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Register a new userSnapping out Walshs Brownian motion and related stiff problem
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Firstly, we shall introduce the so-called snapping out Walshs Brownian motion and present its relation with Walshs Brownian motion. Then the stiff problem related to Walshs Brownian motion will be described and we shall build a phase transition for it. The snapping out Walshs Brownian motion corresponds to the so-called semi-permeable pattern of this stiff problem.
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In this paper, we will first give the numerical simulation of the sub-fractional Brownian motion through the relation of fractional Brownian motion instead of its representation of random walk. In order to verify the rationality of this simulation, we propose a practical estimator associated with the LSE of the drift parameter of mixed sub-fractional Ornstein-Uhlenbeck process, and illustrate the asymptotical properties according to our method of simulation when the Hurst parameter $H>1/2$.
In this paper, we study the well-posedness of multi-dimensional backward stochastic differential equations driven by $G$-Brownian motion ($G$-BSDEs) with diagonal generators, the $z$ parts of whose $l$-th components only depend on the $l$-th columns. The existence and uniqueness of solutions are obtained via a contraction argument for $Y$ component and a backward iteration of local solutions. Furthermore, we show that, the solution of multi-dimensional $G$-BSDE in a Markovian framework provides a probabilistic formula for the viscosity solution of a system of nonlinear parabolic partial differential equations.
In this paper we will study a stiff problem in two-dimensional space and especially its probabilistic counterpart. Roughly speaking, the heat equation with a parameter $varepsilon>0$ is under consideration: [ partial_t u^varepsilon(t,x)=frac{1}{2} abla cdot left(mathbf{A}_varepsilon(x) abla u^varepsilon(t,x) right),quad tgeq 0, xin mathbb{R}^2, ] where $mathbf{A}_varepsilon(x)=text{Id}_2$, the identity matrix, for $x otin Omega_varepsilon:={x=(x_1,x_2)in mathbb{R}^2: |x_2|<varepsilon}$ while $$mathbf{A}_varepsilon(x):=begin{pmatrix} a_varepsilon^- & 0 0 & a^shortmid_varepsilon end{pmatrix}$$ with two positive constants $a^-_varepsilon, a^shortmid_varepsilon$ for $xin Omega_varepsilon$. There exists a diffusion process $X^varepsilon$ on $mathbb{R}^2$ associated to this heat equation in the sense that $u^varepsilon(t,x):=mathbf{E}^xu^varepsilon(0,X_t^varepsilon)$ is its unique weak solution. Note that $Omega_varepsilon$ collapses to the $x_1$-axis, a barrier of zero volume, as $varepsilondownarrow 0$. The main purpose of this paper is to derive all possible limiting process $X$ of $X^varepsilon$ as $varepsilondownarrow 0$. In addition, the limiting flux $u$ of the solution $u^varepsilon$ as $varepsilondownarrow 0 $ and all possible boundary conditions satisfied by $u$ will be also characterized.
This is a summary (in French) of my work about brownian motion and Kac-Moody algebras during the last seven years, presented towards the Habilitation degree.
To extend several known centered Gaussian processes, we introduce a new centered mixed self-similar Gaussian process called the mixed generalized fractional Brownian motion, which could serve as a good model for a larger class of natural phenomena. This process generalizes both the well known mixed fractional Brownian motion introduced by Cheridito [10] and the generalized fractional Brownian motion introduced by Zili [31]. We study its main stochastic properties, its non-Markovian and non-stationarity characteristics and the conditions under which it is not a semimartingale. We prove the long range dependence properties of this process.
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