Numerical studies of stochastic resonance

A new numerical technique is proposed to study the stochastic resonance (SR) phenomenon. The proposed numerical approach allows to find characteristics of SR faster than the previous ones. The signal-to-noise ratio and phase shifts for a system of noisy coupled oscillators are simulated. The spatiotemporal synchronization is shown by means of trajectory analysis

Numerical solution of Dirichlet problems for nonlinear parabolic equations by probability approach

A number of new layer methods solving Dirichlet problems for semilinear parabolic equations is constructed by using probabilistic representations of their solutions. The methods exploit the ideas of weak sense numerical integration of stochastic differential equations in bounded domain. In spite of the probabilistic nature these methods are nevertheless deterministic. Some convergence theorems are proved. Numerical tests are presented

Unidirectional transport in stochastic ratchets

Constructive conditions for existence of the unidirectional transport are given for systems with state-dependent noise and for forced thermal ratchets. Using them, domains of parameters corresponding to the unidirectional transport are indicated. Some results of numerical experiments are presented

Numerical analysis of Monte Carlo finite difference evaluation of Greeks

An error analysis of approximation of derivatives of the solution to the Cauchy problem for parabolic equations by finite differences is given taking into account that the solution itself is evaluated using weak-sense numerical integration of the corresponding system of stochastic differential equations together with the Monte Carlo technique. It is shown that finite differences are effective when the method of dependent realizations is exploited in the Monte Carlo simulations. This technique is applicable to evaluation of Greeks. In particular, it turns out that it is possible to evaluate both the option price and deltas by a single simulation run that reduces the computational costs. Results of some numerical experiments are presented

Regular oscillations in systems with stochastic resonance

Constructive sufficient conditions for regular oscillations in systems with stochastic resonance are given. For bistable systems, they rely on the fact that the probability of transition of a point from one well to the other with subsequent stay there during the half-period of the periodic forcing is close to 1. Using these conditions, domains of parameters corresponding to the regular oscillations are indicated. The regular oscillations are considered in bistable and monostable systems with additive and multiplicative noise. Special attention is paid to numerical methods. Algorithms based on numerical integration of stochastic differential equations turn out to be most natural both for simulation of sample trajectories and for solution of related boundary value problems of parabolic type. Results of numerical experiments are presented

Numerical methods for nonlinear parabolic equations with small parameter based on probability approach

The probabilistic approach is used for constructing special layer methods to solve the Cauchy problem for semilinear parabolic equations with small parameter. In spite of the probabilistic nature these methods are nevertheless deterministic. The algorithms are tested by simulating the Burgers equation with small viscosity and the generalized KPP-equation with a small parameter

Numerical methods for Langevin type equations based on symplectic integrators

Langevin type equations are an important and fairly large class of systems close to Hamiltonian ones. The constructed mean-square and weak quasi-symplectic methods for such systems degenerate to symplectic methods when a system degenerates to a stochastic Hamiltonian one. In addition, quasi-symplectic methods' law of phase volume contractivity is close to the exact law. The methods derived are based on symplectic schemes for stochastic Hamiltonian systems. Mean-square symplectic methods were obtained in \cite{hadd,hmul} while symplectic methods in the weak sense are constructed in this paper. Special attention is paid to Hamiltonian systems with separable Hamiltonians, with additive noise, and with colored noise. Some numerical tests of both symplectic and quasi-symplectic methods are presented. They demonstrate superiority of the proposed methods in comparison with standard ones

Approximation of Wiener integrals with respect to the Brownian bridge by simulation of SDEs

Numerical integration of stochastic differential equations together with the Monte Carlo technique is used to evaluate conditional Wiener integrals of exponential-type functionals. An explicit Runge-Kutta method of order four and implicit Runge-Kutta methods of order two are constructed. The corresponding convergence theorems are proved. To reduce the Monte Carlo error, a variance reduction technique is considered. Results of numerical experiments are presented

Numerical solution of the Neumann problem for nonlinear parabolic equations by probability approach

A number of new layer methods solving the Neumann problem for semilinear parabolic equations is constructed by using probabilistic representations of their solutions. The methods exploit the ideas of weak sense numerical integration of stochastic differential equations in bounded domain. In spite of the probabilistic nature these methods are nevertheless deterministic. Some convergence theorems are proved. Numerical tests are presented