Central Limit Results for Jump-Diffusions with Mean Field Interaction and a Common Factor

A system of $N$ weakly interacting particles whose dynamics is given in terms of jump-diffusions with a common factor is considered. The common factor is described through another jump-diffusion and the coefficients of the evolution equation for each particle depend, in addition to its own state value, on the empirical measure of the states of the $N$ particles and the common factor. A Central Limit Theorem, as $N \to \infty$, is established. The limit law is described in terms of a certain Gaussian mixture. An application to models in Mathematical Finance of self-excited correlated defaults is described

Infinite Dimensional Forward-Backward Stochastic Differential Equations and the KPZ Equation

Kardar-Parisi-Zhang (KPZ) equation is a quasilinear stochastic partial differential equation(SPDE) driven by a space-time white noise. In recent years there have been several works directed towards giving a rigorous meaning to a solution of this equation. Bertini, Cancrini and Giacomin have proposed a notion of a solution through a limiting procedure and a certain renormalization of the nonlinearity. In this work we study connections between the KPZ equation and certain infinite dimensional forward-backward stochastic differential equations. Forward-backward equations with a finite dimensional noise have been studied extensively, mainly motivated by problems in mathematical finance. Equations considered here differ from the classical works in that, in addition to having an infinite dimensional driving noise, the associated SPDE involves a non-Lipschitz (namely a quadratic) function of the gradient. Existence and uniqueness of solutions of such infinite dimensional forward-backward equations is established and the terminal values of the solutions are then used to give a new probabilistic representation for the solution of the KPZ equation

A Numerical Scheme for Invariant Distributions of Constrained Diffusions

Reflected diffusions in polyhedral domains are commonly used as approximate models for stochastic processing networks in heavy traffic. Stationary distributions of such models give useful information on the steady state performance of the corresponding stochastic networks and thus it is important to develop reliable and efficient algorithms for numerical computation of such distributions. In this work we propose and analyze a Monte-Carlo scheme based on an Euler type discretization of the reflected stochastic differential equation using a single sequence of time discretization steps which decrease to zero as time approaches infinity. Appropriately weighted empirical measures constructed from the simulated discretized reflected diffusion are proposed as approximations for the invariant probability measure of the true diffusion model. Almost sure consistency results are established that in particular show that weighted averages of polynomially growing continuous functionals evaluated on the discretized simulated system converge a.s. to the corresponding integrals with respect to the invariant measure. Proofs rely on constructing suitable Lyapunov functions for tightness and uniform integrability and characterizing almost sure limit points through an extension of Echeverria's criteria for reflected diffusions. Regularity properties of the underlying Skorohod problems play a key role in the proofs. Rates of convergence for suitable families of test functions are also obtained. A key advantage of Monte-Carlo methods is the ease of implementation, particularly for high dimensional problems. A numerical example of a eight dimensional Skorohod problem is presented to illustrate the applicability of the approach

Inviscid Large deviation principle and the 2D Navier Stokes equations with a free boundary condition

Using a weak convergence approach, we prove a LPD for the solution of 2D stochastic Navier Stokes equations when the viscosity converges to 0 and the noise intensity is multiplied by the square root of the viscosity. Unlike previous results on LDP for hydrodynamical models, the weak convergence is proven by tightness properties of the distribution of the solution in appropriate functional spaces

Large Deviations for Stochastic Evolution Equations with Small Multiplicative Noise

The Freidlin-Wentzell large deviation principle is established for the distributions of stochastic evolution equations with general monotone drift and small multiplicative noise. As examples, the main results are applied to derive the large deviation principle for different types of SPDE such as stochastic reaction-diffusion equations, stochastic porous media equations and fast diffusion equations, and the stochastic p-Laplace equation in Hilbert space. The weak convergence approach is employed in the proof to establish the Laplace principle, which is equivalent to the large deviation principle in our framework.Comment: 31 pages, published in Appl. Math. Opti

Ultrasonic Bioreactor as a Platform for Studying Cellular Response

The need for tissue-engineered constructs as replacement tissue continues to grow as the average age of the world’s population increases. However, additional research is required before the efficient production of laboratory-created tissue can be realized. The multitude of parameters that affect cell growth and proliferation is particularly daunting considering that optimized conditions are likely to change as a function of growth. Thus, a generalized research platform is needed in order for quantitative studies to be conducted. In this article, an ultrasonic bioreactor is described for use in studying the response of cells to ultrasonic stimulation. The work is focused on chondrocytes with a long-term view of generating tissue-engineered articular cartilage. Aspects of ultrasound (US) that would negatively affect cells, including temperature and cavitation, are shown to be insignificant for the US protocols used and which cover a wide range of frequencies and pressure amplitudes. The bioreactor is shown to have a positive influence on several factors, including cell proliferation, viability, and gene expression of select chondrocytic markers. Most importantly, we show that a total of 138 unique proteins are differentially expressed on exposure to ultrasonic stimulation, using mass-spectroscopy coupled proteomic analyses. We anticipate that this work will serve as the basis for additional research which will elucidate many of the mechanisms associated with cell response to ultrasonic stimulation

Ultrasonic Bioreactor as a Platform for Studying Cellular Response

The need for tissue-engineered constructs as replacement tissue continues to grow as the average age of the world’s population increases. However, additional research is required before the efficient production of laboratory-created tissue can be realized. The multitude of parameters that affect cell growth and proliferation is particularly daunting considering that optimized conditions are likely to change as a function of growth. Thus, a generalized research platform is needed in order for quantitative studies to be conducted. In this article, an ultrasonic bioreactor is described for use in studying the response of cells to ultrasonic stimulation. The work is focused on chondrocytes with a long-term view of generating tissue-engineered articular cartilage. Aspects of ultrasound (US) that would negatively affect cells, including temperature and cavitation, are shown to be insignificant for the US protocols used and which cover a wide range of frequencies and pressure amplitudes. The bioreactor is shown to have a positive influence on several factors, including cell proliferation, viability, and gene expression of select chondrocytic markers. Most importantly, we show that a total of 138 unique proteins are differentially expressed on exposure to ultrasonic stimulation, using mass-spectroscopy coupled proteomic analyses. We anticipate that this work will serve as the basis for additional research which will elucidate many of the mechanisms associated with cell response to ultrasonic stimulation