Convergence to Equilibrium for Time-Inhomogeneous Jump Diffusions with State-Dependent Jump Intensity
- PDF / 483,077 Bytes
- 35 Pages / 439.37 x 666.142 pts Page_size
- 38 Downloads / 208 Views
Convergence to Equilibrium for Time-Inhomogeneous Jump Diffusions with State-Dependent Jump Intensity E. Löcherbach1 Received: 13 May 2019 / Revised: 9 September 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract We consider a time-inhomogeneous Markov process X = (X t )t with jumps having state-dependent jump intensity, with values in Rd , and we are interested in its longtime behavior. The infinitesimal generator of the process is given for any sufficiently smooth test function f by L t f (x) =
d ∂f (x)bi (t, x) + [ f (x + c(t, z, x)) − f (x)]γ (t, z, x)μ(dz), ∂ xi Rm i=1
where μ is a σ -finite measure on (Rm , B(Rm )) describing the jumps of the process. We give conditions on the coefficients b(t, x), c(t, z, x) and γ (t, z, x) under which the longtime behavior of X can be related to the longtime behavior of a time-homogeneous limit process X¯ . Moreover, we introduce a coupling method for the limit process which is entirely based on certain of its big jumps and which relies on the regeneration method. We state explicit conditions in terms of the coefficients of the process allowing control of the speed of convergence to equilibrium both for X and for X¯ . Keywords Diffusions with position-dependent jumps · Nummelin splitting · Total variation coupling · Continuous-time Markov processes · Convergence to equilibrium · Asymptotic pseudotrajectories Mathematics Subject Classification (2010) 60J55 · 60J35 · 60F05
B 1
E. Löcherbach [email protected] SAMM, Université de Paris 1 Panthéon Sorbonne, 90 rue de Tolbiac, 75013 Paris, France
123
Journal of Theoretical Probability
1 Introduction In this paper, we study a rather general class of jump processes taking values in Rd and evolving according to
t
Xt = x +
b(s, X s )ds +
0
[0,t] Rm ×R+
c(s, z, X s− )1u≤γ (s,z,X s− ) N (ds, dz, du),
(1.1) with x ∈ Rd . In the above equation, N (ds, dz, du) is a Poisson random measure which is defined on a fixed probability space (, A, P), with (s, z, u) ∈ R+ × Rm × R+ , having intensity measure dsμ(dz)du. Here, μ is some σ -finite measure on (Rm , B(Rm )). The associated infinitesimal generator at time t is given by d ∂f i (x)b (t, x)+ [ f (x +c(t, z, x))− f (x)]γ (t, z, x)μ(dz). (1.2) L t f (x) = ∂ xi Rm i=1
The coefficients of the system are the measurable functions b : R+ × Rd → Rd , c : R+ × Rm × R+ → Rd and γ : R+ × Rm × R+ → R+ . We shall always work under conditions ensuring that (1.1) admits a unique strong non-explosive adaptive (to the filtration generated by the Poisson random measure) solution which is Markov, having càdlàg trajectories which are of finite variation (see Assumption 3.1). Let us give some comments on (1.1). If the jump rate γ is a constant function, then the above process is a classical jump process. But if γ is not constant, then the jump intensity and also the jump amplitude depend on the current position of the process. This is a natural assumption in many modeling issues (see, e.g., [10,33] or [24] for the modeling of biologica
Data Loading...
