Large deviation principle for diffusion processes under a sublinear expectation
The authors consider the one-dimensional stochastic differential equation (SDE) \(dx^{\varepsilon}_t=b(x^{\varepsilon}_t)dt +\sigma(x^{\varepsilon}_t)dW(t)\) on \([0,T]\), where \(W\) is the standard Brownian motion on \((\Omega,F,\operatorname{P})\). On \((\Omega,F)\), they define for each \(k \geq 0\) a class of probability measures \({\mathcal{P}}_k\) containing \(\operatorname{P}\) and the associated sub-linear probability \(\operatorname{P}_k\) by \(\operatorname{P}_k(A):=\sup_{Q \in {\mathcal{P}}_k} \operatorname{E}_Q 1_A\). The authors show how \(\operatorname{P}_k\) can be defined via an associated backward SDE. They establish a large deviation property for \(\operatorname{P}_k(X_T \in .)\) similar to the classical case using the Laplace principle approach.
backward stochastic differential equation
scientific article; zbMATH DE number 6131941
scientific article