Application of normal cones to the computation of solutions of the nonlinear Kolmogorov backward equation

The Arsenal of Perturbation Bounds for Finite Continuous-Time Markov Chains: A Perspective

“…However, there is another important conceptual difference between Equations (3) and (4). Specifically, in Equation (3), the condition numbers κ 1 (T) and κ 2 (T) depend on the parameters of X via ∥Q∥; in other words, Equation (3) distinguishes between more well-conditioned and less well-conditioned Markov chains. However, Equation (4) does not make that distinction, and its condition numbers are the same for all Markov chains.…”

“…However, Equation (4) does not make that distinction, and its condition numbers are the same for all Markov chains. Ideally, we would like to combine the tightness of the bound in Equation ( 4) with the chain-specific nature of the bound in Equation (3). How can this be achieved?…”

“…Perturbation bounds and related approaches for continuous-time Markov chains have been applied in research fields as diverse as reliability theory [1][2][3], queuing theory [4][5][6][7][8], quantum physics [9][10][11][12], climate science [13], biochemical kinetics [14][15][16][17][18][19], economics [20], population genetics [21], and health insurance modeling [22]. In principle, such bounds can be useful in any field where continuous-time Markov chains and their generalizations are used as mathematical models.…”