A Robust Weak Taylor Approximation Scheme for Solutions of Jump-Diffusion Stochastic Delay Differential Equations

and Applied Analysis 3 Now we present some lemmas to be used later for the proof of the convergence theorem. Consider a right continuous process YΔ l = {YΔ (t), t ∈ [−γ, T]}. YΔ l is called a discrete-time numerical approximationwithmaximum step size Δ l , if it is obtained by using a time discretization t Δ l , and the random variable YΔ l tn is F tn -measurable for n ∈ {1, . . . , N}. Further, YΔ l tn+1 can be expressed as a function of YΔ l t−l , Y Δ l t−l+1 , . . . , Y Δ l tn and the discrete-time t n . Because of dealing with the approximation of solutions of jump-diffusion SDDEs, we introduce a concept of weak order convergence due to Kloeden and Platen [15]. Definition 3. A discrete-time approximation YΔ l converges weakly towards X at time T with order β > 0 if for each g ∈ C p there is a constant C, independent of Δ l , such that 󵄨󵄨󵄨󵄨 E (g (X (T))) − E (g (Y Δ l (T))) 󵄨󵄨󵄨󵄨 ≤ C(Δ l ) β , (8) whereC p denotes the set of all polynomials g : Rd → R. We now give some auxiliary results to prepare for the proof of the weak convergence theorem to be presented. Lemma 4. For n ∈ {−l + 1, . . . , 0, 1, . . . , N} and z ∈ R, we have E(u (n,Xn−1,z n − ) − u (n − 1, z)