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In this paper we study the parabolic Anderson equation $\partial u (x, t) / \partial t = κ 𝛥 u (x, t) + ξ (x, t) u (x, t)$ , $x \in ℤ^{d}$ , $t \geq 0$ , where the $u$ -field and the $ξ$ -field are $ℝ$ -valued, $κ \in [0, \infty)$ is the diffusion constant, and $𝛥$ is the discrete Laplacian. The $ξ$ -field plays the role of adynamic random environmentthat drives the equation. The initial condition $u (x, 0) = u_{0} (x)$ , $x \in ℤ^{d}$ , is taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump...

The partial Malliavin calculus

David Nualart, Moshe Zakai (1989)

Séminaire de probabilités de Strasbourg

The Picard boundary value problem for a third order stochastic difference equation

Marco Ferrante (1996)

Rendiconti del Seminario Matematico della Università di Padova

The pricing of credit risky securities under stochastic interest rate model with default correlation

Anjiao Wang, Zhong Xing Ye (2013)

Applications of Mathematics

In this paper, we study the pricing of credit risky securities under a three-firms contagion model. The interacting default intensities not only depend on the defaults of other firms in the system, but also depend on the default-free interest rate which follows jump diffusion stochastic differential equation, which extends the previous three-firms models (see R. A. Jarrow and F. Yu (2001), S. Y. Leung and Y. K. Kwok (2005), A. Wang and Z. Ye (2011)). By using the method of change of measure and...

The principle of truncations in applied probability

Eugene Seneta (1968)

Commentationes Mathematicae Universitatis Carolinae

The probabilistic approach to the analysis of the limiting behavior of an integro-differential equation depending on a small parameter, and its application to stochastic processes.

Borisenko, O.V., Borisenko, A.D., Malyshev, I.G. (1994)

Journal of Applied Mathematics and Stochastic Analysis

The realization of positive random variables via absolutely continuous transformations of measure on Wiener space.

Feyel, D., Üstünel, A.S., Zakai, M. (2006)

Probability Surveys [electronic only]

The representation of Poisson functionals

Sheng-Wu He (1983)

Séminaire de probabilités de Strasbourg

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

Laure Dumaz, Bálint Virág (2013)

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

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 $P ({𝑇𝑊}_{β} g t; a) = a^{- (3 / 4) β + o (1)} exp (- \frac{2}{3} β a^{3 / 2}) .$

The set of probability distribution solutions of a linear functional equation

Janusz Morawiec, Ludwig Reich (2008)

Annales Polonici Mathematici

Let (Ω,,P) be a probability space and let τ: ℝ×Ω → ℝ be a function which is strictly increasing and continuous with respect to the first variable, measurable with respect to the second variable. Given the set of all continuous probability distribution solutions of the equation $F (x) = \int_{Ω} F (τ (x, ω)) d P (ω)$ we determine the set of all its probability distribution solutions.

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The multi-dimensional super-replication problem under gamma constraints

The multiple stochastic integrals and “non-Poissonian” transformations of the gamma measure.

The noise made by a Poisson snake.

The non-linear stochastic wave equation in high dimensions.

The norm of the product of a large matrix and a random vector.

The ordinary differential equation approach to asymptotically efficient schemes for solution of stochastic differential equations

The parabolic Anderson model in a dynamic random environment: Basic properties of the quenched Lyapunov exponent