We extend the taming techniques for explicit Euler approximations of stochastic differential equations (SDEs) driven by Lévy noise with super-linearly growing drift coefficients. Strong convergence results are presented for the case of locally Lipschitz coefficients. Moreover, rate of convergence results are obtained in agreement with classical literature when the local Lipschitz continuity assumptions are replaced by global and, in addition, the drift coefficients satisfy polynomial Lipschitz continuity. Finally, we further extend these techniques to the case of delay equations. equations (SDDEs) driven by Lévy noise. The link between delay equations and random coefficients utilises ideas from [7]. The aforementioned results are derived under the assumptions of one-sided local Lipschitz condition on drift and local Lipschitz conditions on both diffusion and jump coefficients with respect to non-delay variables, whereas these coefficients are only asked to be continuous with respect to arguments corresponding to delay variables. It is worth mentioning here that our approach allows one to use our schemes to approximate SDDEs with jumps when drift coefficients can have super-linear growth in both delay and non-delay arguments. Thus, the proposed tamed Euler schemes provide significant improvements over the existing results available on numerical techniques of SDDEs, for example, [1,15]. It should also be noted that, by adopting the approach of [7], we prove the existence of a unique solution to the SDDEs driven by Lévy noise under more relaxed conditions than those existing in the literature, for example, [13] whereby we ask for the local Lipschitz continuity only with respect to the non-delay variables.Finally, rate of convergence results are obtained (which are in agreement with classical literature) when the local Lipschitz continuity assumptions are replaced by global and, in addition, the drift coefficients satisfy polynomial Lipschitz continuity. Similar results are also obtained for delay equations when the following assumptions hold -(a) drift coefficients satisfy one-sided Lipschitz and polynomial Lipschitz conditions in non-delay variables whereas polynomial Lipschitz conditions in delay variables and (b) diffusion and jump coefficients satisfy Lipschitz conditions in non-delay variables whereas polynomial Lipschitz conditions in delay variables. This finding is itself a significant improvement over recent results in the area, see for example [1] and references therein.We conclude this section by introducing some basic notation. For a vector x ∈ R d , we write |x| for its Euclidean norm and for a d × m matrix σ, we write |σ| for its Hilbert-Schmidt norm and σ * for its transpose. Also for x, y ∈ R d , xy denotes the inner product of these two vectors. Further, the indicator function of a set A is denoted by I A , whereas [x] stands for the integer part of a real number x. Let P be the predictable sigma-algebra on Ω × R + and B(V ), the sigma-algebra of Borel sets of a topological space V . Also, let T > 0...

A new class of explicit Milstein schemes, which approximate stochastic differential equations (SDEs) with superlinearly growing drift and diffusion coefficients, is proposed in this article. It is shown, under very mild conditions, that these explicit schemes converge in L p to the solution of the corresponding SDEs with optimal rate.

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In this paper we estimate the expected tracking error of a fixed gain stochastic approximation scheme. The underlying process is not assumed Markovian, a mixing condition is required instead. Furthermore, the updating function may be discontinuous in the parameter.MSC 2010 subject classification: Primary: 62L20; secondary: 93E15, 93E35 with some fixed measurable function g : X − → m . Clearly, X is a (strongly) stationary m -valued process, see Lemma 10.1 of [24].Remark 3.1. We remark that, in the present setting, the CLC property holds if, for all θ 1 , θ 2 ∈ D,due to the fact that the law of (X k+1 , ǫ k , ǫ k−1 , . . .) is the same as that of (X 1 , ǫ 0 , ǫ −1 , . . .), for all k ∈ .Define G(θ ) := EH(θ , X 0 ). Note that, by stationarity of X , G(θ ) = EH(θ , X t ) for all t ∈ . We need some stability hypotheses formulated in terms of an ordinary differential equation related to G. Assumption 3.2. On D, the function G is twice continuously differentiable and bounded, together with its first and second derivatives.

We extend the taming techniques developed in [3,19] to construct explicit Milstein schemes that numerically approximate Lévy driven stochastic differential equations with super-linearly growing drift coefficients. The classical rate of convergence is recovered when the first derivative of the drift coefficient satisfies a polynomial Lipschitz condition.