Applications of the Malliavin Calculus, Part I

Harnack Inequalities and Gaussian Estimates for a Class of Hypoelliptic Operators

“…We recall the well-known Gaussian upper and lower bounds for heat kernels due to Jerison and Sánchez-Calle [5], Kusuoka and Stroock [10], Varopoulos, Saloff-Coste and Coulhon [20]. These results apply to Lie groups which are not necessarily homogeneous.…”

A cubature based algorithm to solve decoupled McKean–Vlasov forward–backward stochastic differential equations

“…A reader familiarized with the cubature method might wonder why we assume uniform ellipticity instead of the weaker UFG condition usually needed to apply the method with Lipschitz boundary conditions. The reason is that the smoothing results of Kusuoka and Stroock [KS87] hold for space dependent vector fields only, and therefore do not apply directly to our framework with a time dependence coming from the McKean term. There is some extension in the time inhomogeneous case that do not include derivatives in the V 0 direction (see for example [CM02] and references therein), but, to the best of our knowledge, there is no result that could be applied to our framework.…”

“…As it is pointed out in [LV04] and [CM10b], the regularity of the terminal condition φ in (1.1) may be relaxed to Lipschitz and the approximation convergence rate preserved, provided that the vector fields are uniformly non-degenerate (in fact, the condition in the given references is weaker, since the vector fields are supposed to satisfy an UFG condition, see [KS87]). This relies on the regularization properties of parabolic and semi-linear parabolic PDEs (see [Fri08] for an overview in the elliptic case and respectively [KS87] and [CD12c] for the UFG case).…”

“…This relies on the regularization properties of parabolic and semi-linear parabolic PDEs (see [Fri08] for an overview in the elliptic case and respectively [KS87] and [CD12c] for the UFG case). We show that this remains true in the McKean-Vlasov case and that the convergence rate still holds when the function φ is Lipschitz only and when the vector fields are uniformly elliptic.…”

A Note on Hypoellipticity of Degenerate Elliptic Operators

“…Concerning operators closely related to L, criteria for the hypoellipticity have been recently given by several authors See Fedii [2], Hoshiro [4,5], Kusuoka and Stroock [6] and Morimoto [7,8,9,10]. In particular, Hoshiro considered the same operator as L with the assumptions (A.I) and (A.2) both of / and g are non-decreasing in [0, S), and non-increasing in (-3,0].…”