60H10 Stochastic ordinary differential equations [See also 34F05]
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- Diffusionsprozess (2)
- Stochastische Differentialgleichung (2)
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This dissertation presents a generalization of the generalized grey Brownian motion with componentwise independence, called a vector-valued generalized grey Brownian motion (vggBm), and builds a framework of mathematical analysis around this process with the aim of solving stochastic differential equations with respect to this process. Similar to that of the one-dimensional case, the construction of vggBm starts with selecting the appropriate nuclear triple, and construct the corresponding probability measure on the co-nuclear space. Since independence of components are essential in constructing vggBm, a natural way to achieve this is to use the nuclear triple of product spaces: \[ \mathcal{S}_d(\mathbb{R}) \subset L^2_d(\mathbb{R}) \subset \mathcal{S}_d'(\mathbb{R}), \]
where \( L^2_d(\mathbb{R}) \) is the real separable Hilbert space of \( \mathbb{R}^d \)-valued square integrable functions on \( \mathbb{R} \) with respect to the Lebesgue measure, \( \mathcal{S}_d(\mathbb{R}) \) is the external direct sum of \(d\) copies of the nuclear space \(\mathcal{S}(\mathbb{R})\) of Schwartz test functions, and \(\mathcal{S}_d'(\mathbb{R})\) is the dual space of \(\mathcal{S}_d(\mathbb{R})\).
The probability measure used is the the \(d\)-fold product measure of the Mittag-Leffler measure, denoted by \(\mu_{\beta}^{\otimes d}\), whose characteristic function is given by \[ \int_{\mathcal{S}_d'(\mathbb{R})} e^{i\langle\omega,\varphi\rangle}\,\text{d}\mu_{\beta}^{\otimes d}(\omega) = \prod_{k=1}^{d}E_\beta\left(-\frac{1}{2}\langle\varphi_k,\varphi_k\rangle\right),\qquad \varphi\in \mathcal{S}_d(\mathbb{R}), \]
where \( \beta\in(0,1] \), and \( E_\beta \) is the Mittag-Leffler function. Vector-valued generalized grey Brownian motion, denoted by \( B^{\beta,\alpha}_{d}:=(B^{\beta,\alpha}_{d,t})_{t\geq 0}\), is then defined as a process taking values in \( L^2(\mu_{\beta}^{\otimes d};\mathbb{R}^d) \) given by
\[ B^{\beta,\alpha}_{d,t}(\omega) := (\langle\omega_1,M^{\alpha/2}_{-}1\!\!1_{[0,t)}\rangle,\dots,\langle\omega_d,M^{\alpha/2}_{-}1\!\!1_{[0,t)}\rangle),\quad \omega\in\mathcal{S}_d'(\mathbb{R}), \]
where \( M^{\alpha/2} \) is an appropriate fractional operator indexed by \( \alpha\in(0,2) \) and \( 1\!\!1_{[0,t)} \) is the indicator function on the interval \( [0,t) \). This process is, in general, not the aforementioned \(d\)-dimensional analogues of ggBm for \(d\geq 2\), since componentwise independence of the latter process holds only in the Gaussian case.
The study of analysis around vggBm starts with accessibility to Appell systems, so that characterizations and tools for the analysis of the corresponding distribution spaces are established. Then, explicit examples of the use of these characterizations and tools are given: the construction of Donsker's delta function, the existence of local times and self-intersection local times of vggBm, the existence of the derivative of vggBm in the sense of distributions, and the existence of solutions to linear stochastic differential equations with respect to vggBm.
We propose and analyze a multiscale model for acid-mediated tumor invasion
accounting for stochastic effects on the subcellular level.
The setting involves a PDE of reaction-diffusion-taxis type describing the evolution of the tumor cell density,
the movement being directed towards pH gradients in the local microenvironment,
which is coupled to a PDE-SDE system characterizing the
dynamics of extracellular and intracellular proton concentrations, respectively.
The global well-posedness of the model is shown and
numerical simulations are performed in order to illustrate the solution behavior.
A nonlocal stochastic model for intra- and extracellular proton dynamics in a tumor is proposed.
The intracellular dynamics is governed by an SDE coupled to a reaction-diffusion
equation for the extracellular proton concentration on the macroscale. In a more general context
the existence and uniqueness of solutions for local and nonlocal
SDE-PDE systems are established allowing, in particular, to analyze the proton dynamics model both,
in its local version and the case with nonlocal path dependence.
Numerical simulations are performed
to illustrate the behavior of solutions, providing some insights into the effects of randomness on tumor acidity.
Das zentrale Thema dieser Arbeit sind vollständig gekoppelte reflektierte Vorwärts-Rückwärts-Stochastische-Differentialgleichungen (FBSDE). Zunächst wird ein Spezialfall, die teilweise gekoppelten FBSDE, betrachtet und deren Verbindung zur Bewertung Amerikanischer Optionen aufgezeigt. Für die Lösung dieser Gleichung wird Monte-Carlo-Simulation benötigt, daher werden verschiedene Varianzreduktionsmaßnahmen erarbeitet und miteinander verglichen. Im Folgenden wird der allgemeinere Fall der vollständig gekoppelten reflektierten FBSDE behandelt. Es wird gezeigt, wie das Problem der Lösung dieser Gleichungen in ein Optimierungsproblem übertragen werden kann und infolgedessen mit numerischen Methoden aus diesem Bereich der Mathematik bearbeitet werden kann. Abschließend folgen Vergleiche der verschiedenen numerischen Ansätze mit bereits existierenden Verfahren.
Diese Diplomarbeit gibt eine kurze Einführung in das Gebiet der Diffusionsprozesse (beschrieben als Lösungen stochastischer Differentialgleichungen) und der großen Abweichungen. Mit Methoden aus dem Gebiet der großen Abweichungen wird dann das asymptotische Verhalten des Bayesrisikos für die unterscheidung zweier Diffusionsprozesse untersucht.
In this text we survey some large deviation results for diffusion processes. The first chapters present results from the literature such as the Freidlin-Wentzell theorem for diffusions with small noise. We use these results to prove a new large deviation theorem about diffusion processes with strong drift. This is the main result of the thesis. In the later chapters we give another application of large deviation results, namely to determine the exponential decay rate for the Bayes risk when separating two different processes. The final chapter presents techniques which help to experiment with rare events for diffusion processes by means of computer simulations.
Nonlinear dissipativity, asymptotical stability, and contractivity of (ordinary) stochastic differential equations (SDEs) with some dissipative structure and their discretizations are studied in terms of their moments in the spirit of Pliss (1977). For this purpose, we introduce the notions and discuss related concepts of dissipativity, growth- bounded and monotone coefficient systems, asymptotical stability and contractivity in wide and narrow sense, nonlinear A-stability, AN-stability, B-stability and BN-stability for stochastic dynamical systems - more or less as stochastic counterparts to deterministic concepts. The test class of in a broad sense interpreted dissipative SDEs as natural analogon to dissipative deterministic differential systems is suggested for stochastic-numerical methods. Then, in particular, a kind of mean square calculus is developed, although most of ideas and analysis can be carried over to general "stochastic Lp-case" (p * 1). By this natural restriction, the new stochastic concepts are theoretically meaningful, as in deterministic analysis. Since the choice of step sizes then plays no essential role in related proofs, we even obtain nonlinear A-stability, AN-stability, B-stability and BN-stability in the mean square sense for this implicit method with respect to appropriate test classes of moment-dissipative SDEs.
Nonlinear stochastic dynamical systems as ordinary stochastic differential equations and stochastic difference methods are in the center of this presentation in view of the asymptotical behaviour of their moments. We study the exponential p-th mean growth behaviour of their solutions as integration time tends to infinity. For this purpose, the concepts of nonlinear contractivity and stability exponents for moments are introduced as generalizations of well-known moment Lyapunov exponents of linear systems. Under appropriate monotonicity assumptions we gain uniform estimates of these exponents from above and below. Eventually, these concepts are generalized to describe the exponential growth behaviour along certain Lyapunov-type functionals.