109 research outputs found
Strong convergence of some drift implicit Euler scheme. Application to the CIR process
We study the convergence of a drift implicit scheme for one-dimensional SDEs
that was considered by Alfonsi for the Cox-Ingersoll-Ross (CIR) process. Under
general conditions, we obtain a strong convergence of order 1. In the CIR case,
Dereich, Neuenkirch and Szpruch have shown recently a strong convergence of
order 1/2 for this scheme. Here, we obtain a strong convergence of order 1
under more restrictive assumptions on the CIR parameters
Strong order 1/2 convergence of full truncation Euler approximations to the Cox-Ingersoll-Ross process
We study convergence properties of the full truncation Euler scheme for the
Cox-Ingersoll-Ross process in the regime where the boundary point zero is
inaccessible. Under some conditions on the model parameters (precisely, when
the Feller ratio is greater than three), we establish the strong order 1/2
convergence in of the scheme to the exact solution. This is consistent
with the optimal rate of strong convergence for Euler approximations of
stochastic differential equations with globally Lipschitz coefficients, despite
the fact that the diffusion coefficient in the Cox-Ingersoll-Ross model is not
Lipschitz.Comment: 16 pages, 1 figur
Switching and diffusion models for gene regulation networks
We analyze a hierarchy of three regimes for modeling gene regulation. The most complete model is a continuous time, discrete state space, Markov jump process. An intermediate 'switch plus diffusion' model takes the form of a stochastic differential equation driven by an independent continuous time Markov switch. In the third 'switch plus ODE' model the switch remains but the diffusion is removed. The latter two models allow for multi-scale simulation where, for the sake of computational efficiency, system components are treated differently according to their abundance. The 'switch plus ODE' regime was proposed by Paszek (Modeling stochasticity in gene regulation: characterization in the terms of the underlying distribution function, Bulletin of Mathematical Biology, 2007), who analyzed the steady state behavior, showing that the mean was preserved but the variance only approximated that of the full model. Here, we show that the tools of stochastic calculus can be used to analyze first and second moments for all time. A technical issue to be addressed is that the state space for the discrete-valued switch is infinite. We show that the new 'switch plus diffusion' regime preserves the biologically relevant measures of mean and variance, whereas the 'switch plus ODE' model uniformly underestimates the variance in the protein level. We also show that, for biologically relevant parameters, the transient behaviour can differ significantly from the steady state, justifying our time-dependent analysis. Extra computational results are also given for a protein dimerization model that is beyond the scope of the current analysis
Learning Nash Equilibria
In the paper, we re-investigate the long run behavior of an adaptive learning process driven
by the stochastic replicator dynamics developed by Fudenberg and Harris (1992). It is demonstrated
that the Nash equilibrium will be the robust limit of the adaptive learning process as long as it
is reachable for the learning dynamics in almost surely finite time. Doobās martingale theory and
Girsanov Theorem play very important roles in confirming the required assertion
Guaranteed Control of Sampled Switched Systems using Semi-Lagrangian Schemes and One-Sided Lipschitz Constants
In this paper, we propose a new method for ensuring formally that a
controlled trajectory stay inside a given safety set S for a given duration T.
Using a finite gridding X of S, we first synthesize, for a subset of initial
nodes x of X , an admissible control for which the Euler-based approximate
trajectories lie in S at t [0,T]. We then give sufficient conditions
which ensure that the exact trajectories, under the same control, also lie in S
for t [0,T], when starting at initial points 'close' to nodes x. The
statement of such conditions relies on results giving estimates of the
deviation of Euler-based approximate trajectories, using one-sided Lipschitz
constants. We illustrate the interest of the method on several examples,
including a stochastic one
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