scholarly journals On a selection problem for small noise perturbation in the multidimensional case

2018 ◽  
Vol 18 (06) ◽  
pp. 1850045 ◽  
Author(s):  
Andrey Pilipenko ◽  
Frank Norbert Proske
Keyword(s):  
Normal Component ◽  
Lower Half ◽  
Jump Discontinuity ◽  
Multidimensional Case ◽  
Small Noise ◽  
Lipschitz Continuous ◽  
Limit Process ◽  
Noise Perturbation ◽  
Discontinuous Drift ◽  
Classical Subject

The problem on identification of a limit of an ordinary differential equation with discontinuous drift that perturbed by a zero-noise is considered in multidimensional case. This problem is a classical subject of stochastic analysis, see, for example, [6, 29, 11, 20]. However the multidimensional case was poorly investigated. We assume that the drift coefficient has a jump discontinuity along a hyperplane and is Lipschitz continuous in the upper and lower half-spaces. It appears that the behavior of the limit process depends on signs of the normal component of the drift at the upper and lower half-spaces in a neighborhood of the hyperplane, all cases are considered.

scholarly journals On regularization by a small noise of multidimensional ODEs with non-Lipschitz coefficients

2020 ◽  
Vol 72 (9) ◽  
pp. 1254-1285
Author(s):  
A. Pilipenko ◽  
A. Kulik
Keyword(s):  
Selection Problem ◽  
Extremal Solutions ◽  
Averaging Principle ◽  
Small Noise ◽  
Lipschitz Continuous ◽  
Limit Process ◽  
Locally Lipschitz ◽  
And Diffusion ◽  
Locally Lipschitz Continuous

UDC 519.21 In this paper we solve a selection problem for multidimensional SDE where the drift and diffusion are locally Lipschitz continuous outside of a fixed hyperplane It is assumed that the drift has a Hoelder asymptotics as approaches and the limit ODE does not have a unique solution.We show that if the drift pushes the solution away from then the limit process with certain probabilities selects some extremal solutions to the limit ODE. If the drift attracts the solution to then the limit process satisfies an ODE with some averaged coefficients. To prove the last result we formulate an averaging principle, which is quite general and new.

scholarly journals An asymptotic expansion of forward-backward SDEs with a perturbed driver

2015 ◽  
Vol 02 (02) ◽  
pp. 1550020 ◽  
Author(s):  
Akihiko Takahashi ◽  
Toshihiro Yamada
Keyword(s):  
Asymptotic Expansion ◽  
Differential Equations ◽  
Error Estimate ◽  
Arbitrary Order ◽  
Nonlinear Pricing ◽  
Small Noise ◽  
Small Diffusion ◽  
Noise Perturbation ◽  
Backward Sdes ◽  
Expansion Scheme

Motivated by nonlinear pricing in finance, this paper presents a mathematical validity of an asymptotic expansion scheme for a system of forward-backward stochastic differential equations (FBSDEs) in terms of a perturbed driver in the BSDE and a small diffusion in the FSDE. In particular, we represent the coefficients of the expansion of the FBSDE up to an arbitrary order, and obtain the error estimate of the expansion with respect to the driver and the small noise perturbation.

Homoclinics in the Reconstruction of Dynamic Systems From Experimental Data

1993 ◽  
Vol 46 (7) ◽  
pp. 361-371
Author(s):  
V. S. Anishchenko ◽  
M. A. Safonova
Keyword(s):  
Probability Measure ◽  
Dynamic Systems ◽  
Reconstruction Algorithm ◽  
External Noise ◽  
Small Noise ◽  
Noise Perturbation ◽  
Experimental Observable ◽  
Model Equations ◽  
System Attractor

The role of homoclinic effects in solution of a reconstruction problem of system attractors and model equations from experimental observable in the presence of external noise is investigated numerically. It is shown that the possibility of reconstruction essentially depends on character of origin system homoclinic trajectories and noise intensity. If the homoclinic structure belongs to the attractor, then the reconstruction results in restoration origin system attractors. A small noise influence causes in this case a small perturbation of attractors probability measure and practically disappears due to filtering properties of the reconstruction algorithm. The homoclinic structure does not belong to the attractor, then in the absence of noise the probability measure concentrates at the attractor, the structure of which is not defined by the homoclinics. The noise perturbation induces new regimes. Then the attractor structure essentially depends on the homoclinics structure and noise level. In this case the model system attractor of which reproduces “invisible” homoclinic structure, is obtained as a result of reconstruction.

scholarly journals The Existence of Strong Solutions for a Class of Stochastic Differential Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-5
Author(s):  
Junfei Zhang
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Strong Solution ◽  
Strong Solutions ◽  
Lipschitz Continuous ◽  
Discontinuous Drift ◽  
Increasing Function

In this paper, we will consider the existence of a strong solution for stochastic differential equations with discontinuous drift coefficients. More precisely, we study a class of stochastic differential equations when the drift coefficients are an increasing function instead of Lipschitz continuous or continuous. The main tools of this paper are the lower solutions and upper solutions of stochastic differential equations.

scholarly journals A note on strong solutions of stochastic differential equations with a discontinuous drift coefficient

2006 ◽  
Vol 2006 ◽  
pp. 1-6 ◽  
Author(s):  
Nikolaos Halidias ◽  
P. E. Kloeden
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Upper And Lower Solutions ◽  
Strong Solutions ◽  
Heaviside Function ◽  
Lipschitz Continuous ◽  
Discontinuous Drift ◽  
Vector Valued ◽  
Increasing Function ◽  
Drift Coefficient

The existence of a mean-square continuous strong solution is established for vector-valued Itô stochastic differential equations with a discontinuous drift coefficient, which is an increasing function, and with a Lipschitz continuous diffusion coefficient. A scalar stochastic differential equation with the Heaviside function as its drift coefficient is considered as an example. Upper and lower solutions are used in the proof.

Stochastic resonance with multiplicative heavy-tailed Lévy noise: Optimal tuning on an algebraic time scale

2017 ◽  
Vol 17 (04) ◽  
pp. 1750027 ◽  
Author(s):  
Isabelle Kuhwald ◽  
Ilya Pavlyukevich
Keyword(s):  
Stochastic Resonance ◽  
Heavy Tails ◽  
Bistable System ◽  
Lévy Noise ◽  
Small Noise ◽  
Optimal Tuning ◽  
Noise Perturbation ◽  
Heavy Tailed ◽  
Synchronization Effect ◽  
Levy Noise

Stochastic resonance is an amplification and synchronization effect of weak periodic signals in nonlinear systems through a small noise perturbation. In this paper we study the dynamics of stochastic resonance in a bistable system driven by multiplicative Lévy noise with heavy tails, e.g., [Formula: see text]-stable Lévy noise. We determine the optimal tuning with respect to a probabilistic synchronization measure for both the jump-diffusion and the reduced two-state Markov chain. These results extend the theory of stochastic resonance to the case of heavy tail Lévy perturbations.

On the stability against shear waves of steady flows of non-linear viscoelastic fluids

1968 ◽  
Vol 33 (1) ◽  
pp. 165-181 ◽  
Author(s):  
Bernard D. Coleman ◽  
Morton E. Gurtin
Keyword(s):  
Normal Component ◽  
Exact Formula ◽  
Tangential Component ◽  
Jump Discontinuity ◽  
Fading Memory ◽  
Steady Flows ◽  
Acceleration Wave ◽  
Critical Amplitude ◽  
Planar Shear ◽  
Non Linear

A shear-acceleration wave is a propagating singular surface across which the velocity vector and the normal component of the acceleration are continuous, while the tangential component$\dot{v}$of the acceleration suffers a jump discontinuity [$\dot{v}$]. We here discuss plane-rectilinear shearing flows of general, non-linear, incompressible simple fluids with fading memory. Working within the framework of such planar motions, we derive a general and exact formula for the time-dependence of the amplitudea= [$\dot{v}$] of a shear-acceleration wave propagating into a region undergoing a steady but not necessarily homogeneous shearing flow. When this expression is specialized to the case in which the velocity gradient is constant in space ahead of the wave, it assumes a form familiar in the theory of longitudinal acceleration waves in compressible materials with fading memory (cf., e.g., Coleman & Gurtin 1965, equation (4.12)).In earlier work (1965) we observed that a planar shear-acceleration wave cannot grow in amplitude if it is propagating into a fluid in a state of equilibrium. It is clear from our present results that if the fluid ahead of the wave is being sheared, |a(t)| not only increases, but can approach infinity in a finite time, provideda(0) is of proper sign and |a(0)| exceeds a certain critical amplitude. We expect this critical amplitude to decrease as the rate of shear ahead of the wave is increased.

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