Solving BSDE with adaptive control variate. A note on the rate of convergence of the operator P

This note is a complement of the paper " Solving BSDE with adaptive control variate " [1]. It deals with the convergence of the approximating operator P, based on a non parametric regression technique called local averaging, and defined in Definition 1.1. Although the computations are quite standard (see [3], [2]), the specificities of the paper are the following • the support of the variables is unbounded; • the error has to be measured using specific L 2-norms; • errors on the gradient are provided. Let us first introduce some notations • Let C k,l b be the set of continuously differentiable functions φ : (t, x) ∈ [0, T ] × R d with continuous and uniformly bounded derivatives w.r.t. t (resp. w.r.t. x) up to order k (resp. up to order l). • C k p denotes the set of C k−1 functions whose k-th derivative is piecewise continuous. • Functions K(T). K(·) denotes a generic function non decreasing in T which may depend on d, µ, β, on the coefficients b and σ (through σ 0 , σ 1 , c 1,3 (σ), c 0,1 (∂ t σ), c 1,3 (b)) and on other constants appearing in [1, Appendix A]. The parameter β is defined in [1, Section 2.1], µ is defined in [1, Section 3.2], σ 0 and σ 1 are defined in [1, Hypothesis 1].. • Functions K 0 (T). K 0 (T) are analogous to K(T) except that they may also depend on the operator P (through c 1 (K t) and c 2 (K x), defined in Section [1, Section 7].