Rough asymptotics of forward-backward stochastic differential equations (Q2712229)

Given a Brownian motion \(W\), the authors of the present paper consider the forward-backward stochastic differential equation [the reader is referred to \textit{J. Ma} and \textit{J. Yong}, ``Forward-backward stochastic differential equations and their applications'' (1999; Zbl 0927.60004) for an overview] NEWLINE\[NEWLINEdX^\varepsilon_t= b(t, \theta^\varepsilon_t) dt+ \sqrt\varepsilon\sigma(t, X^\varepsilon_t, Y^\varepsilon_t) dW_t,\quad dY^\varepsilon_t= -f(t, \theta^\varepsilon_t) dt+ Z^\varepsilon_t dW_t,NEWLINE\]NEWLINE \(t\in [0,T]\), \(X_0= x\), \(Y_T= g(X_T)\), with \(\theta^\varepsilon= (X^\varepsilon_t, Y^\varepsilon_t, Z^\varepsilon_t)\), and show under standard assumptions, namely the strict ellipticity of \(\sigma\), and some ``hypothesis A'' that the distributions of \(\{\theta^\varepsilon\}\) satisfy the large deviation principle, as \(\varepsilon\to 0\). They prove that their ``hypothesis A'' is satisfied when the time horizon \(T> 0\) is small enough, or, for arbitrary \(T> 0\), if the supplementary condition \(b(t,x,y,z)\equiv 0\) holds. As an application the authors apply their large deviation principle results for the existence of an asymptotically efficient estimator in the class importance sampling estimators.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00050].