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

Alina Semrau-Giłka

Displaying similar documents to “Euler's Approximations of Solutions of Reflecting SDEs with Discontinuous Coefficients”

Discrete Approximations of Strong Solutions of Reflecting SDEs with Discontinuous Coefficients

Alina Semrau (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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We study $L^{p}$ convergence for the Euler scheme for stochastic differential equations reflecting on the boundary of a general convex domain D ⊆ ℝd. We assume that the equation has the pathwise uniqueness property and its coefficients are measurable and continuous almost everywhere with respect to the Lebesgue measure. In the case D=[0,∞) new sufficient conditions ensuring pathwise uniqueness for equations with possibly discontinuous coefficients are given.

On Stochastic Differential Equations with Reflecting Boundary Condition in Convex Domains

Weronika Łaukajtys (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let D be an open convex set in $ℝ^{d}$ and let F be a Lipschitz operator defined on the space of adapted càdlàg processes. We show that for any adapted process H and any semimartingale Z there exists a unique strong solution of the following stochastic differential equation (SDE) with reflection on the boundary of D: $X_{t} = H_{t} + \int_{0}^{t} ⟨ F {(X)}_{s -}, d Z_{s} ⟩ + K_{t}$ , t ∈ ℝ⁺. Our proofs are based on new a priori estimates for solutions of the deterministic Skorokhod problem.

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 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...

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,...

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...

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 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...

Vectorial quasilinear diffusion equation with dynamic boundary condition

Nakayashiki, Ryota

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In this paper, we consider a class of initial-boundary value problems for quasilinear PDEs, subject to the dynamic boundary conditions. Each initial-boundary problem is denoted by (S) $_{ε}$ with a nonnegative constant $ε$ , and for any $ε \geq 0$ , (S) $_{ε}$ can be regarded as a vectorial transmission system between the quasilinear equation in the spatial domain $Ω$ , and the parabolic equation on the boundary $Γ : = \partial Ω$ , having a sufficient smoothness. The objective of this study is to establish a mathematical method,...

Metastability in reversible diffusion processes I: Sharp asymptotics for capacities and exit times

Anton Bovier, Michael Eckhoff, Véronique Gayrard, Markus Klein (2004)

Journal of the European Mathematical Society

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We develop a potential theoretic approach to the problem of metastability for reversible diffusion processes with generators of the form $- ϵ Δ + \nabla F (\cdot) \nabla$ on $ℝ^{d}$ or subsets of $ℝ^{d}$ , where $F$ is a smooth function with finitely many local minima. In analogy to previous work on discrete Markov chains, we show that metastable exit times from the attractive domains of the minima of $F$ can be related, up to multiplicative errors that tend to one as $ϵ ↓ 0$ , to the capacities of suitably constructed sets. We show that...

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...

Continuous pluriharmonic boundary values

Per Åhag, Rafał Czyż (2007)

Annales Polonici Mathematici

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Let $D_{j}$ be a bounded hyperconvex domain in $ℂ^{n_{j}}$ and set $D = D ₁ \times \dots \times D_{s}$ , j=1,...,s, s≥ 3. Also let ₙ be the symmetrized polydisc in ℂⁿ, n ≥ 3. We characterize those real-valued continuous functions defined on the boundary of D or ₙ which can be extended to the inside to a pluriharmonic function. As an application a complete characterization of the compliant functions is obtained.

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 $𝐌_{n, m}$ .

Nonlinear diffusion equations with perturbation terms on unbounded domains

Kurima, Shunsuke

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This paper considers the initial-boundary value problem for the nonlinear diffusion equation with the perturbation term $u_{t} + (- Δ + 1) β (u) + G (u) = g in Ω \times (0, T)$ in an unbounded domain $Ω \subset ℝ^{N}$ with smooth bounded boundary, where $N \in ℕ$ , $T > 0$ , $β$ , is a single-valued maximal monotone function on $ℝ$ , e.g., $β (r) = {| r |}^{q - 1} r (q > 0, q \neq 1)$ and $G$ is a function on $ℝ$ which can be regarded as a Lipschitz continuous operator from ${(H^{1} (Ω))}^{*}$ to ${(H^{1} (Ω))}^{*}$ . The present work establishes existence and estimates for the above problem.

A symmetry problem in the calculus of variations

Graziano Crasta (2006)

Journal of the European Mathematical Society

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We consider the integral functional $J (u) = \int_{Ω} [f (| D u |) - u] d x$ , $u \in W_{0}^{1, 1} (Ω)$ , where $Ω \subset ℝ^{n}$ , $n \geq 2$ , is a nonempty bounded connected open subset of $ℝ^{n}$ with smooth boundary, and $ℝ ∋ s \mapsto f (| s |)$ is a convex, differentiable function. We prove that if $J$ admits a minimizer in $W_{0}^{1, 1} (Ω)$ depending only on the distance from the boundary of $Ω$ , then $Ω$ must be a ball.

Weakly nonlinear stochastic CGL equations

Sergei B. Kuksin (2013)

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

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We consider the linear Schrödinger equation under periodic boundary conditions, driven by a random force and damped by a quasilinear damping: $\frac{d}{d t} u + i (- Δ + V (x)) u = ν (Δ u - γ_{R} {| u |}^{2 p} u - i γ_{I} {| u |}^{2 q} u) + \sqrt{ν} η (t, x) . (*)$ The force $η$ is white in time and smooth in $x$ ; the potential $V (x)$ is typical. We are concerned with the limiting, as $ν \to 0$ , behaviour of solutions on long time-intervals $0 \leq t \leq ν^{- 1} T$ , and with behaviour of these solutions under the double limit $t \to \infty$ and $ν \to 0$ . We show that these two limiting behaviours may be described in terms of solutions for the( $*$ ) which is a well...

Generalized Lebesgue points for Sobolev functions

Nijjwal Karak (2017)

Czechoslovak Mathematical Journal

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In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point $x$ in a metric measure space $(X, d, μ)$ is called a generalized Lebesgue point of a measurable function $f$ if the medians of $f$ over the balls $B (x, r)$ converge to $f (x)$ when $r$ converges to $0$ . We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function....