On Stochastic Differential Equations with Reflecting Boundary Condition in Convex Domains

Weronika Łaukajtys

Displaying similar documents to “On Stochastic Differential Equations with Reflecting Boundary Condition in Convex Domains”

Euler's Approximations of Solutions of Reflecting SDEs with Discontinuous Coefficients

Alina Semrau-Giłka (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let D be either a convex domain in $ℝ^{d}$ or a domain satisfying the conditions (A) and (B) considered by Lions and Sznitman (1984) and Saisho (1987). We investigate convergence in law as well as in $L^{p}$ for the Euler and Euler-Peano schemes for stochastic differential equations in D with normal reflection at the boundary. The coefficients are measurable, continuous almost everywhere with respect to the Lebesgue measure, and the diffusion coefficient may degenerate on some subsets of the domain. ...

Stochastic dynamical systems with weak contractivity properties II. Iteration of Lipschitz mappings

Marc Peigné, Wolfgang Woess (2011)

Colloquium Mathematicae

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In this continuation of the preceding paper (Part I), we consider a sequence ${(F ₙ)}_{n \geq 0}$ of i.i.d. random Lipschitz mappings → , where is a proper metric space. We investigate existence and uniqueness of invariant measures, as well as recurrence and ergodicity of the induced stochastic dynamical system (SDS) $X ₙ^{x} = F ₙ \circ . . . \circ F ₁ (x)$ starting at x ∈ . The main results concern the case when the associated Lipschitz constants are log-centered. Principal tools are local contractivity, as considered in detail in Part I,...

Stochastic convolution in separable Banach spaces and the stochastic linear Cauchy problem

Zdzisław Brzeźniak, Jan van Neerven (2000)

Studia Mathematica

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Let H be a separable real Hilbert space and let E be a separable real Banach space. We develop a general theory of stochastic convolution of ℒ(H,E)-valued functions with respect to a cylindrical Wiener process ${W_{t}^{H}}_{t \in [0, T]}$ with Cameron-Martin space H. This theory is applied to obtain necessary and sufficient conditions for the existence of a weak solution of the stochastic abstract Cauchy problem (ACP) $d X_{t} = A X_{t} d t + B d W_{t}^{H}$ (t∈ [0,T]), $X_{0} = 0$ almost surely, where A is the generator of a $C_{0}$ -semigroup ${S (t)}_{t \geq 0}$ of bounded linear...

On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes

Nicolas Fournier (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We study a one-dimensional stochastic differential equation driven by a stable Lévy process of order $α$ with drift and diffusion coefficients $b$ , $σ$ . When $α \in (1, 2)$ , we investigate pathwise uniqueness for this equation. When $α \in (0, 1)$ , we study another stochastic differential equation, which is equivalent in law, but for which pathwise uniqueness holds under much weaker conditions. We obtain various results, depending on whether $α \in (0, 1)$ or $α \in (1, 2)$ and on whether the driving stable process is symmetric or not. Our...

Initial measures for the stochastic heat equation

Daniel Conus, Mathew Joseph, Davar Khoshnevisan, Shang-Yuan Shiu (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider a family of nonlinear stochastic heat equations of the form $\partial_{t} u = ℒ u + σ (u) \dot{W}$ , where $\dot{W}$ denotes space–time white noise, $ℒ$ the generator of a symmetric Lévy process on $𝐑$ , and $σ$ is Lipschitz continuous and zero at 0. We show that this stochastic PDE has a random-field solution for every finite initial measure $u_{0}$ . Tight a priori bounds on the moments of the solution are also obtained. In the particular case that $ℒ f = c f^{''}$ for some $c g t; 0$ , we prove that if $u_{0}$ is a finite measure of compact support, then the...

A continuous mapping theorem for the argmin-set functional with applications to convex stochastic processes

Dietmar Ferger (2021)

Kybernetika

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For lower-semicontinuous and convex stochastic processes $Z_{n}$ and nonnegative random variables $ϵ_{n}$ we investigate the pertaining random sets $A (Z_{n}, ϵ_{n})$ of all $ϵ_{n}$ -approximating minimizers of $Z_{n}$ . It is shown that, if the finite dimensional distributions of the $Z_{n}$ converge to some $Z$ and if the $ϵ_{n}$ converge in probability to some constant $c$ , then the $A (Z_{n}, ϵ_{n})$ converge in distribution to $A (Z, c)$ in the hyperspace of Vietoris. As a simple corollary we obtain an extension of several argmin-theorems in the literature. In particular,...

On risk reserve under distribution constraints

Mariusz Michta (2000)

Discussiones Mathematicae Probability and Statistics

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The purpose of this work is a study of the following insurance reserve model: $R (t) = η + \int_{0}^{t} p (s, R (s)) d s + \int_{0}^{t} σ (s, R (s)) d W_{s} - Z (t)$ , t ∈ [0,T], P(η ≥ c) ≥ 1-ϵ, ϵ ≥ 0. Under viability-type assumptions on a pair (p,σ) the estimation γ with the property: $i n f_{0 \leq t \leq T} P R (t) \geq c \geq γ$ is considered.

Quasi-diffusion solution of a stochastic differential equation

Agnieszka Plucińska, Wojciech Szymański (2007)

Applicationes Mathematicae

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We consider the stochastic differential equation $X_{t} = X ₀ + \int_{0}^{t} (A_{s} + B_{s} X_{s}) d s + \int_{0}^{t} C_{s} d Y_{s}$ , where $A_{t}$ , $B_{t}$ , $C_{t}$ are nonrandom continuous functions of t, X₀ is an initial random variable, $Y = (Y_{t}, t \geq 0)$ is a Gaussian process and X₀, Y are independent. We give the form of the solution ( $X_{t}$ ) to (0.1) and then basing on the results of Plucińska [Teor. Veroyatnost. i Primenen. 25 (1980)] we prove that ( $X_{t}$ ) is a quasi-diffusion proces.

On smoothing properties of transition semigroups associated to a class of SDEs with jumps

Seiichiro Kusuoka, Carlo Marinelli (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We prove smoothing properties of nonlocal transition semigroups associated to a class of stochastic differential equations (SDE) in $ℝ^{d}$ driven by additive pure-jump Lévy noise. In particular, we assume that the Lévy process driving the SDE is the sum of a subordinated Wiener process $Y$ (i.e. $Y = W \circ T$ , where $T$ is an increasing pure-jump Lévy process starting at zero and independent of the Wiener process $W$ ) and of an arbitrary Lévy process independent of $Y$ , that the drift coefficient is continuous...

Stochastic optimization problems with second order stochastic dominance constraints via Wasserstein metric

Vlasta Kaňková, Vadim Omelčenko (2018)

Kybernetika

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Optimization problems with stochastic dominance constraints are helpful to many real-life applications. We can recall e. g., problems of portfolio selection or problems connected with energy production. The above mentioned constraints are very suitable because they guarantee a solution fulfilling partial order between utility functions in a given subsystem $𝒰$ of the utility functions. Especially, considering $𝒰 : = 𝒰_{1}$ (where $𝒰_{1}$ is a system of non decreasing concave nonnegative utility functions)...

On reliability analysis of consecutive $k$ -out-of- $n$ systems with arbitrarily dependent components

Ebrahim Salehi (2016)

Applications of Mathematics

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In this paper, we consider the linear and circular consecutive $k$ -out-of- $n$ systems consisting of arbitrarily dependent components. Under the condition that at least $n - r + 1$ components ( $r \leq n$ ) of the system are working at time $t$ , we study the reliability properties of the residual lifetime of such systems. Also, we present some stochastic ordering properties of residual lifetime of consecutive $k$ -out-of- $n$ systems. In the following, we investigate the inactivity time of the component with lifetime...

Systems of Bellman Equations to Stochastic Differential Games with Discount Control

Alain Bensoussan, Jens Frehse (2008)

Bollettino dell'Unione Matematica Italiana

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We consider two dimensional diagonal elliptic systems $\Delta u+au=H(x,u,\nabla u)$ which arise from stochastic differential games with discount control. The Hamiltonians $H$ have quadratic growth in $\nabla u$ and a special structure which has notyet been covered by regularity theory. Without smallness condition on $H$ , the existence of a regular solution is established.

Some Results on Stochastic Porous Media Equations

Viorel Barbu, Giuseppe Da Prato, Michael Röckner (2008)

Bollettino dell'Unione Matematica Italiana

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Some recent results about nonnegative solutions of stochastic porous media equations in bounded open subsets of $\mathbb{R}^{3}$ are considered. The existence of an invariant measure is proved.

The right tail exponent of the Tracy–Widom $β$ distribution

Laure Dumaz, Bálint Virág (2013)

Annales de l'I.H.P. Probabilités et statistiques

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The Tracy–Widom $β$ distribution is the large dimensional limit of the top eigenvalue of $β$ random matrix ensembles. We use the stochastic Airy operator representation to show that as $a \to \infty$ the tail of the Tracy–Widom distribution satisfies $P ({𝑇𝑊}_{β} g t; a) = a^{- (3 / 4) β + o (1)} exp (- \frac{2}{3} β a^{3 / 2}) .$

Stochastic differential equations with Sobolev drifts and driven by $α$ -stable processes

Xicheng Zhang (2013)

Annales de l'I.H.P. Probabilités et statistiques

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In this article we prove the pathwise uniqueness for stochastic differential equations in $ℝ^{d}$ with time-dependent Sobolev drifts, and driven by symmetric $α$ -stable processes provided that $α \in (1, 2)$ and its spectral measure is non-degenerate. In particular, the drift is allowed to have jump discontinuity when $α \in (\frac{2 d}{d + 1}, 2)$ . Our proof is based on some estimates of Krylov’s type for purely discontinuous semimartingales.

Explicit construction of a unitary double product integral

R. L. Hudson, Paul Jones (2011)

Banach Center Publications

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In analogy with earlier work on the forward-backward case, we consider an explicit construction of the forward-forward double stochastic product integral $\prod^{\to \to} (1 + d r)$ with generator $d r = λ (d A^{†} \otimes d A - d A \otimes d A^{†})$ . The method of construction is to approximate the product integral by a discrete double product $\prod_{(j, k) \in ℕ_{m} \times ℕ ₙ}^{\to \to} Γ (R_{m, n}^{(j, k)}) = Γ (\prod_{(j, k) \in ℕ_{m} \times ℕ ₙ}^{\to \to} (R_{m, n}^{(j, k)}))$ of second quantised rotations $R_{m, n}^{(j, k)}$ in different planes using the embedding of $ℂ^{m} \oplus ℂ ⁿ$ into L²(ℝ) ⊕ L²(ℝ) in which the standard orthonormal bases of $ℂ^{m}$ and ℂⁿ are mapped to the orthonormal sets consisting of normalised indicator...

Doubly stochastic matrices and the Bruhat order

Richard A. Brualdi, Geir Dahl, Eliseu Fritscher (2016)

Czechoslovak Mathematical Journal

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The Bruhat order is defined in terms of an interchange operation on the set of permutation matrices of order $n$ which corresponds to the transposition of a pair of elements in a permutation. We introduce an extension of this partial order, which we call the stochastic Bruhat order, for the larger class $Ω_{n}$ of doubly stochastic matrices (convex hull of $n \times n$ permutation matrices). An alternative description of this partial order is given. We define a class of special faces of $Ω_{n}$ induced by permutation...

G-tridiagonal majorization on $𝐌_{n, m}$

Ahmad Mohammadhasani, Yamin Sayyari, Mahdi Sabzvari (2021)

Communications in Mathematics

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For $X, Y \in 𝐌_{n, m}$ , it is said that $X$ is majorized by $Y$ (and it is denoted by $X ≺_{g t} Y$ ) if there exists a tridiagonal g-doubly stochastic matrix $A$ such that $X = A Y$ . In this paper, the linear preservers and strong linear preservers of $≺_{g t}$ are characterized on $𝐌_{n, m}$ .

Monotone substochastic operators and a new Calderón couple

Karol Leśnik (2015)

Studia Mathematica

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An important result on submajorization, which goes back to Hardy, Littlewood and Pólya, states that b ⪯ a if and only if there is a doubly stochastic matrix A such that b = Aa. We prove that under monotonicity assumptions on the vectors a and b the matrix A may be chosen monotone. This result is then applied to show that $(\tilde{L^{p}}, L^{\infty})$ is a Calderón couple for 1 ≤ p < ∞, where $\tilde{L^{p}}$ is the Köthe dual of the Cesàro space $C e s_{p^{'}}$ (or equivalently the down space $L_{↓}^{p^{'}}$ ). In particular, $(\tilde{L ¹}, L^{\infty})$ is a Calderón couple, which...