A short proof of a martingale representation result (Q1103266)

Let \(W=(W^ 1\),..., \(W^ n)\) be an n-dimensional Brownian motion on (\(\Omega\), F, P) and consider the stochastic differential equation \[ (1)\quad dx_ t=f(t,x_ t)dt+\sigma (t,x_ t)dw_ t \] where \(f: [0,\infty)\times R^ n\to R^ n\) and \(\sigma\) : \([0,\infty)\times R^ n\to R^ n\times R^ n\) are measurable and three times differentiable in \(x\in R^ n\). If \(\xi_{t,s}(x)\) is the solution of (1) for \(t\geq s\), having initial condition \(\xi_{t,t}(x)=x\), then results of \textit{J. M. Bismut} [Mécanique aléatoires. (1981; Zbl 0457.60002)] prove that there is a null set \(N\subset \Omega\) such that for \(\omega\not\in N\), there is a version of \(\xi_{t,s}(x)\) twice differentiable in x, continuous in t and s. Now consider \(0\leq t\leq T\), \(x_ 0\in R^ n\), initial condition at \(t=0\), and a function \(c(\xi_{0,T}(x_ 0))\) where c is differentiable and such that \(c(\xi_{0,T}(x_ 0))\) is integrable. If \(\underset {=} F_ t\) is the right-continuous complete family of \(\sigma\)-fields generated by \(\sigma (w_ s:\) \(s\leq t)\), we obtain by the martingale representation theorem \[ M_ t=E(c(\xi_{0,T}(x_ 0))/\underset {=} F_ t)=M_ 0+\int^{t}_{0}\gamma_ sdw_ s. \] The authors give then in this case a very short proof of the expression of \(\gamma_ s\), using the properties of stochastic flows and the unique decomposition of special semi-martingales.