888 research outputs found
Stability of Numerical Methods for Jump Diffusions and Markovian Switching Jump Diffusions
This work focuses on stability analysis of numerical solutions to jump
diffusions and jump diffusions with Markovian switching. Due to the use of
Poisson processes, using asymptotic expansions as in the usual approach of
treating diffusion processes does not work. Different from the existing
treatments of Euler-Maurayama methods for solutions of stochastic differential
equations, we use techniques from stochastic approximation. We analyze the
almost sure exponential stability and exponential -stability. The benchmark
test model in numerical solutions, namely, one-dimensional linear scalar jump
diffusion is examined first and easily verifiable conditions are presented.
Then Markovian regime-switching jump diffusions are dealt with. Moreover,
analysis on stability of numerical methods for linearizable and
multi-dimensional jump diffusions is carried out.Comment: This paper has been withdrawn by the author due to a private reaso
Stability of Nonlinear Regime-switching Jump Diffusions
Motivated by networked systems, stochastic control, optimization, and a wide
variety of applications, this work is devoted to systems of switching jump
diffusions. Treating such nonlinear systems, we focus on stability issues.
First asymptotic stability in the large is obtained. Then the study on
exponential p-stability is carried out. Connection between almost surely
exponential stability and exponential p-stability is exploited. Also presented
are smooth-dependence on the initial data. Using the smooth-dependence,
necessary conditions for exponential p-stability are derived. Then criteria for
asymptotic stability in distribution are provided. A couple of examples are
given to illustrate our results
Stability of Regime-Switching Diffusion Systems with Discrete States Belonging to a Countable Set
This work focuses on stability of regime-switching diffusions consisting of
continuous and discrete components, in which the discrete component switches in
a countably infinite set and its switching rates at current time depend on the
continuous component. In contrast to the existing approach, this work provides
more practically viable approach with more feasible conditions for stability. A
classical approach for asymptotic stabilityusing Lyapunov function techniques
shows the Lyapunov function evaluated at the solution process goes to 0 as time
. A distinctive feature of this paper is to obtain estimates of
path-wise rates of convergence, which pinpoints how fast the aforementioned
convergence to 0 taking place. Finally, some examples are given to illustrate
our findings.Comment: 25page
Recurrence and Ergodicity for A Class of Regime-Switching Jump Diffusions
This work develops asymptotic properties of a class of switching jump
diffusion processes. The processes under consideration may be viewed as a
number of jump diffusion processes modulated by a random switching mechanism.
The underlying processes feature in the switching process depends on the jump
diffusions. In this paper, conditions for recurrence and positive recurrence
are derived. Ergodicity is examined in detail. Existence of invariant
probability measures is proved.Comment: To appear in Applied Mathematics and Optimizatio
Stabilization of regime-switching processes by feedback control based on discrete time observations II: state-dependent case
This work investigates the almost sure stabilization of a class of
regime-switching systems based on discrete-time observations of both continuous
and discrete components. It develops Shao's work [SIAM J. Control Optim.,
55(2017), pp. 724--740] in two aspects: first, to provide sufficient conditions
for almost sure stability in lieu of moment stability; second, to investigate a
class of state-dependent regime-switching processes instead of
state-independent ones. To realize these developments, we establish an
estimation of the exponential functional of Markov chains based on the spectral
theory of linear operator. Moreover, through constructing order-preserving
coupling processes based on Skorokhod's representation of jumping process, we
realize the control from up and below of the evolution of state-dependent
switching process by state-independent Markov chains.Comment: 33 page
Stability and recurrence of regime-switching diffusion processes
We provide some criteria on the stability of regime-switching diffusion
processes. Both the state-independent and state-dependent regime-switching
diffusion processes with switching in a finite state space and an infinite
countable state space are studied in this work. We provide two methods to deal
with switching processes in an infinite countable state space. One is a finite
partition method based on the nonsingular M-matrix theory. Another is an
application of principal eigenvalue of a bilinear form. Our methods can deal
with both linear and nonlinear regime-switching diffusion processes. Moreover,
the method of principal eigenvalue is also used to study the recurrence of
regime-switching diffusion processes
Strong solutions and strong Feller properties for regime-switching diffusion processes in an infinite state space
We establish the existence and pathwise uniqueness of regime-switching
diffusion processes in an infinite state space, which could be
time-inhomogeneous and state-dependent. Then the strong Feller properties of
these processes are investigated by using the theory of parabolic differential
equations and dimensional-free Harnack inequalities
Regime-Switching Jump Diffusions with Non-Lipschitz Coefficients and Countably Many Switching States: Existence and Uniqueness, Feller, and Strong Feller Properties
This work focuses on a class of regime-switching jump diffusion processes,
which is a two component Markov processes , where
is a component representing discrete events taking values in a
countably infinite set. Considering the corresponding stochastic differential
equations, our main focus is on treating those with non-Lipschitz coefficients.
We first show that there exists a unique strong solution to the corresponding
stochastic differential equation. Then Feller and strong Feller properties are
investigated
Jump-diffusion processes in random environments
In this paper we investigate jump-diffusion processes in random environments
which are given as the weak solutions to SDE's. We formulate conditions
ensuring existence and uniqueness in law of solutions. We investigate Markov
property. To prove uniqueness we solve a general martingale problem for \cadlag
processes. This result is of independent interest. In the last section we
present application of our results considering generalized exponential Levy
model
Regime-switching diffusion processes: strong solutions and strong Feller property
We investigate the existence and uniqueness of strong solutions up to an
explosion time for regime-switching diffusion processes in an infinite state
space. Instead of concrete conditions on coefficients, our existence and
uniqueness result is established under the general assumption that the
diffusion in every fixed environment has a unique non-explosive strong
solution. Moreover, non-explosion conditions for regime-switching diffusion
processes are given. The strong Feller property is proved by further assuming
that the diffusion in every fixed environment generates a strong Feller
semigroup
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