Approximation of Markov semigroups in total variation distance

The Girsanov Theorem Without (So Much) Stochastic Analysis

“…Remark 3. For a closely related approach with semi-groups, the article [5] provides us with some conditions in a more general context to exhibit the rate of convergence of Euler schemes.…”

“…With the discrete time approximation, both expressions leads to different expressions for the weights, whose limit is however the same. The expression obtained by the Euler scheme is a discretization of the stochastic integral in the exponential weight given by (5). The ones obtained using the splitting is a discretization of the SDE (6) the exponential weights solve.…”

Nonlinear trend exclusion procedure for models defined by stochastic differential and difference equations

“…Markov Chain (20) obtained by application of the procedure of the excluding nonlinear trend is different from the original Markov chain (12).The difference is that trend and diffusion now depend on innovation ε(t n k+1 ). This more general case of Markov chains studied in [BaR16]. These authors considered the following class of Markov chains…”

“…The main result of [BaR16] obtained on the assumption (1.9) -(1.11) that are not made for the model (12) because of an unbounded function F .Let α = γ = 0, β = 1 in (1.10) then the condition is not satisfied. For the model (20) we can specify a class of models with infinitely differentiable functions F, m and σ, for which these conditions are satisfied and we can use the results of [BaR16] for this model.…”

Absolute continuity of the invariant measure in piecewise deterministic Markov Processes having degenerate jumps

“…The main difference with his work is that he also imposes the preservation of smoothness property for the transition kernel. Using splitting methods to create noise has also been applied in Bally and Rey (2015) [4], however in a different context.…”