60H30.
math0505195.tex
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\begin{document}

\title*{Generalized It$\hat {\rm o}$ Formulae and Space-Time Lebesgue-Stieltjes Integrals of Local Times}
\titlerunning{Generalized It$\hat {\rm o}$ Formulae } 
\author{K.D. Elworthy\inst{1},
A. Truman\inst{2} and H.Z. Zhao\inst{3}}
% Use \authorrunning{Short Title} for an abbreviated version of
% your contribution title if the original one is too long
\institute{Mathematics Institute, University of Warwick, Coventry CV4 7AL,UK.
\texttt{kde@maths.warwick.ac.uk}
\and Department of Mathematics, 
University of Wales Swansea,
Singleton Park, Swansea SA2 8PP, UK.
\texttt{A.Truman@swansea.ac.uk}
\and Department of Mathematical Sciences, Loughborough 
University, LE11 3TU, UK.
\texttt{H.Zhao@lboro.ac.uk}}
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\begin{abstract}
Generalised It${\hat{\rm o}}$ formulae are proved for time dependent  
functions of continuous real valued semi-martingales.The conditions involve 
left space and time first  derivatives, with the left space derivative required to have locally 
bounded 2-dimensional variation. In particular a class of functions with discontinuous first 
derivative is included. An estimate of Krylov allows further weakening of these conditions 
when the semi-martingale is a diffusion.
\vskip5pt

Keywords: Local time, continuous semi-martingale, generalized It$\hat {\rm o}$'s formula, 
two-dimensional Lebesgue-Stieltjes integral.
\vskip5pt

AMS 2000 subject classifications: 60H05, 60H30.

\end{abstract}


\section{Introduction}
Extensions of It${\hat {\rm o}}$ formula to less smooth functions are useful in
studying many problems such as partial differential equations with some 
singularities, see below, and in the mathematics of finance.
The first extension was obtained for $|X(t)|$ by Tanaka \cite{tanaka} with a beautiful use of local time. 
The generalized It${\hat {\rm o}}$ formula in one-dimension for time independent 
convex functions was developed in \cite{meyer} and for superharmonic functions in multidimensions 
in \cite{brosamler} and for distance functions in \cite{kendall}. Extensions of  It${\hat {\rm o}}$'s
formula have also been studied by \cite{krylov}, \cite{follmer}, \cite{nualart} and \cite{frw}.
In \cite{frw},  It${\hat {\rm o}}$'s formula for $W^{1,2}_{loc}$ functions was
studied using Lyons-Zheng's backward and forward stochastic integrals
\cite{lz}. In \cite{yor1}, It${\hat {\rm o}}$'s formula was extended to absolutely continuous 
functions with locally bounded derivative using the integral $\int _{-\infty}^{\infty}
\nabla f(x){\rm d}_xL  _s(x)$. This integral was defined 
through the existence of the expression  $f(X(t))-f(X(0))-\int _0^t{\partial \over 
\partial x}f(X(s)){\rm d}X(s)$; it was extended to $\int_0^t\int _{-\infty}^{\infty}
\nabla f(s,x){\rm d}_{s,x}L  _s(x)$ for a time dependent function $f(s,x)$ using forward and backward
integrals  for Brownian motion in \cite{eisenbaum}.  Recent activities in this 
direction have been to look for minimal assumptions on $f$ to make 
this integral well defined for semi-martingales other
than Brownian motion \cite{eisenbaum2}. However, our motivation in 
establishing generalized It$\hat {\rm o}$ formulae was to use them to describe the 
 asymptotics of the solution of heat equations in the presence of a caustic. Due to the 
 appearance of caustics, the solution of the Hamilton-Jacobi equation, the leading term in the asymptotics, is no longer differentiable, but has a jump in the gradient  across 
 the shock wave front of the associated Burgers' equation.  Therefore, 
the local time of continuous semimartingales in a neighbourhood of the shock wave front of the Burgers equation and the jump of the derivatives of the Hamilton-Jacobi function 
(or equivalently the jump in the Burgers' velocity) appear naturally in the semi-classical representation of the corresponding solution to the heat equation \cite{etz}. None of the earlier versions of 
 It$\hat {\rm o}$'s formula apply directly to this situation.  
 
 
In this paper, we first  generalize It${\hat {\rm o}}$'s formula to the case of a continuous semimartingale and a left continuous and locally bounded function $f(t,x)$ which satisfies (1) its left derivative ${\partial ^- \over \partial t}f(t,x)$ exists and
 is  left continuous, (2)
$f(t,x)=f_h(t,x)+f_v(t,x)$ with $f_h(t,x)$ being $C^1$ in $x$ and  
 $\nabla f_h(t,x)$ having left continuous and locally bounded left derivative 
 $\Delta ^-f_h(t,x)$, and $f_v$ having 
left derivative $\nabla  ^-f_v(t,x)$ which is left continuous and of locally bounded variation in $(t, x)$. 
Here we use the two-dimensional Lebesgue-Stieltjes 
integral of local time with respect to $\nabla^- f(t,x)$.
The main result of this paper is formula (\ref{krylov6}). Formula
(\ref{ito11}) follows from (\ref{krylov6}) easily as a special case. 
These formulae appear to be new and in a good form for extensions to 
two dimensions (\cite{fz1}). Moreover, in \cite{fz2}, Feng and Zhao observed that the local time 
$L_t(x)$ can be considered as a  rough path in $x$ of finite
 2-variation and therefore defined $\int _0^t\int_{-\infty}
^{\infty} \nabla ^-f(s,x){\rm d_{s,x}}L_s(x)$ pathwise by extending Young and Lyons' profound
 idea of rough path
integration (\cite{lq}, \cite{young}) to two parameters. 
When this paper was nearly completed, we received two preprints concerning a generalized 
It${\hat {\rm o}}$'s formula for a continuous function $f(t,x)$ with jump 
derivative $\nabla^- f(t,x)$, (\cite{pe}, \cite{gp}). We remark that 
 formula (\ref{ito11}) was also observed by \cite{pe} independently.
 
 In section 3, we consider diffusion processes $X(t)$. We prove the generalized It${\hat{\rm o}}$
 formula for a function $f$ with 
generalized derivative ${\partial\over \partial t} f$ in $L^2_{loc}(dtdx)$ and generalized derivative $\nabla f(t,x)$ being of locally 
bounded variation in $(t,x)$. We use an inequality from Krylov \cite{krylov}.

\section{The continuous semimartingale case}

We need the following definitions (see e.g. \cite{ash}, \cite{mcshane}): A two-variable function $f(s,x)$ is called monotonically increasing if whenever $s_2\geq s_1$, $x_2\geq x_1$,
\begin{eqnarray*}
f(s_2,x_2)-f(s_2,x_1)-f(s_1,x_2)+f(s_1,x_1)\geq 0.
\end{eqnarray*}
It is called monotonically decreasing if $-f$ is monotonically increasing. The function $f$ is called left continuous iff it is left continuous in both variables together, in other words, for any sequence 
$(s_1,x_1)\leq (s_2,x_2)\leq \cdots \leq (s_k,x_k)\to (s,x)$, we have $f(s_k,x_k)\to f(s,x)$ as 
$k\to \infty$. Here $(s,x)\leq (t,y)$ means $s\leq t$ and $x\leq y$.
For a monotonically
increasing and left continuous function $f(s,x)$, we can define a Lebesgue-Stieltjes measure by setting
\begin{eqnarray*}
\mu ([s_1,s_2)\times [x_1,x_2))=f(s_2,x_2)-f(s_2,x_1)-f(s_1,x_2)+f(s_1,x_1),
\end{eqnarray*}
for $s_2>s_1$ and $x_2>x_1$. So for a measurable function $g(s,x)$, we can define the 
Lebesgue-Stieltjes integral by 
\begin{eqnarray*}
\int _{t_1}^{t_2}\int _a^b g(s,x){\rm d}_{s,x} f(s,x)=\int _{t_1}^{t_2}\int _a^b g(s,x)d\mu.
\end{eqnarray*}
Denote a partition ${\cal P}$ of $[t,s]\times 
[a,x]$ by $t=s_1<s_2<\cdots <s_m=s$, $a=x_1<x_2<\cdots <x_n=x$ and the variation of $f$ associated with ${\cal P}$  by 
\begin{eqnarray*}
&&
V_{\cal P}(f,[t,s]\times [a,x])\\
&=&\sum _{i=1}^{m-1}\sum _{j=1}^{n-1} |f(s_{i+1},x_{j+1})-f(s_{i+1},x_j)-f(s_i,x_{j+1})+f(s_i,x_j)| 
\end{eqnarray*}
and the variation of $f$ on $[t,s]\times [a,x]$ by 
\begin{eqnarray*}
V_f([t,s]\times [a,x])=\sup _{\cal P}V_{\cal P}(f,[t,s]\times [a,x]).
\end{eqnarray*}

One can find Proposition \ref{prop2}, its proof and definition of the multidimensional 
Lebesgue-Stieltjes integral with respect to measures generated by functions of bounded variation
in \cite{mcshane}. 
For the convenience of the reader, we include them here briefly. 
\begin{prop} \label {prop1}(Additivity of variation) For $s_2\geq s_1\geq t$, and $a_2\geq a_1\geq a$,
\begin{eqnarray}
V_f([t,s_2]\times [a,a_2])&=&V_f([t,s_1]\times [a,a_2])+V_f([t,s_2]\times [a,a_1])\nonumber\\
&&+V_f([s_1,s_2]\times [a_1,a_2])-V_f([t,s_1]\times [a,a_1]).
\end{eqnarray}
\end{prop}
\vskip5pt
{\em Proof}. We only need to prove that for $a\leq a_1< a_2$ and $t\leq s_1$,
\begin{eqnarray}\label{variation1}
V_f([t,s_1]\times [a,a_2])=V_f([t,s_1]\times [a,a_1])+V_f([t,s_1]\times [a_1,a_2]).
\end{eqnarray}
Our proof is similar to the case of one-dimension. We can always refine a partition ${\cal P}$ of 
$[t,s_1]\times [a,a_2]$ to include $a_1$. The refined partition is denoted by
${\cal P}^{\prime}$. Then 
\begin{eqnarray*}
V_{\cal P}(f,[t,s_1]\times [a,a_2])\leq V_{{\cal P}^{\prime}}(f,[t,s_1]\times [a,a_2]).
\end{eqnarray*}
Then (\ref{variation1}) follows easily. \hfill$\diamond$

\begin{prop}\label{prop2}
A function $f(s,x)$ of locally bounded variation can be decomposed as the 
difference of two increasing functions $f_1(s,x)$ and $f_2(s,x)$, in any quarter space 
$s\geq t, x\geq a$. 
Moreover, if $f$ is also left
continuous, then $f_1$ and $f_2$ can be taken left continuous.
\end{prop}
\vskip5pt
{\em Proof}. For any $(t,x)\in R^2$, define for $s\geq t$ and $x\geq a$,
\begin{eqnarray*}
2\tilde f_1(s,x)&=&V_f([t,s]\times [a,x])+f(s,x),\\
2\tilde f_2(s,x)&=&V_f([t,s]\times [a,x])-f(s,x).
\end{eqnarray*}
Then $f(s,x)=\tilde f_1(s,x)-\tilde f_2(s,x)$. We need to prove that $\tilde f_1$ and $\tilde 
f_2$ are increasing functions. For this,
let $s_2\geq s_1\geq t$, $a_2\geq a_1\geq a$,
then use Proposition \ref{prop1}, 
\begin{eqnarray*}&&
2(\tilde f_1(s_2,a_2)-\tilde f_1(s_1,a_2)-\tilde f_1(s_2,a_1)+\tilde f_1(s_1,a_1))\\
&=&V_f([t,s_2]\times [a,a_2])-V_f([t,s_1]\times [a,a_2])-V_f([t,s_2]\times [a,a_1])\\
&&
+V_f([t,s_1]\times [a,a_1])+f(s_2,a_2)-f(s_1,a_2)-f(s_2,a_1)+f(s_1,a_1)\\
&=&V_f([s_1,s_2]\times [a_1,a_2])+f(s_2,a_2)-f(s_1,a_2)-f(s_2,a_1)+f(s_1,a_1)\\
&\geq & 0.
\end{eqnarray*}
So $\tilde f_1(s,x)$ is an increasing function. Similarly one can prove that $\tilde 
f_2(s,x)$ is an increasing function.

Define 
\begin{eqnarray*}
f_i(s,x)&=&\lim \limits_{t\uparrow s,y\uparrow x}\tilde f_i(t,y),\ \ i=1,2.
\end{eqnarray*}
Then since $f$ is left continuous, so 
\begin{eqnarray}\label{andrew1}
f(s,x)=f_1(s,x)-f_2(s,x),
\end{eqnarray}
and $f_1$ and $f_2$ are as required.
\hfill $\diamond$
\bigskip

%If $f(s,x)$ is of bounded variation,  it is not difficult to prove using similar method as in the 
%one-dimensional case that $f(s,x)$ can be decomposed as the difference of two increasing functions $f_1(s,x)$ and $f_2(s,x)$. 

From Proposition \ref{prop2}, 
the two-dimensional Lebesgue-Stieltjes integral of a measurable function $g$ 
with respect to the left continuous function $f$ of bounded variation 
can be defined by 
\begin{eqnarray*}
\int _{t_1}^{t_2}\int _a^b g(s,x){\rm d}_{s,x} f(s,x)&=&\int _{t_1}^{t_2}\int _a^b g(s,x){\rm d}_{s,x} f_1(s,x)
\\
&&
-\int _{t_1}^{t_2}\int _a^b g(s,x){\rm d}_{s,x} f_2(s,x)\  {\rm for } \  t_2\geq t_1, b\geq a.
\end{eqnarray*}
Here $f_1$ and $f_2$ are taken to be left continuous.
\smallskip

It is worth pointing out that it is possible that a function $f(s,x)$ is of locally bounded variation in 
$(s,x)$ but not of locally bounded variation in $x$ for fixed $s$. 
For instance consider $f(s,x)=b(x)$, where $b(x)$ is not of locally 
bounded variation, then $V_f=0$. However it is easy to see that when a function
$f(s,x)$ is of locally bounded variation in $(s,x)$ and of locally bounded variation
in $x$ for a fixed $s=s_0$, then it is of locally bounded variation in $x$ 
for all $s$. We denote by $V_{f(s)}[a,b]$ the variation of $f(s,x)$ on $[a,b]$
as a function of $x$ for a fixed $s$. 
\smallskip


Now we recall some well-known results of local time which will be used later in this paper.
Let $X(s)$ be a continuous semimartingale $X(s)=X(0)+M_s+V_s$ on a probability space $\{\Omega,{\cal F},P\}$. Here $M_s$ is a 
continuous local martingale and $V_s$ is a continuous process of bounded variation. Let $L _t(a)$ be the local time introduced by P. L\'evy
\begin{eqnarray}
L  _t(a)=\lim_{\epsilon\downarrow 0} {1\over 2\epsilon}\int _0^t1_{[a,a+\epsilon)}(X(s))d<M,M>_s \ \ 
a.s.,
\end{eqnarray}
for each $t$ and $a$.
 Then it is well known that for each fixed $a\in R$, 
 $L _t(a,\omega)$
is continuous, and nondecreasing in $t$ and right continuous with left limit (cadlag) with respect to $a$ 
(\cite{ks}, \cite{yor}). Therefore we can consider the Lebesgue-Stieltjes integral 
$\int _0^{\infty}\phi(s)dL  _s(a,\omega)$
for each $a$ for any Borel-measurable function $\phi$. In particular
\begin{eqnarray}
\int _0^{\infty}1_{R-\{a\}}(X(s))dL _s(a,\omega)=0 \  \ a.s.
\end{eqnarray}
Furthermore if $\phi$ is in $L^{1,1}_{loc}(ds)$, i.e. $\phi$ has locally integrable generalized 
derivative, 
then we have the following integration by parts formula
\begin{eqnarray}
\int _0^t\phi(s)dL _s(a,\omega)=\phi(t)L _t(a,\omega)-\int _0^t\phi^{\prime}(s)L _s(a,\omega)ds\ \ a.s.
\end{eqnarray}
Moreover, if $g(s,x)$ is Borel measurable in $s$ and $x$ and bounded, by the occupation times formula (e.g. see \cite{ks}, \cite{yor})), 
\begin{eqnarray*}
\int _0^tg(s,X(s))d<M,M> _s=2\int _{-\infty}^{\infty}\int _0^tg(s,a)dL _s(a,\omega)da\ \ a.s.
\end{eqnarray*}
If further $g(s,x)$ is in $L^{1,1}_{loc}(ds)$ for almost all $x$, then using the integration by parts formula, we have
\begin{eqnarray*}
\int _0^tg(s,X(s))d<M,M> _s&=&2\int _{-\infty}^{\infty}\int _0^tg(s,a)dL _s(a,\omega)da\nonumber\\
&=&2\int _{-\infty}^{\infty}g(t,a)L _t(a,\omega)da\nonumber\\
&&-2\int _{-\infty}^{\infty}\int _0^t{\partial \over \partial s}g(s,a)L _s(a,\omega)dsda\ \ a.s.
\end{eqnarray*}
%Consider a function which is differentiable almost everywhere. Denote
%by ${\partial \over \partial t}f(t,x)=\lim\limits_{s\to t-} {\partial\over \partial s}f(s,x)$ and
%${\nabla  }f(t,x)=\lim\limits_{y\to x-} \nabla f(t,y)$. Here we assume the limits exist. 

We first prove a theorem with $f_h= 0$. The result with a term $f_h$ is a trivial 
generalization of Theorem \ref{tom100}.

\begin{thm} \label{tom100}
Assume $f: [0,\infty)\times R\to R$ satisfies
\vskip2pt

(i) $f$ is left continuous and locally bounded, with $f(t,x)$ jointly continuous from the right in $t$ 
and left in $x$ at each point $(0,x)$, 
\vskip2pt

(ii) the left derivatives ${\partial ^-\over \partial t}f$ and  $\nabla ^-f$ exist at all points of $(0,
\infty )\times R$ and $[0,\infty)\times R$ respectively,
\vskip2pt

(iii)  ${\partial ^-\over \partial t}f$ and  $\nabla ^-f$ are left continuous and locally bounded,
\vskip2pt

(iv)  $\nabla ^-f$ is of locally bounded variation in $(t, x)$ and $\nabla ^-f(0,x)$ is of locally 
bounded variation in $x$. 
\vskip2pt
 
 \noindent
 Then for any continuous semimartingale $\{X(t), t\geq 0\}$
 \begin{eqnarray}\label{ito10}
&&
 f(t,X(t))\nonumber\\
&=&f(0,X(0))+\int _0^t{\partial^- \over \partial s} f(s,X(s))ds+\int _0^t\nabla ^-f(s,X(s))dX(s)\\
&&
+
\int _{-\infty}^{\infty }L  _t(x){\rm d}_x\nabla ^- f(t,x)-\int _{-\infty}^{+\infty}\int _0^{t}L  _s(x)
{\bf \rm d}_{s,x}\nabla ^-f(s,x)\ \ a.s. \nonumber
\end{eqnarray}
\end{thm}
 \vskip5pt
 {\em Proof}. By a standard localization argument we can assume that $X$ and its quadratic
 variation are bounded processes and that $f$,  ${\partial ^-\over \partial t}f$, $\nabla ^-f$,
 $V_{\nabla ^-f(t)}$ and $V_{\nabla ^-f}$ are bounded (note here $V_{\nabla ^-f(0)}<\infty$ and 
 $V_{\nabla ^-f}<\infty$ imply $V_{\nabla ^-f(t)}<\infty$ for all $t\geq 0$). Note first that from (i) to (iii) 
 the left partial derivatives of $f$ agree with the distributional
 derivatives and so (iii) implies that $f$ is absolutely continuous in each variable.
 We use standard regularizing mollifiers (e.g. see \cite{ks}). 
Define 
\begin{eqnarray*}
\rho(x)=\cases {c{\rm e}^{{1\over (x-1)^2-1}}, {\rm \  if } \ x\in (0,2),\cr
0, \ \ \ \ \ \ \ \ \ \ \ {\rm otherwise.}}
\end{eqnarray*}
Here $c$ is chosen such that $\int _0^2\rho(x)dx=1$.
Take $\rho_n(x)=n\rho(nx)$ as mollifiers. Define
\begin{eqnarray}\label{krylov5}
f_n(s,x)=\int _{-\infty}^{+\infty}\int _{-\infty}^{+\infty}\rho_n(x-y)\rho_n(s-\tau)f(\tau,y)d\tau dy, \ \ n\geq 1,
\end{eqnarray}
 where we set $f(\tau,y)=f(-\tau,y)$ if $\tau<0$.
 Then $f_n(s,x)$ are smooth and 
\begin{eqnarray}\label{truman41}
f_n(s,x)=\int _0^2\int _0^2\rho(\tau)\rho(z)f(s-{\tau\over n},x-{z\over n})d\tau dz, \ \ n\geq 1.
\end{eqnarray}
Because of the absolutely continuity mentioned above, we can differentiate 
under the integral in (\ref{truman41}) to see 
that  ${\partial \over \partial t} f_n(t,x)$, 
$\nabla f_n(t,x)$, $V_{\nabla f_n(t)}$ and $V_{\nabla f_n}$ are uniformly bounded. Moreover 
using Lebesgue's dominated convergence theorem, one can prove that as $n\to \infty$, 
for each $(t,x)$ with $t\geq 0$,
\begin{eqnarray}
f_n (t,x)&\to & f (t,x).\label{zhao1}
\end{eqnarray}
Also
\begin{eqnarray}
{\partial \over \partial t}f_n(t,x)&\to & {\partial  ^-\over \partial t} f(t,x),\ \ t>0\label{zhao2}\\
\nabla f_n(t,x)&\to & \nabla ^-f(t,x),\ \ t\geq 0.\label{zhao3}
\end{eqnarray}
Note the convergence 
in (\ref{zhao1}), (\ref{zhao2}), (\ref{zhao3}) is also in $L^p_{loc}$, $1\leq p<\infty$.

It turns out for any $g(t,x)$ being continuous in $t$ and 
$C^1$ in $x$ and having a compact support, using the integration by 
parts formula and Lebesgue's dominated convergence theorem,
\begin{eqnarray}
\lim_{n\to +\infty}\int _{-\infty}^{+\infty} g(t,x)\Delta f_n(t,x)dx
&=&
-\lim_{n\to +\infty}\int _{-\infty}^{\infty}
\nabla  g(t,x)\nabla f_n(t,x)dx
\nonumber\\
&=&
-\int _{-\infty}^{\infty}\nabla  g(t,x)\nabla ^-f(t,x)dx.
\end{eqnarray}
Note $\nabla ^-f(t,x)$ is of bounded variation in $x$ and $\nabla  g(t,x)$ 
has a compact support, so 
\begin{eqnarray}
-\int _{-\infty}^{+\infty} \nabla  g(t,x)\nabla^- f(t,x)dx=\int _{-\infty}^{
+\infty} g(t,x){\rm d}_x\nabla ^-f(t,x).
\end{eqnarray}
%Similarly
%\begin{eqnarray}
%&&\lim\limits_{\delta \to 0}\int _{\delta}^{\infty} g(t,x){\partial ^2\over \partial x^2}f(t,x+l(t))dx
%= \int _{0}^{\infty} g(t,x) {\rm d}_x\nabla ^+f(t,x+l(t))\nonumber\\
%&=&
 %\int _{0}^{\infty} g(t,x) {\rm d}_x\nabla f(t,x+l(t))-g(t,0)(\nabla ^+f(t,l(t))
%-\nabla f(t,l(t))).
%\end{eqnarray}
Thus
\begin{eqnarray}\label{elworthy2}
\lim_{n\to +\infty}\int _{-\infty}^{+\infty} g(t,x)\Delta f_n(t,x)dx
=\int _{-\infty}^{\infty } g(t,x){\rm d}_x\nabla ^-f(t,x).
\end{eqnarray}
Similarly, one can easily see from the integration by parts formula and 
Lebesgue's dominated convergence theorem, if $g(s,x)$ is 
$C^1$ in $x$ with ${\partial \over \partial s}\nabla g(s,x)$ 
being continuous and has a compact support in $x$,
\begin{eqnarray}
&&
\lim_{n\to +\infty}\int _{0}^{t}\int _{-\infty}^{+\infty}g(s,x)\Delta
{\partial \over \partial s}f_n(s,x)dxds\nonumber\\
&=&
-\lim_{n\to +\infty}\int _{0}^{t}\int _{-\infty}^{+\infty}
\nabla g(s,x)\nabla {\partial \over \partial s}f_n(s,x)dxds\nonumber\\
&=&-\lim_{n\to +\infty}\int _{-\infty}^{\infty}
\left [\nabla g(s,x)\nabla f_n(s,x)\right ] _0^t dx\nonumber\\
&&
+\lim_{n\to +\infty}\int _{0}^{t}\int _{-\infty}^{+\infty}
{\partial \over \partial s}\nabla g(s,x)\nabla f_n(s,x)dxds\nonumber\\
&=&-\int _{-\infty}^{+\infty}
\left [\nabla g(s,x)\nabla ^-f(s,x)\right ] _0^t dx\nonumber\\
&&
+\int _{0}^{t}\int _{-\infty}^{\infty}
{\partial \over \partial s}\nabla g(s,x)\nabla  ^-f(s,x)dxds.
\end{eqnarray}
Thus 
\begin{eqnarray}\label{zhao6}
&&
\lim_{n\to +\infty}\int _{0}^{t}\int _{-\infty}^{+\infty}g(s,x)\Delta
{\partial \over \partial s}f_n(s,x)dxds\nonumber\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\int _{0}^{t}\int _{-\infty}^{\infty}g(s,x){\bf \rm d}_{s,x}\nabla ^-f(s,x).
\end{eqnarray}
Now suppose $g(s,x)$ is continuous in $s$ and cadlag in $x$ jointly, and has compact support.
(In particular, $g$ is bounded). We claim (\ref{elworthy2}) and (\ref{zhao6}) still valid. 
For this define
\begin{eqnarray*}
g_m(t,x)=\int _{-\infty}^{\infty}\rho_m(y-x)g(t,y)dy=\int _0^2\rho (z)g(x+{z\over m})dz.
\end{eqnarray*}
To see (\ref{elworthy2}),  using Lebesgue's dominated convergence theorem, 
note that there is a compact set $G_t\subset R^1$ such that
\begin{eqnarray*}
&\max \limits _{x\in G_t}|g_m(t,x)-g(t,x)|\to 0 & {\rm as } \ \ m\to +\infty,\\
&g_m(t,x)=g(t,x) =0 & {\rm for } \ \ x\notin G_t.
\end{eqnarray*}
Note 
\begin{eqnarray}\label{elworthy1}
\int _{-\infty}^{+\infty}g(t,x)\Delta f_n(t,x)dx&=&
\int _{-\infty}^{+\infty}g_m(t,x)\Delta f_n(t,x)dx\nonumber\\
&+&\int _{-\infty}^{+\infty}(g(t,x)-g_m(t,x))\Delta  f_n(t,x)dx.
\end{eqnarray}
It is easy to see from ({\ref{elworthy2}) and using Lebesgue's dominated convergence theorem, that 
\begin{eqnarray}\label{elworthy3}
\lim\limits_{m\to \infty}\lim\limits _{n\to \infty}
\int _{-\infty}^{+\infty}g_m(t,x)\Delta f_n(t,x)dx
&=&\lim\limits _{m\to \infty}\int _{-\infty}^{\infty } g_m(t,x){\rm d}_x\nabla ^-f(t,x)\nonumber\\
&=&\int _{-\infty}^{\infty } g(t,x){\rm d}_x\nabla ^-f(t,x).
\end{eqnarray}
%We assume temporarily that 
%$f_n(t,x)$ is convex. This condition will be dropped soon. 
% Then 
%$\Delta f_n(t,x)\geq 0$. 
%Thus
Moreover,
\begin{eqnarray}\label{elworthy6}
&&
|\int _{-\infty}^{+\infty}(g(t,x)-g_m(t,x))\Delta f_n(t,x)dx|\nonumber\\
&\leq & |\int _{-\infty}^{+\infty}(g(t,x)-g_m(t,x)){\rm d_x}\nabla f_n(t,x)|\nonumber\\
&\leq & 
\max\limits _{x\in G} |g(t,x)-g_m(t,x)| V_{\nabla f_n(t)}(G).
\end{eqnarray}
This leads easily to 
\begin{eqnarray}\label{elworthy7}
\lim\limits_{m\to \infty}\limsup\limits_{n\to \infty}
|\int _{-\infty}^{+\infty}(g(t,x)-g_m(t,x))\Delta f_n(t,x)dx|=0.
\end{eqnarray}
 % Since  $\nabla^- f$ is of locally bounded variation and $f$ is absolutely continuous in $x$,
% we can left
%differentiate under the integral sign in (\ref{truman41}) to see that the functions $\nabla^-(f_n)$ are %uniformly of local bounded variation and so from the formulae in Proposition \ref{prop2} they %decompose into
%the difference of two increasing functions  each of which has local bounds independent of n.
%Thus (\ref{elworthy7}) is true for the function $f$
%with $\nabla^- f(t,x)$ being of bounded variation.
Now we use (\ref{elworthy1}), (\ref{elworthy3}), (\ref{elworthy7})
\begin{eqnarray*}
&&
\limsup\limits_{n\to \infty}\int _{-\infty}^{+\infty}g(t,x)\Delta f_n(t,x)dx\nonumber\\
&=&
\lim\limits_{m\to \infty}\limsup\limits_{n\to \infty}\int _{-\infty}^{+\infty}g_m(t,x)\Delta f_n(t,x)dx\nonumber\\
&&
+\lim\limits_{m\to \infty}\limsup\limits_{n\to \infty}\int _{-\infty}^{+\infty}(g(t,x)-g_m(t,x))\Delta f_n(t,x)dx\nonumber
\\
&=& \int _{-\infty}^{\infty } g(t,x){\rm }{\rm d}_x\nabla^- f(t,x).
\end{eqnarray*}
Similarly we also have 
\begin{eqnarray*}
\liminf\limits_{n\to \infty}\int _{-\infty}^{+\infty}g(t,x)\Delta f_n(t,x)dx&=&
\int _{-\infty}^{\infty } g(t,x){\rm d}_x\nabla  ^-f(t,x).
\end{eqnarray*}
So (\ref{elworthy2}) holds for a cadlag function $g$ with a compact support.

Similarly we can prove (\ref{zhao6}) holds for a cadlag function $g$ with a compact support.
That is there exists a compact $G\subset R^1$ such that $g(s,x)=0$ 
for $x\notin G$ and $s\in [0, t]$.

To complete the proof of (\ref{ito10}), use It${\hat {\rm o}}$'s formula for the smooth function $f_n(s,X(s))$, then a.s. 
\begin{eqnarray}\label{zhao11}
f_n(t,X(t))-f_n(0,X(0))&=&\int _0^t{\partial \over \partial s} f_n(s,X(s))ds+\int _0^t\nabla f_n(s,X(s))dX(s)\nonumber\\
&&
+{1\over 2}
\int _0^t\Delta f_n(s,X(s))d<M,M>_s.
\end{eqnarray}
As $n\to \infty$, for all $t\geq 0$,
\begin{eqnarray*}
f_n(t,X(t))-f_n(0,X(0))\to f(t,X(t))-f(0,X(0))\ \ a.s.,
\end{eqnarray*}
and 
\begin{eqnarray*}
\int _0^t{\partial \over \partial s} f_n(s,X(s))ds\to \int _0^t{\partial ^-\over \partial s} f(s,X(s))ds\ \ a.s.,
\end{eqnarray*}
\begin{eqnarray*}
\int _0^t \nabla f_n(s,X(s))dV_s\to \int _0^t\nabla^- f(s,X(s))dV_s\ \ a.s.
\end{eqnarray*}
and 
\begin{eqnarray*}
E\int _0^t(\nabla f_n(s,X(s)))^2d<M,M>_s\to E \int _0^t(\nabla^- f(s,X(s)))^2d<M,M>_s.
\end{eqnarray*}
Therefore in $L^2(\Omega,P)$,
\begin{eqnarray*}
\int _0^t \nabla f_n(s,X(s))dM_s\to \int _0^t\nabla^- f(s,X(s))dM_s.
\end{eqnarray*}
To see the convergence of the last term, we recall the well-known result that the local time 
$L  _s(x)$ is jointly continuous in $s$ and cadlag with respect to $x$ and
has a compact support in space $x$ for each $s$. As $L  _s(x)$ is an increasing
function of $s$ for each $x$ , so if $G\subset R^1$ is the support of $L  _t$, then
$L  _s(x)=0$ for all $x\notin G$ and $s\le t$.  Now we use the occupation times formula, the integration by parts formula
and (\ref{elworthy2}), (\ref{zhao6}) for the case when $g$ is cadlag with compact support in $x$,
\begin{eqnarray*}
&&
{1\over 2}\int _0^t\Delta f_n(s,X(s))d<M,M>_s\nonumber\\
 &=&\int _{-\infty}^{+\infty}\int _0^t\Delta f_n(s,x)d_sL  _s(x)dx\nonumber\\
&=&
\int _{-\infty}^{+\infty}\Delta f_n(t,x)L  _t(x)dx-\int _{-\infty}^{+\infty}\int _0^{t}{d \over ds}
\Delta f_n(s,x)L  _s(x)dsdx\nonumber\\
&\to &
\int _{-\infty}^{\infty }L  _t(x){\rm d}_x\nabla^- f(t,x)-\int _{-\infty}^{+\infty}\int _0^{t}L  _s(x)
{\bf \rm d}_{s,x}\nabla^- f(s,x)\ \ a.s.,
\end{eqnarray*}
as $n\to \infty$.
This proves the desired formula. 
$\hfill\diamond$
\bigskip
 
 %Recall the following extension of It${\hat {\rm o}}$'s formula due to Krylov (\cite{krylov}): 
 The smoothing procedure can easily be modified to prove that  if $f: 
 R^+\times R\to R$ satisfies (i), (ii) and (iii) of Theorem \ref{tom100}, 
 is also  $C^1$ in $x$ and the left derivative
 $\Delta ^-f(t,x)$ exists at all points of $[0,\infty)\times R$ and is jointly left continuous and locally 
 bounded, then $\Delta f_n(t,x)\to \Delta ^-f(t,x)$ as 
 $n\to \infty$, $t>0$. Thus
  \begin{eqnarray}\label{krylov7}
 f(t,X(t))
&=&f(0,X(0))+\int _0^t{\partial ^-\over \partial s} f(s,X(s))ds+\int _0^t\nabla f(s,X(s))dX(s)\nonumber\\
&&
 +{1\over 2}\int _0^t\Delta ^-f(s,X(s))d<X>_s
\ \ a.s.
 \end{eqnarray}
 
 The next theorem is an easy extension of Theorem \ref{tom100} and formula (\ref{krylov7}).
 
\begin{thm} \label{tom102}
Assume $f: R^+\times R\to R$ satisfies conditions (i), (ii) and (iii) of Theorem \ref{tom100}. 
Further suppose $f(t,x)=f_h(t,x)+f_v(t,x)$ where
\vskip2pt

(i)  $f_h(t,x)$ is $C^1$ in $x$ with
 $\nabla f_h(t,x)$ having left partial derivative 
 $\Delta ^-f_h(t,x)$, (with respect to $x$), which is left continuous and locally bounded, 
\vskip2pt
 
 (ii)
 $f_v(t,x)$ has a left continuous derivative $\nabla ^-f_v(t,x)$ at all points $(t,x)$ $[0,\infty)\times R$, which is of locally bounded variation in $(t, x)$ and of locally bounded in $x$ for $t=0$. 
\vskip2pt
 
 \noindent 
Then for any continuous semi-martingale $\{X(t), t\geq 0\}$,
\begin{eqnarray}\label{krylov6}
f(t,X(t))
&=&f(0,X(0))+\int _0^t{\partial ^- \over \partial s} f(s,X(s))ds+\int _0^t\nabla^- f(s,X(s))dX(s)\nonumber\\
&&
 +{1\over 2}\int _0^t\Delta ^-f_h(s,X(s))d<X>_s+
\int _{-\infty}^{\infty }L  _t(x){\rm d}_x\nabla  ^-f_v(t,x)\nonumber\\
 &&
-\int _{-\infty}^{+\infty}\int _0^{t}L  _s(x)
{\bf \rm d}_{s,x}\nabla ^- f_v(s,x)\ \ a.s.
\end{eqnarray}
\end{thm}
 \vskip5pt
 {\em Proof}. Mollify $f_h$ and $f_v$, and so $f$, as in the proof of Theorem \ref{tom100}.
 Apply It${\hat {\rm o}}$'s formula to the mollification of $f$ and take the limits as in the proofs of
 Theorem \ref{tom100} and (\ref{krylov7}). \hfill $\diamond$
 
 If $f$ has discontinuity of first and second order derivatives across a curve $x=l(t)$, where 
 $l(t)$ is a continuous function of locally bounded 
 variation, it will be convenient to consider the continuous semi-martingale 
\begin{eqnarray*}
X^*(s)=X(s)-l(s),
\end{eqnarray*}
 and let $L  _s^*(a)$ be its local time. We can prove the following version of our main results:

\begin{thm}\label{ito2}
 Assume $f: R^+\times R\to R$ satisfies conditions (i), (ii) and (iii) of Theorem \ref{tom100}. 
Moreover,  suppose $f(t,x)=f_h(t,x)+f_v(t,x)$, where $f_h(t,x)$ is $C^1$ in $x$ and  
 $\nabla f_h(t,x)$ has left derivative 
 $\Delta ^-f_h(t,x)$ which is left continuous and locally bounded, and 
there exists a curve $x=l(t)$, $t\geq 0$, a continuous 
function of locally bounded variation such that 
$\nabla  ^-f_v(t,x+l(t))$ as a function of $(t,x)$ 
is of locally bounded variation in $(t,x)$ and of locally bounded in $x$ for $t=0$. Then 
\begin{eqnarray}\label{ito13}
&&
f(t,X(t))\nonumber\\
&=&f(0,z)+\int _0^t{\partial  \over \partial s} f(s,X(s))ds+\int _0^t\nabla^- f(s,X(s))dX(s)\nonumber \\
&&
 +{1\over 2}\int _0^t\Delta ^-f_h(s,X(s))d<X>_s
+
\int _{-\infty}^{\infty }L  _t^*(x){\rm d}_x\nabla  ^-f_v(t,x+l(t))\nonumber\\
 &&
 -\int _{-\infty}^{+\infty}\int _0^{t}L  _s^*(x)
{\bf \rm d}_{s,x}\nabla  ^-f_v(s,x+l(s)))\ \ a.s.
\end{eqnarray}
\end{thm}
\vskip5pt
{\em Proof}. We only need to consider the case
when $f_h=0$ as the general case will follow
easily. We basically follow the proof of Theorem \ref{tom100} and apply It${\hat {\rm o}}$'s
formula to $f_n$ and $X(s)$. We still have (\ref{zhao11}). But by the occupation times formula, a.s.
\begin{eqnarray*}
&&
{1\over 2}\int _0^t\Delta f_n(s,X(s))d<M,M>_s\nonumber\\
&=&{1\over 2}\int _0^t\Delta f_n(s,X^*(s)+l(s))d<M,M>_s\nonumber\\
 &=&\int _{-\infty}^{+\infty}\int _0^t\Delta f_n(s,x+l(s))d_sL  ^*_s(x)dx\nonumber\\
&=&
\int _{-\infty}^{+\infty}\Delta f_n(t,x+l(t))L  ^* _t(x)dx-\int _{-\infty}^{+\infty}\int _0^{t}{d \over ds}
\Delta f_n(s,x+l(s))L  ^*_s(x)dsdx\nonumber\\
&\to &
\int _{-\infty}^{\infty }L  ^* _t(x){\rm d}_x\nabla^- f(t,x+l(t))-\int _{-\infty}^{+\infty}\int _0^{t}L  ^*_s(x)
{\bf \rm d}_{s,x}\nabla^- f(s,x+l(s)),
\end{eqnarray*}
as $n\to \infty$ as in the proof of Theorem \ref{tom100}.
This proves the desired formula. \hfill $\diamond$

\begin{cor} 
Assume $f: R^+\times R\to R$ satisfies condition (i) of Theorem \ref{tom100} and its left derivative
${\partial ^-\over \partial t}f$ exists on $(0,\infty)\times R$ and is left continuous. Further suppose that there exists a curve $x=l(t)$ of locally 
bounded variation such that 
f is $C^1$ in x off the curve with $\nabla f$ having left and right limits in $x$ at each point $(t,x)$ and  a left continuous and locally bounded left derivative $\Delta^- f$
on $x$ not equal to $l(t)$. Also assume $\nabla f(t,l(t)+y-)$ as a function of $t$ and $y$ 
is locally bounded and jointly left continuous if $y\leq 0$, and $\nabla f(t,l(t)+y+)$ is locally
bounded and jointly left continuous in $t$ and right continuous in $y$ if $y\geq 0$.
 Then for any continuous semi-martingale $\{X(t), t\geq 0\}$,
\begin{eqnarray}\label{ito11}
&&
f(t,X(t))\nonumber\\
&=&f(0,X(0))+\int _0^t{\partial  ^-\over \partial s} f(s,X(s))ds+\int _0^t\nabla^- f(s,X(s))dX(s)\nonumber\\
&&
+{1\over 2}\int _0^t\Delta  ^-f(s,X(s))d<X,X>_s\nonumber\\
&&+
\int _0^t(\nabla f(s,l(s)+)-\nabla f(s,l(s)-))d_sL  _s^*(0)\ \ a.s. 
\end{eqnarray}
\end{cor}
\vskip5pt
{\em Proof}. At first we assume temporarily that $(\nabla f(t,l(t)+)-\nabla f(t,l(t)-))$ is of bounded variation. 
This condition will be dropped later.
Formula (\ref{ito11}) can be read from (\ref{ito13}) by considering
 \begin{eqnarray*}
f_h(t,x)&=&f(t,x)+(\nabla f(t,l(t)-)-\nabla f(t,l(t)+))(x-l(t))^+,
\\
f_v(t,x)&=&(\nabla f(t,l(t)+)-\nabla f(t,l(t)-))(x-l(t))^+,
 \end{eqnarray*} 
 and integration by parts formula and noticing $\nabla ^-f_v(t,x+l(t))$ is of locally bounded 
 variation in $(t,x)$.  Let 
$g(t,y)=f(t,y+l(t))$.
 In terms of $X^*$, (\ref{ito11}) can be rewritten as 
 \begin{eqnarray}\label{krylov10}
&&
g(t,X^*(t))\nonumber\\
&=&g(0,X^*(0))+\int _0^t g(ds,X^*(s))
+\int _0^t\nabla^- g(s,X^*(s))dX^*(s)\nonumber\\
&&
+{1\over 2}\int _0^t\Delta  ^-g(s,X^*(s))d<X^*,X^*>_s\nonumber\\
&&+
\int _0^t(\nabla g(s,0+)-\nabla g(s,0-))d_sL  _s^*(0)\ \ a.s. 
\end{eqnarray}
Here 
\begin{eqnarray*}
g(ds,y)=d_sg(s,y)={\partial^-\over \partial s}f(s,y+l(s))ds+\nabla ^-f(s,y+l(s))dl(s).
\end{eqnarray*}

 Now without assuming that $(\nabla f(t,l(t)+)-\nabla f(t,l(t)-))$ is of bounded variation,
 we can prove the formula by a smoothing procedure in the variable $t$. To see this, let  
 \begin{eqnarray*}
g_n(t,y)=\int _0^2\rho (\tau)g(t-{\tau\over n},y)d\tau
=\int _0^2\rho (\tau)f(t-{\tau\over n},y+l(t-{\tau\over n}))d\tau,
 \end{eqnarray*}
 with $l(s)=l(0)$ if $s<0$ and $f(s,x)=f(-s,x)$ for $s<0$ as usual.
 Then as $n\to \infty$,
  \begin{eqnarray}\label{ky1}
\int _0^t g_n(ds,X^*(s))
&=& \int _0^t \int _0^2\rho (\tau){\partial^- \over \partial s}f(s-{\tau\over n},X^*(s)+l(s-{\tau\over n}))d\tau ds\nonumber\\
&&+\int _0^t\int _0^2\rho (\tau)\nabla ^-f(s-{\tau\over n},X^*(s)+l(s-{\tau\over n}))dl(s-{\tau\over n})d\tau
\nonumber\\
&\to & \int _0^t{\partial^- \over \partial s}f(s,X^*(s)+l(s))ds
+\nabla ^-f(s,X^*(s)+l(s))dl(s)\nonumber\\
&=&\int _0^tg(ds,X^*(s))\ \ a.s.
 \end{eqnarray}
 It is easy to see that for all $(t,y)$
 \begin{eqnarray}\label{ky2}
 g_n(t,y)\to g(t,y)
 \end{eqnarray}
 and 
 for all $y\ne 0$, 
 \begin{eqnarray}\label{ky3}
 \nabla g_n(t,y)\to \nabla g(t,y),  \Delta ^-g_n(t,y)\to \Delta ^-g(t,y),
 \end{eqnarray}
 with uniform local bounds.
 Moreover, we can see that as  $y\to 0\pm$ and $n\to \infty$,
  \begin{eqnarray}\label{ky4}
\nabla ^{\pm}g_n(t,y)=\int _0^2\rho (\tau)\nabla ^{\pm}g(t-{\tau\over n},y)d\tau \to \nabla g(t,0\pm).
\end{eqnarray}
 
 Since $\nabla g_n(t,0\pm )$ are smooth in $t$ then are of locally bounded variation.  
 From (\ref{krylov10}),
  \begin{eqnarray}\label{krylov11}
&&
g_n(t,X^*(t))\nonumber\\
&=&g_n(0,X^*(0))+\int _0^t g_n(ds,X^*(s))
+\int _0^t\nabla^- g_n(s,X^*(s))dX^*(s)\nonumber\\
&&
+{1\over 2}\int _0^t\Delta  ^-g_n(s,X^*(s))d<X^*,X^*>_s\nonumber\\
&&+
\int _0^t(\nabla g_n(s,0+)-\nabla g_n(s,0-))d_sL  _s^*(0)\ \ a.s. 
\end{eqnarray}
We obtain the desired formula by passing to 
the limits using (\ref{ky1}), (\ref{ky2}), (\ref{ky3}) and (\ref{ky4}).  
 $\hfill\diamond$
 
 \begin{rmk} (i) Formula (\ref{ito11}) was also observed by Peskir in \cite{pe} 
 and \cite{gp} independently.
 
(ii) From the proof of Theorem \ref{tom100}, one can take different mollifications, 
e.g. one can take (\ref{truman41}) as 
\begin{eqnarray*}
f_n(s,x)=\int _0^2\int _0^2\rho(\tau)\rho(z)f(s+{\tau\over n},x+{z\over n})d\tau dz, \ \ n\geq 1.
\end{eqnarray*}
This will lead to as $n\to \infty$,
\begin{eqnarray*}
{\partial\over \partial s} f_n(s,x)\to {\partial^+\over \partial s} f(s,x)
\end{eqnarray*}
instead of (\ref{zhao2}), if $ {\partial^+\over \partial s} f(s,x)$ is jointly right continuous. 
 Therefore we have the following more general It$\hat  o$'s formula
\begin{eqnarray*}\label{truman42}
f(t,X(t))
&=&f(0,z)+\int _0^t{\partial ^{s_1}\over \partial s} f(s,X(s))ds+\int _0^t\nabla ^{s_2}f(s,X(s))dX(s)\nonumber\\
&&
+{1\over 2}\int _0^t\Delta ^{s_2} f_h(s,X(s))d<X>_s\nonumber\\
&&
+\int _{-\infty}^{\infty }L  _t(x){\rm d}_x\nabla ^{s_2}f_v(t,x)
-\int _{-\infty}^{+\infty}\int _0^{t}L  _s(x)
{\bf \rm d}_{s,x}\nabla ^{s_2}f_v(s,x)\ \ a.s.,
\end{eqnarray*}
where $s_1=\pm$ and $s_2=\pm $. 
\end{rmk}
\vskip10pt
 
 Formula (\ref{krylov6}) is in a very general form. It includes
the  classical It$\hat{\rm o}$ formula, Tanaka's formula,
 Meyer's formula for convex functions, the formula given by Az\'ema, Jeulin, Knight and Yor \cite{azema}
 and formula (\ref{ito11}).   In the following we will give some
 examples for which (\ref{ito11}) and some known  generalized It${\hat {\rm o}}$ formulae do
 not immediately apply, but formula (\ref{krylov6}) can be applied.
 These examples can be presented in different forms to include local times on curves.
 \vskip10pt

 \begin{exap}\label{example1} Consider the function 
 \begin{eqnarray*}
 f(t,x)=(\sin \pi x\sin\pi t)^+.
 \end{eqnarray*}
 Then
 \begin{eqnarray*}
 \nabla ^-f(t,x)=\pi \cos \pi x\sin\pi t1_{\sin \pi x\sin\pi t>0}.
 \end{eqnarray*}
 One can verify that $\nabla ^-f(t,x)$ is of locally bounded variation in $(t,x)$. This can be easily seen
 from Proposition \ref{prop1} and the simple fact that 
 %\begin{eqnarray*}
\begin{eqnarray*}
&&  \cos \pi x\sin\pi t 1_{\sin \pi x\sin \pi t>0}\\
& =&
\left\{ \begin{array}{l}
\cos \pi x\sin\pi t, \ {\rm if}\  i\leq t< i+1, 
 j\leq x< j+1, i+j \ {\rm is \ even}
\\
0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
{\rm otherwise}
 \end{array} \right.
\end{eqnarray*}
 %\end{eqnarray*}
 Therefore 
 \begin{eqnarray*}
 (\sin \pi X(t)\sin\pi t)^+&=&\pi \int _0^t \cos \pi s\sin\pi X(s)1_{\sin \pi X(s)\sin\pi s>0}ds
 \\
 &&+\pi \int _0^t \cos \pi X(s)\sin\pi s1_{\sin \pi X(s)\sin\pi s>0}dX(s)\\
 &&
+\pi \sin \pi t \int _{-\infty}^{\infty}L  _t(a){\rm d_a}(\cos \pi a 1_{\sin \pi a\sin\pi t>0})\\
&&-\pi
\int _0^t \int _{-\infty}^{\infty}L  _s(a){\rm d_{s,a}}(\cos \pi a\sin\pi s1_{\sin \pi a\sin\pi s>0}).
 \end{eqnarray*}
 One can expand the last two integrals to see the jump of $\cos \pi a\sin\pi s1_{\sin \pi a\sin\pi s>0}$.
  \end{exap}
 
 Note in example \ref{example1}, $\nabla ^-f(t,x)$ has jump on the boundary of each interval
 $i\leq t<i+1$, $j\leq x< j+1$. One can use this example as a prototype to construct 
 many other examples with other types of derivative jumps.
 
 \begin{exap} Consider the function 
 \begin{eqnarray*}
 f(t,x)=(\sin \pi x)^{1\over 3} (\sin \pi x\sin\pi t)^+.
 \end{eqnarray*}
 Then
 \begin{eqnarray*}
 \nabla ^-f(t,x)&=&{1\over 3}\pi \cos \pi x(\sin\pi x)^{-{2\over 3}} (\sin \pi x\sin\pi t)^+\\
 && +
 \pi (\sin \pi x)^{1\over 3}\cos \pi x\sin\pi t1_{\sin \pi x\sin\pi t>0}.
 \end{eqnarray*}
 One can verify that $\nabla ^-f(t,x)$ is of locally bounded variation in $(t,x)$ and continuous. In fact, 
 \begin{eqnarray*}
&&
 \cos \pi x(\sin\pi x)^{-{2\over 3}} (\sin \pi x\sin\pi t)^+ \\
 &=&\left\{ \begin{array}{l}
 \cos \pi x(\sin\pi x)^{{1\over 3}} \sin\pi t , \ {\rm if}\  i\leq t< i+1,\\
 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ j\leq x< j+1, i+j \ {\rm is \ even}
\\
0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
{\rm otherwise}.
 \end{array} \right.
\end{eqnarray*}
then it is easy to see that $\cos \pi x(\sin\pi x)^{-{2\over 3}} (\sin \pi x\sin\pi t)^+$ is 
of locally bounded variation in $(t,x)$ using proposition \ref{prop1}. Similarly one can see that 
$(\sin \pi x)^{1\over 3}\linebreak \cos \pi x \sin\pi t1_{\sin \pi x\sin\pi t>0}$ is of locally 
bounded variation in $(t,x)$ as well. 

 Note 
 $\Delta ^-f(t,x)$ blows up when $x$ is near an integer value, and their left and right limits
 also blow up. However one can apply our generalized It${\hat{\rm o}}$'s formula (\ref{krylov6})
 to this function so that 
 \begin{eqnarray*}
 &&
 (\sin \pi X(t))^{1\over 3} (\sin \pi X(t)sin\pi t)^+\nonumber\\
 &=&\pi \int _0^t  (\sin \pi X(s))^{4\over 3} \cos \pi s 1_{\sin \pi X(s)\sin\pi s>0}{\rm ds}\nonumber\\
&& +\int_{-\infty}^{\infty}L_t(a){\rm d_a}({1\over 3}\pi \cos \pi a(\sin\pi a)^{-{2\over 3}} (\sin \pi a\sin\pi t)^+\nonumber \\
&& \hskip3cm +
 \pi (\sin \pi a)^{1\over 3}\cos \pi a\sin\pi t1_{\sin \pi a\sin\pi t>0})\nonumber\\
 &&
 -\int _0^t\int _{-\infty}^{\infty}L_s(a){\rm d_{s,a}}({1\over 3}\pi \cos \pi a(\sin\pi a)^{-{2\over 3}} (\sin \pi a\sin\pi s)^+\nonumber\\
 &&\hskip3cm +
 \pi 
 (\sin \pi a)^{1\over 3} \cos \pi a\sin\pi s1_{\sin \pi a\sin\pi s>0}).
 \end{eqnarray*}
 
  \end{exap}

\section{The case for It${\hat {\rm o}}$ processes}

\setcounter{equation}{0}

For It${\hat {\rm o}}$ processes, we can allow some of the generalized derivatives of $f$ to be 
only in 
$L^2_{loc}(dtdx)$. Consider
\begin{eqnarray}\label{krylov8}
X(t)=X(0)+\int_0^t\sigma_rdW_r+\int _0^tb_rdr.
\end{eqnarray}
Here $W_r$ is a one-dimensional Brownian motion on a filtered probability space $(\Omega,{\cal F},
\{{\cal F}_r\}_{r\geq 0}, P)$ and $\sigma_r$ and $b_r$ are progressively measurable with respect to 
$\{{\cal F}_r\}$ and satisfy the following conditions: 
for all $t>0$
\begin{eqnarray}\label{krylov1}
\int _0^t|\sigma _r|^2dr<\infty, \ \ \int _0^t|b _r|dr<\infty \ \ a.s.
\end{eqnarray}
Under condition (\ref{krylov1}), the process (\ref{krylov8}) is well defined.
For any $N>0$, define $\tau_N=\inf\{s:|X(s)|\geq N\}$. Assume 
there exist constants $\delta >0$ and $K>0$ 
such that  ,
\begin{eqnarray}\label{krylov9}
\sigma _t(\omega)\geq \delta >0,\ \ |\sigma _t(\omega)|+|b_t(\omega)|\leq K,\ \ 
{\rm for\ all}\  (t,\omega) {\rm \ with \ } t\leq \tau _N.
\end{eqnarray}
The following inequality due to Krylov \cite{krylov} plays an important role. 

\begin{lem}\label{krylov2}
Assume condition (\ref{krylov1}) and (\ref{krylov9}). Then there exists a constant $M>0$, depending 
only on $\delta$ and $K$ such that
\begin{eqnarray}\label{krylov4}
E\int _0^{t\wedge\tau_N} |f(r,X(r))|dr\leq M(\int_0^t\int _{-N}^{+N}(f(r,x))^2drdx)^{1\over 2}.
\end{eqnarray}
\end{lem} 
%{\em Proof}. The first one (\ref{krylov3}) is due to 
%Kurylov \cite{krylov} and the second one (\ref{krylov4}) is an easy corollary of (\ref{krylov3})
%by noting
%\begin{eqnarray}
%E\int _0^{t\wedge\tau_N}|f(r,X(r))|dt=E\int _0^{t}1_{|X(r)|\leq N}|f(r,X(r))|dr.
%\end{eqnarray}
%$\hfill\diamond$

Denote again by $L_t(x)$ the local time of the diffusion process $X(t)$ at level $x$.
We can prove the following theorem.

\begin{thm}\label{krylov8}
Assume $f(t,x)$ is continuous with 
generalized derivative ${\partial \over \partial t}f$  in $L^2_{loc}(dtdx)$
and generalized derivative $\nabla f$ of 
 locally bounded variation in $(t, x)$ and of locally bounded variation in $x$ 
 for $t=0$. Consider  an It${\hat {\rm o}}$
 process $X(t)$ given by (\ref{krylov8}) with $\sigma$ and $b$ satisfying (\ref{krylov1}) and (\ref{krylov9}).
Then  a.s. 
\begin{eqnarray}
f(t,X(t))
&=&f(0,X(0))+\int _0^t{\partial \over \partial s} f(s,X(s))ds+\int _0^t
 \nabla f(s,X(s))dX(s)\nonumber\\
&&
+
\int _{-\infty}^{\infty }L  _t(x){\rm d}_x\nabla f(t,x)-\int _{-\infty}^{+\infty}\int _0^{t}L  _s(x)
{\bf \rm d}_{s,x}\nabla f(s,x).
\end{eqnarray}
\end{thm}
 \vskip5pt
{\em Proof}. Define $f_n$ by (\ref{krylov5}). From a well-known result on Sobolev
spaces (see Theorem 3.16, p.52 in \cite{adams}),  we know that as  $n\to \infty$, 
\begin{eqnarray*}
f_n(t,x) &\to & f(t,x) ,
\end{eqnarray*}
for all $(t,x)$ and for any $N>0$
\begin{eqnarray*}
{\partial \over \partial t}f_n&\to & {\partial  \over \partial t} f,\label{zhao02} {\rm \ in}\  L^2([0,t]\times [-N,N])\\
\nabla f_n&\to & \nabla f,\label{zhao03} {\rm \ in}\   L^4([0,t]\times [-N,N]).
\end{eqnarray*}
% Then by Chebyshev inequality, we know that above
 %convergences hold in measure. Therefore there is a subsequence, still
 %denoted by $n$, such that convergences (\ref{zhao01}), (\ref{zhao02}) and
 %(\ref{zhao03}) hold for almost all $s$ and $x$.
%It is also easy to check that $f_n(s,x)$, ${\partial \over \partial s} f_n(s,x)$, 
%$\nabla f_n(s,x)$ are bounded uniformly in $n$.

As in the proof of Theorem \ref{tom100}, we have the It$\hat {\rm o}$ formula (\ref{zhao11}) for 
$f_n(t\wedge \tau _N, X(t\wedge \tau _N))$. The convergence of the terms 
$f_n(t\wedge \tau _N, X(t\wedge \tau _N))$, and ${1\over 2}
\int _0^{t\wedge \tau _N}\sigma _s^2\Delta f_n(s,X(s))ds$ is the same as before. Now by using Lemma \ref{krylov2},
\begin{eqnarray*}
&&
E|\int _0^{t\wedge\tau_N}{\partial \over \partial s} f_n(s,X(s))ds-\int _0^{t\wedge\tau_N}{\partial \over \partial s} f(s,X(s))ds|\nonumber\\
&\leq &
E\int _0^{t\wedge\tau_N}|{\partial \over \partial s} f_n(s,X(s))-{\partial \over \partial s} f(s,X(s))|ds\nonumber\\
&\leq &
M(\int _0^t\int _{-N}^{N}({\partial \over \partial s} f_n(s,x)-{\partial \over \partial s} f(s,x))^2dsdx)^{1\over 2}\to 0
\end{eqnarray*}
as $n\to \infty$. 
%Therefore 
%$
%\int _0^{t\wedge\tau_N}{\partial \over \partial s} f_n(s,X(s))ds\to \int _0^{t\wedge\tau_N}{\partial \over %\partial s} f(s,X(s))ds$ 
%in probability. Thus, there exists a subsequence, still denote by $n$, such that 
%\begin{eqnarray}
%\int _0^{t\wedge\tau_N}{\partial \over \partial s} f_n(s,X(s))ds\to \int _0^{t\wedge\tau_N}{\partial \over %\partial s} f(s,X(s))ds
%\end{eqnarray}
%as $n\to \infty$ almost surely. 
% Then by the standard localization argument we have as $n\to \infty$, 
%\begin{eqnarray}
%\int _0^t{\partial \over \partial s} f_n(s,X(s))ds\to \int _0^t{\partial \over \partial s} f(s,X(s))ds,
%\end{eqnarray}
%almost surely. 
Similarly one can prove
\begin{eqnarray*}
\int _0^{t\wedge\tau_N} b_s\nabla f_n(s,X(s))ds\to \int _0^{t\wedge\tau_N}b_s\nabla f(s,X(s))ds \ \ in \ \ L^1(dP).
\end{eqnarray*}
Moreover, there exists a constant $M>0$ such that 
\begin{eqnarray*}
&&
E(\int _0^{t\wedge\tau_N}\sigma _s \nabla f_n(s,X(s))dW_s-\int _0^t\sigma _s\nabla f(s,X(s))dW_s)^2\nonumber
\\
&=&
E(\int _0^{t\wedge\tau_N}\sigma _s ^2(\nabla f_n(s,X(s))-\nabla f(s,X(s)))^2ds\nonumber\\
&\leq &
M(\int _0^t\int _{-N}^{N}(\nabla f_n(s,x)-\nabla f(s,x))^4dsdx\to 0
\end{eqnarray*}
as $n\to \infty$. 
%Therefore there exists subsequence, still denote by $n$, such that as $n\to \infty$,
%\begin{eqnarray}
%\int _0^{t\wedge\tau_N}\sigma _s \nabla f_n(s,X(s))dW_s\to \int _0^{t\wedge\tau_N}\sigma _s\nabla %f(s,X(s))dW_s
%\end{eqnarray} almost surely. 
%By standard localization argument again,
%\begin{eqnarray}
%\int _0^t\sigma _s \nabla f_n(s,X(s))dW_s\to \int _0^t\sigma _s\nabla f(s,X(s))dW_s,
%\end{eqnarray}
%almost surely.
Therefore we have proved that 
\begin{eqnarray*}
&&
f(t\wedge\tau_N,X(t\wedge\tau_N))\nonumber\\
&=&f(0,X(0))+\int _0^{t\wedge\tau_N}{\partial \over \partial s} f(s,X(s))ds+\int _0^{t\wedge\tau_N}
 \nabla f(s,X(s))dX(s)\\
&&
+
\int _{-\infty}^{\infty }L  _{t\wedge\tau_N}(x){\rm d}_x\nabla f(t\wedge\tau_N,x)-\int _{-\infty}^{+\infty}\int _0^{t\wedge\tau_N}L  _s(x)
{\bf \rm d}_{s,x}\nabla f(s,x).\nonumber
\end{eqnarray*}
The desired formula follows. 
$\hfill\diamond$
\bigskip

Recall the following extension of It${\hat {\rm o}}$'s formula due to Krylov (\cite{krylov}): 
   if $f: 
 R^+\times R$ is $C^1$ in $x$ and $\nabla f$ is absolutely continuous with respect 
 to $x$ for each $t$ and the generalized derivatives ${\partial \over \partial s}f(s,x)$ and
 $\Delta f$ are in $L^2_{loc}(dtdx)$, then
 \begin{eqnarray}\label{ito14}
 f(t,X(t))
&=&f(0,z)+\int _0^t{\partial \over \partial s} f(s,X(s))ds+\int _0^t\nabla f(s,X(s))dX(s)\nonumber\\
&&
 +{1\over 2}\int _0^t\sigma _s^2\Delta f(s,X(s))ds
\ \ a.s.
 \end{eqnarray}
 
 The next theorem is an easy consequence of the method of proof of 
 Theorem \ref{krylov8} and of formula (\ref{ito14}).
 
\begin{thm} 
Assume $f(t,x)$ is continuous and 
its generalized derivative ${\partial \over \partial t} f$ is in $L^2_{loc}(dtdx)$. 
Moreover $f(t,x)=f_h(t,x)+f_v(t,x)$ with $f_h(t,x)$ being $C^1$ in $x$ and  
 $\nabla f_h(t,x)$ having generalized derivative 
 $\Delta f_h(t,x)$ in $L^2_{loc}(dtdx)$, and $f_v$ having generalized derivative
 $\nabla f_v(t,x)$ being of locally bounded variation in $(t, x)$ and of locally
 bounded variation in $x$ for $t=0$. Suppose $X(t)$ is an It${\hat {\rm o}}$
 process given by (\ref{krylov8}) 
with $\sigma$ and $b$ satisfying (\ref{krylov1}) and (\ref{krylov9}). 
Then,
\begin{eqnarray}\label{ito12}
f(t,X(t))
&=&f(0,X(0))+\int _0^t{\partial \over \partial s} f(s,X(s))ds+\int _0^t\nabla f(s,X(s))dX(s)\nonumber\\
&&
 +{1\over 2}\int _0^t\Delta f_h(s,X(s))d<X>_s+
\int _{-\infty}^{\infty }L  _t(x){\rm d}_x\nabla  f_v(t,x)\nonumber\\
 &&
-\int _{-\infty}^{+\infty}\int _0^{t}L  _s(x)
{\bf \rm d}_{s,x}\nabla f_v(s,x)\ \ a.s.
\end{eqnarray}
\end{thm}

%  \begin{exap} Let $\phi_i(s,x)$ be a $C^{1,2}$ function and of locally bounded variation
%   in $(s,x)$ with support $[i,i+1)$ in $s$, $i=0,1,2,\cdots$ and $\theta_i(x)$ is a function with
%   \begin{eqnarray*}
 %  \theta _i^{\prime}(x)=1_{i\leq x<i+1}.
 %  \end{eqnarray*}  
%Consider a function $f(s,x)=\sum_{i}\theta_i(x)\phi_i(s,x)$. Then
%\begin{eqnarray*}
%\nabla f(s,x)=\sum_{i}(1_{i\leq x<i+1}\phi _i(s,x)+\theta_i(x)\nabla \phi_i(s,x))
%\end{eqnarray*}
  % It is easy to see that $\sum_{i}1_{i\leq x<i+1}\phi _i(s,x)$ is of bounded variation. So the generalized 
  % It${\hat {\rm o}}$ formula, for a continuous semi-martingale $X(t)$, gives, 
   %\begin{eqnarray*}
   %&&
   %f(t,(X(t))\\
  % &=&f(0,X(0))+\sum_{i}\int _0^t\theta_i(X(s)){\partial \over \partial s}\phi_i(s,X(s))ds\nonumber\\
  % &&+\sum_{i}\int _0^t(1_{i\leq %X(s)<i+1}\phi_i(s,X(s))+\theta_i(X(s))\nabla\phi_i(s,X(s)))dX(s)\nonumber\\
  % &&+{1\over 2}\sum _{i}\int _0^t \nabla(\theta_i(X(s))\nabla\phi_i(s,X(s))d<X>_s\nonumber\\
  % &&+\sum_{i}\int_{-\infty}^{\infty}L  _t(a){\rm d}_a(\phi _i(t,a)1_{i\leq a<i+1})-
  % \sum_{i}\int _0^t\int_{-\infty}^{\infty}L  _s(a){\rm d}_{s,a}(\phi _i(s,a)1_{i\leq a<i+1})\nonumber\\
 %  &=&f(0,X(0))+\int _0^t\theta_{[s]}(X(s)){\partial \over \partial s}\phi_{[s]}(s,X(s))ds\nonumber\\
%   &&+\int _0^t(1_{{[s]}\leq %X(s)<{[s]}+1}\phi_{[s]}(s,X(s))+\theta_{[s]}(X(s))\nabla\phi_{[s]}(s,X(s)))dX(s)\nonumber\\
  % &&+{1\over 2}\int _0^t \nabla(\theta_{[s]}(X(s))\nabla\phi_{[s]}(s,X(s)))d<X>_s\nonumber\\
 %  &&+\int_{-\infty}^{\infty}L  _t(a){\rm d}_a(\phi _{[t]}(t,a)1_{[t]\leq a<{[t]}+1})-
 % \int _0^t\int_{-\infty}^{\infty}L  _{[s]}(a){\rm d}_{s,a}(\phi _{[s]}(s,a)1_{{[s]}\leq a<{[s]}+1}).
  % \end{eqnarray*}
  % Here by $[s]$ we denote the integer part of $s$.
  % \end{exap}
 %There has been progress of generalized It$\hat {\rm o}$'s formula in two dimensions and
  %stochastic Lebesgue-Stieltjes integrals. See
  %\cite{fz1}.
  \bigskip
  
{\bf Acknowledgment}
\vskip5pt

\footnotesize

It is our great pleasure to thank M. Chen, M. Freidlin, Z. Ma, S. Mohammed, B. \O ksendal, S. Peng, 
L.M. Wu, J.A. Yan, 
M. Yor and W.A. Zheng for useful conversations. 
 The first version of this paper was presented in the Kautokeino
Stochastic Analysis Workshop in July 2001 organized by B. \O ksendal.
We would also like to thank S. Albeverio, Z. Ma, 
M. R\"ockner for inviting us to the Sino-German Stochastic Analysis
Meeting (Beijing 2002). HZ would like to thank T.S. Zhang and 
T. Lyons for invitations to Manchester and Oxford respectively to
present the results of this paper.  One version of this paper was also presented in
Swansea Workshop on Probabilistic Methods in Fluid in April 2002 and the final  version 
in Warwick SPDEs Workshop in August 2003 and Mini-workshop of Local Time-Space
Calculus with Applications in Obverwolfach in May 2004. We would like to thank
G. Peskir and N. Eisenbaum for invitations to the Oberwolfach conference.
We would like to thank Y. Liu, C.R. Feng and B. Zhou for reading the manuscript and 
making some valuable suggestions. It is our pleasure to thank the referee
for useful comments. This project is partially supported by EPSRC grants
GR/R69518 and GR/R93582.

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%(Springer, Berlin-New York,  pp. 237--285 (1982)).

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\bibitem{brosamler} Brosamler, G.A.: Quadratic variation of potentials and harmonic functions.
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%equations and their singularities I--geometrical properties}, \Journal{Journal of 
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