backward and forward operators

2010-07-24T02:27:49Z

consider stochastic differential equation : $$d\xi(t)=b(t,\xi(t))dt+\sigma(t,\xi(t))dw$$ $t\in[s,T]$ and $\xi(s)=y$

where $w$ is a d-dimensional standard Brownian motion, $b\in \mathbb{R}^n$ and $\sigma\in \mathbb{R}^{n\times d}$. The backward operator is defined to be $$\frac{\partial}{\partial s}+A(s)=\frac{\partial}{\partial s}+\frac{1}{2}\sum_{i,j}a_{i,j}(s,y)\frac{\partial^2}{\partial y_i\partial y_j}+\sum_{i}b_{i}(s,y)\frac{\partial}{\partial y_i}$$

the forward operator is defined to be :

$$-\frac{\partial}{\partial s}+A^*(s)$$

where $$A^*(s)q:=\frac{1}{2}\sum_{i,j}\frac{\partial^2}{\partial y_i\partial y_j}[a_{i,j}(s,y)q]+\sum_{i}\frac{\partial}{\partial y_i}[b_{i}(s,y)q]$$

asumme that $P_{s,y}(\xi(t)\in B)=\int_{B}p(s,y,t,x)dx$ for all $B\in$ B$(\mathbb{R}^n)$ : the set of all Borel sets of $\mathbb{R}^n$

let $Q_{s,T}=(s,T)\times\mathbb{R}^{n}$ and $\phi\in C^{1,2}$ that has compact support in $Q_{s,T}$

CLAIM : $$ \int_{Q_{s,T}}[A(t)\phi] p dxdt=\int_{Q_{s,T}}\phi[ A^*(t)p]dxdt$$

so far I could showed that the above identity is true in case of 1-dimension, but not so satisfied with my argument. I am wondering that whether there is a neat way to see it?

Thanks for your time

backward and forward operators - MathOverflow

backward and forward operators