Convergence Rate of Markov Chains and Hybrid Numerical Schemes to Jump-Diffusion with Application to the Bates Model

Fractional partial differential equations with time-space fractional derivatives describe some important physical phenomena. For example, the subdiffusion equation (time order 0<α<1) is more suitable to describe the phenomena of charge carrier transport in amorphous semiconductors, nuclear magnetic resonance (NMR) diffusometry in percolative, Rouse, or reptation dynamics in polymeric systems, the diffusion of a scalar tracer in an array of convection rolls, or the dynamics of a bead in a polymeric network, and so on. However, the superdiffusion case (1<α<2) is more accurate to depict the special domains of rotating flows, collective slip diffusion on solid surfaces, layered velocity fields, Richardson turbulent diffusion, bulk-surface exchange controlled dynamics in porous glasses, the transport in micelle systems and heterogeneous rocks, quantum optics, single molecule spectroscopy, the transport in turbulent plasma, bacterial motion, and even for the flight of an albatross (for more physical applications of fractional sub-super diffusion equations, one can see Metzler and Klafter in 2000). In this work, we establish two fully discrete numerical schemes for solving a class of nonlinear time-space fractional subdiffusion/superdiffusion equations by using backward Euler difference 1<α<2 or second-order central difference 1<α<2/local discontinuous Galerkin finite element mixed method. By introducing the mathematical induction method, we show the concrete analysis for the stability and the convergence rate under the L2 norm of the two LDG schemes. In the end, we adopt several numerical experiments to validate the proposed model and demonstrate the features of the two numerical schemes, such as the optimal convergence rate in space direction is close to Ohk+1. The convergence rate in time direction can arrive at Oτ2−α when the fractional derivative is 0<α<1. If the fractional derivative parameter is 1<α<2 and we choose the relationship as h=C′τ (h denotes the space step size, C′ is a constant, and τ is the time step size), then the time convergence rate can reach to Oτ3−α. The experiment results illustrate that the proposed method is effective in solving nonlinear time-space fractional subdiffusion/superdiffusion equations.

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Conductance bounds on the L 2 convergence rate of Metropolis algorithms on unbounded state spaces

Advances in Applied Probability ◽

10.1017/s0001867800012957 ◽

2004 ◽

Vol 36 (01) ◽

pp. 243-266

Author(s):

Søren F. Jarner ◽

Wai Kong Yuen

Keyword(s):

Convergence Rate ◽

Markov Chains ◽

Spectral Gap ◽

Unbounded Domains ◽

Decomposition Theorem ◽

Metropolis Algorithm ◽

The State ◽

Bounded Domains ◽

Target Density ◽

State Spaces

In this paper we derive bounds on the conductance and hence on the spectral gap of a Metropolis algorithm with a monotone, log-concave target density on an interval of ℝ. We show that the minimal conductance set has measure ½ and we use this characterization to bound the conductance in terms of the conductance of the algorithm restricted to a smaller domain. Whereas previous work on conductance has resulted in good bounds for Markov chains on bounded domains, this is the first conductance bound applicable to unbounded domains. We then show how this result can be combined with the state-decomposition theorem of Madras and Randall (2002) to bound the spectral gap of Metropolis algorithms with target distributions with monotone, log-concave tails on ℝ.

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A comparison of the ordinary and a varying parameter exponential smoothing

Journal of Applied Probability ◽

10.2307/3214383 ◽

1989 ◽

Vol 26 (4) ◽

pp. 784-792 ◽

Cited By ~ 7

Author(s):

Heikki Bonsdorff

Keyword(s):

Convergence Rate ◽

Markov Chains ◽

Finite Interval ◽

Exponential Smoothing ◽

Uniform Ergodicity ◽

Geometric Convergence ◽

Monotonically Increasing Function ◽

Increasing Function

An adaptive-type exponential smoothing, motivated by an insurance tariff problem, is treated. We consider the process Zn = ß(Zn –1)Xn +(1 – ß (Zn–1))Zn–1, where Xn are i.i.d. taking values in the interval [0, M], M ≦ ∞ and ß is a monotonically increasing function [0, M] → [c, d], 0 < c < d < 1.Together with (Zn), we consider the ordinary exponential smoothing Yn = αXn + (1 – α)Yn –1 where α is a constant, 0 < α < 1. We show that (Yn) and (Zn) are geometrically ergodic Markov chains (in the case of finite interval we even have uniform ergodicity) and that EYn, EZn converge to limits EY, EZ, respectively, with a geometric convergence rate. Moreover, we show that Ez is strictly less than EY = EXn.

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Exponential convergence rate of ruin probabilities for level-dependent Lévy-driven risk processes

Journal of Applied Probability ◽

10.1017/jpr.2019.71 ◽

2019 ◽

Vol 56 (4) ◽

pp. 1244-1268 ◽

Cited By ~ 1

Author(s):

Pierre-Olivier Goffard ◽

Andrey Sarantsev

Keyword(s):

Convergence Rate ◽

Ruin Probability ◽

Lyapunov Functions ◽

Risk Model ◽

Exponential Convergence ◽

Jump Diffusion ◽

Ruin Probabilities ◽

Exponential Rate ◽

Risk Processes

AbstractWe find explicit estimates for the exponential rate of long-term convergence for the ruin probability in a level-dependent Lévy-driven risk model, as time goes to infinity. Siegmund duality allows us to reduce the problem to long-term convergence of a reflected jump-diffusion to its stationary distribution, which is handled via Lyapunov functions.

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Computable bounds of an 𝓁²-spectral gap for discrete Markov chains with band transition matrices

Journal of Applied Probability ◽

10.1017/jpr.2016.53 ◽

2016 ◽

Vol 53 (3) ◽

pp. 946-952

Author(s):

Loï Hervé ◽

James Ledoux

Keyword(s):

Convergence Rate ◽

Markov Chains ◽

Spectral Radius ◽

Spectral Gap ◽

Transition Probability ◽

Transition Probability Matrix ◽

Geometric Ergodicity ◽

Transition Matrices ◽

Essential Spectral Radius ◽

Band Transition

AbstractWe analyse the 𝓁²(𝜋)-convergence rate of irreducible and aperiodic Markov chains with N-band transition probability matrix P and with invariant distribution 𝜋. This analysis is heavily based on two steps. First, the study of the essential spectral radius ress(P|𝓁²(𝜋)) of P|𝓁²(𝜋) derived from Hennion’s quasi-compactness criteria. Second, the connection between the spectral gap property (SG2) of P on 𝓁²(𝜋) and the V-geometric ergodicity of P. Specifically, the (SG2) is shown to hold under the condition α0≔∑m=−NNlim supi→+∞(P(i,i+m)P*(i+m,i)1∕2<1. Moreover, ress(P|𝓁²(𝜋)≤α0. Effective bounds on the convergence rate can be provided from a truncation procedure.

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