Stochastic Processes with Applications

Bhattacharya, Rabi; Waymire, Edward C.

doi:10.1137/1.9780898718997

Cited by 152 publications

(160 citation statements)

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“…Then if the p j s are independent and identically distributed with variance σ 2 > 0 and mean m, the functional central limit theorem, see Theorem 8.1 in [9], says that the stochastic processes {x n t ,t ≥ 0} converge (in distribution) to a Brownian motion b t starting at the origin with zero drift and diffusion coefficent 1, as n → ∞. This must be done in the direction of any Fourier component (e k = exp(2πik · x)) that form a basis in the infinite dimensional space L 2 and the result is the differential of an infinite dimensional Brownian motion…”

Section: The Deterministic Navier-stokes Equationmentioning

confidence: 99%

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The Kolmogorov–Obukhov Statistical Theory of Turbulence

Birnir

2013

J Nonlinear Sci

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In 1941 Kolmogorov and Obukhov proposed that there exists a statistical theory of turbulence that should allow the computation of all the statistical quantities that can be computed and measured in turbulent systems. These are quantities such as the moments, the structure functions and the probability density functions (PDFs) of the turbulent velocity field. In this paper we will outline how to construct this statistical theory from the stochastic Navier-Stokes equation. The additive noise in the stochastic Navier-Stokes equation is generic noise given by the central limit theorem and the large deviation principle. The multiplicative noise consists of jumps multiplying the velocity, modeling jumps in the velocity gradient. We first estimate the structure functions of turbulence and establish the Kolmogorov-Obukhov '62 scaling hypothesis with the She-Leveque intermittency corrections. Then we compute the invariant measure of turbulence writing the stochastic Navier-Stokes equation as an infinite-dimensional Ito process and solving the linear Kolmogorov-Hopf functional differential equation for the invariant measure. Finally we project the invariant measure onto the PDF. The PDFs turn out to be the normalized inverse Gaussian (NIG) distributions of Barndorff-Nilsen, and compare well with PDFs from simulations and experiments.

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Section: The Deterministic Navier-stokes Equationmentioning

confidence: 99%

“…Now squaring and summing in k we get (13). Thus for non-degenerate fluid velocities u that satisfy (9),…”

Section: Integral Equation and Spectrum Of The Navierstokes Operatormentioning

confidence: 99%

The Kolmogorov–Obukhov Statistical Theory of Turbulence

Birnir

2013

J Nonlinear Sci

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“…Because we assume, that the reader of this work is perfectly acquainted with the terms and definitions of stochastic calculus we just state the theorems we have used in the Appendix. For detailed introduction to martingale theory and stochastic analysis we refer to Protter [60], Bhattacharya and Waymire [9], Karatzas and Shreve [49] or Ethier and Kurtz [23].…”

Section: Measuring Risksmentioning

confidence: 99%

“…In this appendix we consider some well known results, which state the processes to be martingales under some special constrains. For detailed introduction to martingales we refer for example to Protter [60], to Bhattacharya and Waymire [9] or to Rogers and Williams [63,64].…”

Section: Remark 528mentioning

confidence: 99%

On optimal control of capital injections by reinsurance and investments

Eisenberg¹

2010

Blätter DGVFM

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AcknowledgementsI would like to express my thanks to my supervisor Prof. Dr. Hanspeter Schmidli. I have greatly profited from his hints and suggestions generously given during our conversations. I am very grateful to Prof. Dr. Josef Steinebach who agreed to be coreferent.I would like to acknowledge Markus Schulz for his listening ear and Mrs Anderka for believing in me.In particular, I would like to thank my family who has supported me all the time and enabled a successful completion of this work. AbstractAn insurance company, having an initial capital x, cashes premiums continuously and pays claims of random sizes at random times. In addition to that, the company can buy reinsurance or/and invest money into a riskless or risky assets. The company holders are confronted with the problem of taking decisions on a business policy of the company. Thus, a measure for the risk connected with an insurance portfolio is sorely needed.The ruin probability, i.e. the probability that the surplus process becomes negative in finite time, is typically the measure for an insurance company's solvency. However, the ruin probability approach has been criticised among other things for not considering the severity of an insolvency and for ignoring the time value of money.An alternative to measure the risk of a surplus process is to consider the value of expected discounted capital injections, which are necessary to keep the process above zero. Naturally, it raises the question how to minimise this value. If the company holders prefer (or are indifferent) investing tomorrow to investing today, it is optimal to inject capital only when the surplus becomes negative and only as much as is necessary to keep the process above zero.In the first part of this work, we solve the problem of minimising the expected discounted capital injections over all dynamic reinsurance strategies for the classical risk model and its diffusion approximation. In the second part, we extend the concept by adding the possibility of investing money, if the surplus remains positive, into a riskless asset. In these two cases we are able to show the existence and uniqueness of the optimal reinsurance strategy and the value function as the minimising value of expected discounted capital injections.In the third part, we consider the surplus process, where the company holders can invest money into a risky asset modeled as a Black-Scholes model. The forth part extends the setup of the third part by possibility of reinsurance. In the last two cases we solve the problem explicitly only for the case of diffusion approximation. In the classical risk model the concept of viscosity solutions introduced by Crandall and Lions has been used.All the studies are illustrated by simulations, written in Java. Als alternatives Risikomaß betrachtet man den Wert der erwarteten diskontierten Kapitalzuführungen, welche notwendig sind damit derÜberschussprozess nichtnegativ bleibt. Es stellt sich die Frage, wie man diesen Wert minimiert. Wir nehmen an, dass die Inhaber der Versicherungsges...

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“…The proof is based on Central Limit Theorem for Markov processes 3 (see Theorem III.8.6 in Bhattacharya and Waymire [4]). Under the assumptions of the theorem The second part of (C) follows along the same lines; one simply has to replace the process (ρ(t)) by (−ρ(t)).…”

Section: Proof Of Theorem 2 Define a New Variablementioning

confidence: 99%