Catalogue Search | MBRL
Are you sure you want to remove the book from the shelf?
{{itemTitle}}
5,229
result(s) for
"Stochastic calculus"
Sort by:
Introduction to stochastic calculus with applications
This text presents a concise and rigorous treatment of stochastic calculus. It also gives its main applications in finance, biology and engineering.
Book
Decoupling on the Wiener Space, Related Besov Spaces, and Applications to BSDEs
We introduce a decoupling method on the Wiener space to define a wide class of anisotropic Besov spaces. The decoupling method is
based on a general distributional approach and not restricted to the Wiener space.
The class of Besov spaces we introduce
contains the traditional isotropic Besov spaces obtained by the real interpolation method, but also new spaces that are designed to
investigate backwards stochastic differential equations (BSDEs). As examples we discuss the Besov regularity (in the sense of our
spaces) of forward diffusions and local times. It is shown that among our newly introduced Besov spaces there are spaces that
characterize quantitative properties of directional derivatives in the Malliavin sense without computing or accessing these Malliavin
derivatives explicitly.
Regarding BSDEs, we deduce regularity properties of the solution processes from the Besov regularity of
the initial data, in particular upper bounds for their
Among other tools, we use methods from harmonic analysis. As a
by-product, we improve the asymptotic behaviour of the multiplicative constant in a generalized Fefferman inequality and verify the
optimality of the bound we established.
eBook
FUNCTIONAL ITÔ CALCULUS AND STOCHASTIC INTEGRAL REPRESENTATION OF MARTINGALES
We develop a nonanticipative calculus for functionals of a continuous semimartingale, using an extension of the Itô formula to path-dependent functionals which possess certain directional derivatives. The construction is based on a pathwise derivative, introduced by Dupire, for functionals on the space of right-continuous functions with left limits. We show that this functional derivative admits a suitable extension to the space of square-integrable martingales. This extension defines a weak derivative which is shown to be the inverse of the Itô integral and which may be viewed as a nonanticipative \"lifting\" of the Malliavin derivative. These results lead to a constructive martingale representation formula for Itô processes. By contrast with the Clark—Haussmann—Ocone formula, this representation only involves nonanticipative quantities which may be computed pathwise.
Journal Article
The SLH framework for modeling quantum input-output networks
Many emerging quantum technologies demand precise engineering and control over networks consisting of quantum mechanical degrees of freedom connected by propagating electromagnetic fields, or quantum input-output networks. Here we review recent progress in theory and experiment related to such quantum input-output networks, with a focus on the SLH framework, a powerful modeling framework for networked quantum systems that is naturally endowed with properties such as modularity and hierarchy. We begin by explaining the physical approximations required to represent any individual node of a network, e.g. atoms in cavity or a mechanical oscillator, and its coupling to quantum fields by an operator triple (S,L,H). Then we explain how these nodes can be composed into a network with arbitrary connectivity, including coherent feedback channels, using algebraic rules, and how to derive the dynamics of network components and output fields. The second part of the review discusses several extensions to the basic SLH framework that expand its modeling capabilities, and the prospects for modeling integrated implementations of quantum input-output networks. In addition to summarizing major results and recent literature, we discuss the potential applications and limitations of the SLH framework and quantum input-output networks, with the intention of providing context to a reader unfamiliar with the field.
Journal Article
A WEAK VERSION OF PATH-DEPENDENT FUNCTIONAL ITÔ CALCULUS
We introduce a variational theory for processes adapted to the multidimensional Brownian motion filtration that provides a differential structure allowing to describe infinitesimal evolution of Wiener functionals at very small scales. The main novel idea is to compute the “sensitivities” of processes, namely derivatives of martingale components and a weak notion of infinitesimal generator, via a finite-dimensional approximation procedure based on controlled inter-arrival times and approximating martingales. The theory comes with convergence results that allow to interpret a large class ofWiener functionals beyond semimartingales as limiting objects of differential forms which can be computed path wisely over finite-dimensional spaces. The theory reveals that solutions of BSDEs are minimizers of energy functionals w.r.t. Brownian motion driving noise.
Journal Article
Absolute Continuity under Time Shift of Trajectories and Related Stochastic Calculus
The text is concerned with a class of two-sided stochastic processes of the form X=W+A. Here W is a two-sided Brownian motion with random initial data at time zero and A\\equiv A(W) is a function of W. Elements of the related stochastic calculus are introduced. In particular, the calculus is adjusted to the case when A is a jump process. Absolute continuity of (X,P) under time shift of trajectories is investigated. For example under various conditions on the initial density with respect to the Lebesgue measure, m, and on A with A_0=0 we verify \\frac{P(dX_{\\cdot -t})}{P(dX_\\cdot)}=\\frac{m(X_{-t})}{m(X_0)}\\cdot \\prod_i\\left|\\nabla_{d,W_0}X_{-t}\\right|_i i.e. where the product is taken over all coordinates. Here \\sum_i \\left(\\nabla_{d,W_0}X_{-t}\\right)_i is the divergence of X_{-t} with respect to the initial position. Crucial for this is the temporal homogeneity of X in the sense that X\\left(W_{\\cdot +v}+A_v \\mathbf{1}\\right)=X_{\\cdot+v}(W), v\\in {\\mathbb R}, where A_v \\mathbf{1} is the trajectory taking the constant value A_v(W). By means of such a density, partial integration relative to a generator type operator of the process X is established. Relative compactness of sequences of such processes is established.
eBook
Analysis of fractional stochastic evolution equations by using Hilfer derivative of finite approximate controllability
The approximate controllability of a class of fractional stochastic evolution equations (FSEEs) are discussed in this study utilizes the Hilbert space by using Hilfer derivative. For different approaches, we remove the Lipschitz or compactness conditions and merely have to assume a weak growth requirement. The fixed point theorem, the diagonal argument, and approximation methods serve as the foundation for the study. The abstract theory is demonstrated using an example. A conclusion is given at the end.
Journal Article
Universal cutoff for Dyson Ornstein Uhlenbeck process
We study the Dyson–Ornstein–Uhlenbeck diffusion process, an evolving gas of interacting particles. Its invariant law is the beta Hermite ensemble of random matrix theory, a non-product log-concave distribution. We explore the convergence to equilibrium of this process for various distances or divergences, including total variation, relative entropy, and transportation cost. When the number of particles is sent to infinity, we show that a cutoff phenomenon occurs: the distance to equilibrium vanishes abruptly at a critical time. A remarkable feature is that this critical time is independent of the parameter beta that controls the strength of the interaction, in particular the result is identical in the non-interacting case, which is nothing but the Ornstein–Uhlenbeck process. We also provide a complete analysis of the non-interacting case that reveals some new phenomena. Our work relies among other ingredients on convexity and functional inequalities, exact solvability, exact Gaussian formulas, coupling arguments, stochastic calculus, variational formulas and contraction properties. This work leads, beyond the specific process that we study, to questions on the high-dimensional analysis of heat kernels of curved diffusions.
Journal Article
Torsion-Induced Quantum Fluctuations in Metric-Affine Gravity Using the Stochastic Variational Method
This review paper comprehensively examines the influence of spatial torsion on quantum fluctuations from the perspectives of metric-affine gravity (MAG) and the stochastic variational method (SVM). We first outline the fundamental framework of MAG, a generalized theory that includes both torsion and non-metricity, and discuss the geometrical significance of torsion within this context. Subsequently, we summarize SVM, a powerful technique that facilitates quantization while effectively incorporating geometrical effects. By integrating these frameworks, we evaluate how the geometrical structures originating from torsion affect quantum fluctuations, demonstrating that they induce non-linearity in quantum mechanics. Notably, torsion, traditionally believed to influence only spin degrees of freedom, can also affect spinless degrees of freedom via quantum fluctuations. Furthermore, extending beyond the results of previous work [Koide and van de Venn, Phys. Rev. A112, 052217 (2025)], we investigate the competitive interplay between the Levi-Civita curvature and torsion within the non-linearity of the Schrödinger equation. Finally, we discuss the structural parallelism between SVM and information geometry, highlighting that the splitting of time derivatives in stochastic processes corresponds to the dual connections in statistical manifolds. These insights pave the way for future extensions to gravity theories involving non-metricity and are expected to deepen our understanding of unresolved cosmological problems.
Journal Article
Quantum analysis of nonlinear optics in Kerr affected saturable nonlinear media and multiplicative noise: a path to new discoveries
Solitons are characterized by their ability to maintain their shape and velocity as they propagate through a medium, and are also known for their stability against mutual collisions. However, it is crucial to investigate the behavior of nonlinear partial differential equations under random environmental conditions, as this has important implications for a wide range of fields and applications. In this study, the stochastic soliton solutions of the saturable nonlinear Schrödinger equation are found using extended generalized Riccati equation mapping method. To our knowledge, the stochastic solitary wave solutions we have obtained are new in the literature. The discovery of these new solutions is of great importance due to the widespread use of the nonlinear Schrödinger equation in fields such as hydrodynamics, nonlinear optics, and nuclear physics. These solutions exhibit a combination of random behavior and soliton structures that differ from one another. This is an exciting development with potential implications for fields such as physics, engineering, and mathematics. The availability of these solutions provides researchers with a valuable tool to better understand and explain a variety of captivating physical phenomenon. To visualize the behavior of the stochastic soliton solutions, 3D and contour graphs are displayed using Matlab in graphical representation section.
Journal Article