scholarly journals Strong Feller property and continuous dependence on initial data for one-dimensional stochastic differential equations with Hölder continuous coefficients

2020 ◽  
Vol 25 (0) ◽  
Author(s):  
Hua Zhang
Keyword(s):  
Initial Data ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Continuous Dependence ◽  
Hölder Continuous ◽  
One Dimensional ◽  
Strong Feller Property ◽  
Feller Property ◽  
Strong Feller

scholarly journals CONTINUOUS DEPENDENCE ON THE INITIAL DATA OF THE SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH FRACTIONAL BROWNIAN MOTIONS

2018 ◽  
Vol 54 (2) ◽  
pp. 193-209
Author(s):  
I. V. Kachan
Keyword(s):  
Initial Data ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Continuous Dependence ◽  
Initial Conditions ◽  
Hölder Continuous ◽  
Brownian Motions ◽  
Fractional Brownian Motions ◽  
Right Hand ◽  
The Right

In the present acticle we consider finite-dimensional stochastic differential equations with fractional Brownian motions having different Hurst indices larger than 1/3 and a drift. These heterogeneous components of the equations are combined into a single process. The solutions of the equations are understood in the integral sense, and the integrals in turn are Gubinelli’s rough path integrals [1] realizing the well-known approach of the rough paths theory [2]. The existence and uniqueness conditions of the solutions of these stochastic differential equations are specified. Such conditions are sufficient to obtain the results related the continuous dependence on the initial data. In this article, we have first proved a continuous dependence on the initial conditions and the right-hand sides of the solutions of the stochastic differential equations under consideration for almost all their trajectories. The result obtained does not depend on the probabilistic properties of fractional Brownian motions, and therefore it can be easily generalized to the case of arbitrary Holder-continuous processes with an exponent greater than 1/3. In this case, the constant arising in the estimates appears to be exponentially dependent on the norms of fractional Brownian motions. Taking into account the last fact and the proved result, an expected logarithmic continuous dependence on the initial conditions and the right-hand sides of the solutions of the stochastic differential equations con - si dered is subsequently derived. This is the major result of this article.

A Hölder continuous ODE related to traffic flow

2003 ◽  
Vol 133 (4) ◽  
pp. 759-772 ◽  
Author(s):  
Rinaldo M. Colombo ◽  
Andrea Marson
Keyword(s):  
Vector Field ◽  
Initial Data ◽  
Differential Equations ◽  
Traffic Flow ◽  
Conservation Law ◽  
Continuous Dependence ◽  
Well Posedness ◽  
Hölder Continuous ◽  
Scalar Conservation Law ◽  
Scalar Conservation

This paper is devoted to the proof of the well posedness of a class of ordinary differential equations (ODEs). The vector field depends on the solution to a scalar conservation law. Forward uniqueness of Filippov solutions is obtained, as well as their Hölder continuous dependence on the initial data of the ODE. Furthermore, we prove the continuous dependence in C0 of the solution to the ODE from the initial data of the conservation law in L1.This problem is motivated by a model of traffic flow.

scholarly journals Aspects of Hadamard well-posedness for classes of non-Lipschitz semilinear parabolic partial differential equations

2016 ◽  
Vol 146 (4) ◽  
pp. 777-832 ◽  
Author(s):  
J. C. Meyer ◽  
D. J. Needham
Keyword(s):  
Partial Differential Equations ◽  
Initial Data ◽  
Differential Equations ◽  
Continuous Dependence ◽  
Parabolic Partial Differential Equations ◽  
Well Posedness ◽  
Partial Differential ◽  
Hölder Continuous ◽  
Lipschitz Continuous ◽  
Semilinear Parabolic

We study classical solutions of the Cauchy problem for a class of non-Lipschitz semilinear parabolic partial differential equations in one spatial dimension with sufficiently smooth initial data. When the nonlinearity is Lipschitz continuous, results concerning existence, uniqueness and continuous dependence on initial data are well established (see, for example, the texts of Friedman and Smoller and, in the context of the present paper, see also Meyer), as are the associated results concerning Hadamard well-posedness. We consider the situations when the nonlinearity is Hölder continuous and when the nonlinearity is upper Lipschitz continuous. Finally, we consider the situation when the nonlinearity is both Hölder continuous and upper Lipschitz continuous. In each case we focus upon the question of existence, uniqueness and continuous dependence on initial data, and thus upon aspects of Hadamard well-posedness.

scholarly journals REFLECTED BROWNIAN MOTION ON SIMPLE NESTED FRACTALS

Fractals ◽  
2019 ◽  
Vol 27 (06) ◽  
pp. 1950104
Author(s):  
KAMIL KALETA ◽  
MARIUSZ OLSZEWSKI ◽  
KATARZYNA PIETRUSKA-PAŁUBA
Keyword(s):  
Brownian Motion ◽  
Transition Probability ◽  
Strong Feller Property ◽  
Feller Property ◽  
Reflected Diffusion ◽  
Geometric Conditions ◽  
Regularity Properties ◽  
Probability Densities ◽  
Free Brownian Motion ◽  
Strong Feller

For a large class of planar simple nested fractals, we prove the existence of the reflected diffusion on a complex of an arbitrary size. Such a process is obtained as a folding projection of the free Brownian motion from the unbounded fractal. We give sharp necessary geometric conditions for the fractal under which this projection can be well defined, and illustrate them by numerous examples. We then construct a proper version of the transition probability densities for the reflected process and we prove that it is a continuous, bounded and symmetric function which satisfies the Chapman–Kolmogorov equations. These provide us with further regularity properties of the reflected process such us Markov, Feller and strong Feller property.

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