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Juan Carlos Jimenez

Publications and source records attributed to Juan Carlos Jimenez.

6 recordsLinked to original sources
arXiv2023

Weak variable step-size schemes for stochastic differential equations based on controlling conditional moments

We address the weak numerical solution of stochastic differential equations driven by independent Brownian motions (SDEs for short). This paper develops a new methodology to design adaptive strategies for determining automatically the step-sizes of the numerical schemes that compute the mean values of smooth functions of the solutions of SDEs. First, we introduce a general method for constructing variable step-size weak schemes for SDEs, which is based on controlling the match between the first conditional moments of the increments of the numerical integrator and the ones corresponding to an additional weak approximation. To this end, we use certain local discrepancy functions that do not involve sampling random variables. Precise directions for designing suitable discrepancy functions and for selecting starting step-sizes are given. Second, we introduce a variable step-size Euler scheme, together with a variable step-size second order weak scheme via extrapolation. Finally, numerical simulations are presented to show the potential of the introduced variable step-size strategy and the adaptive scheme to overcome known instability problems of the conventional fixed step-size schemes in the computation of diffusion functional expectations.

math.PR↗
arXiv2018

An estimation of distribution algorithm for the computation of innovation estimators of diffusion processes

Estimation of Distribution Algorithms (EDAs) and Innovation Method are recognized methods for solving global optimization problems and for the estimation of parameters in diffusion processes, respectively. Well known is also that the quality of the Innovation Estimator strongly depends on an adequate selection of the initial value for the parameters when a local optimization algorithm is used in its computation. Alternatively, in this paper, we study the feasibility of a specific EDA - a continuous version of the Univariate Marginal Distribution Algorithm (UMDAc) - for the computation of the Innovation Estimators. Numerical experiments are performed for two different models with a high level of complexity. The numerical simulations show that the considered global optimization algorithms substantially improves the effectiveness of the Innovation Estimators for different types of diffusion processes with complex nonlinear and stochastic dynamics.

math.NA↗
arXiv2013

Locally Linearized Runge Kutta method of Dormand and Prince

In this paper, the effect that produces the local linearization of the embedded Runge-Kutta formulas of Dormand and Prince for initial value problems is studied. For this, embedded Locally Linearized Runge-Kutta formulas are defined and their performance is analyzed by means of exhaustive numerical simulations. For a variety of well-known physical equations with different dynamics, the simulation results show that the locally linearized formulas exhibit significant higher accuracy than the original ones, which implies a substantial reduction of the number of time steps and, consequently, a sensitive reduction of the overall computation cost of their adaptive implementation.

math.NA↗
arXiv2013

Simplified formulas for the mean and variance of linear stochastic differential equations

Explicit formulas for the mean and variance of linear stochastic differential equations are derived in terms of an exponential matrix. This result improved a previous one by means of which the mean and variance are expressed in terms of a linear combination of higher dimensional exponential matrices. The important role of the new formulas for the system identification as well as numerical algorithms for their practical implementation are pointed out.

math.OC↗
arXiv2013

Approximate linear minimum variance filters for continuous-discrete state space models: convergence and practical algorithms

In this paper, approximate Linear Minimum Variance (LMV) filters for continuous-discrete state space models are introduced. The filters are obtained by means of a recursive approximation to the predictions for the first two moments of the state equation. It is shown that the approximate filters converge to the exact LMV filter when the error between the predictions and their approximations decreases. As particular instance, the order-$β$ Local Linearization filters are presented and expounded in detail. Practical algorithms are also provided and their performance in simulation is illustrated with various examples. The proposed filters are intended for the recurrent practical situation where a nonlinear stochastic system should be identified from a reduced number of partial and noisy observations distant in time.

math.OC↗