backward and forward operators - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T09:12:06Z http://mathoverflow.net/feeds/question/33163 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/33163/backward-and-forward-operators backward and forward operators Steven 2010-07-24T02:27:49Z 2010-07-24T02:27:49Z <p>consider stochastic differential equation : $$d\xi(t)=b(t,\xi(t))dt+\sigma(t,\xi(t))dw$$ $t\in[s,T]$ and $\xi(s)=y$</p> <p>where $w$ is a d-dimensional standard Brownian motion, $b\in \mathbb{R}^n$ and $\sigma\in \mathbb{R}^{n\times d}$. The <em>backward operator</em> is defined to be $$\frac{\partial}{\partial s}+A(s)=\frac{\partial}{\partial s}+\frac{1}{2}\sum_{i,j}a_{i,j}(s,y)\frac{\partial^2}{\partial y_i\partial y_j}+\sum_{i}b_{i}(s,y)\frac{\partial}{\partial y_i}$$</p> <p>the <em>forward operator</em> is defined to be :</p> <p>$$-\frac{\partial}{\partial s}+A^*(s)$$</p> <p>where $$A^*(s)q:=\frac{1}{2}\sum_{i,j}\frac{\partial^2}{\partial y_i\partial y_j}[a_{i,j}(s,y)q]+\sum_{i}\frac{\partial}{\partial y_i}[b_{i}(s,y)q]$$</p> <p>asumme that $P_{s,y}(\xi(t)\in B)=\int_{B}p(s,y,t,x)dx$ for all $B\in$ <strong><em>B</em></strong>$(\mathbb{R}^n)$ : the set of all Borel sets of $\mathbb{R}^n$</p> <p>let $Q_{s,T}=(s,T)\times\mathbb{R}^{n}$ and $\phi\in C^{1,2}$ that has compact support in $Q_{s,T}$</p> <p>CLAIM : $$\int_{Q_{s,T}}[A(t)\phi] p dxdt=\int_{Q_{s,T}}\phi[ A^*(t)p]dxdt$$</p> <p>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? </p> <p>Thanks for your time</p>