Model of Continuous Random Cascade Processes in Financial Markets

3 - Time varying signals

Annals of operations research (vol. 28 - 1991) Markov decision processes — Part I Annals of operations research (vol. 29 - 1991) Markov decision processes — Part II: O. Hernandez-Lerma and J.B. Lasserre (eds.) Baltzer AG, Basel, 1991, 649 pages, ISSN 0254 5330

Many types of financial time series, most notably market returns, have been found to exhibit long-range memory as well as dramatic day-to-day swings that cannot be adequately represented by light-tailed distributions such as the normal distribution. In particular, this means that for such time series, the usual variance parameter (i.e., the sum of covariances at all time lags) is not defined because the covariance function does not converge to zero fast enough as the time lag increases. Moreover in such time series, often the tails of the marginal density converge to zero so slowly that higher-order marginal moments such as skewness and kurtosis fail to exist. Therefore conventional methods for analyzing simulation-generated time series cannot generally be applied to high-fidelity simulations of financial markets.Building on earlier work in fractal geometry and fractal time series, Mandelbrot et al. proposed the multifractal model of asset returns (MMAR) as an alternative to the ARCH models for analyzing time series exhibiting volatility clustering, long-range dependence, and heavy-tailed returns (Mandelbrot et al. 1997). They defined the multifractal spectrum as the renormalized probability density function of the Holder exponents observed in the time series; and they used the multifractal spectrum to measure the ability of MMAR to match the statistical properties of real data. In 2002 Kantelhardt et al. formulated multifractal detrended fluctuation analysis (MF-DFA), an algorithm for extracting the multifractal spectrum from a time series (Kantelhardt et al. 2002).Many economists have recently adopted agent-based simulation for modeling financial markets. Fads based on new products, shocks related to world events, scandals involving company leaders, or outright criminal activity can drastically change the processes and relationships governing a market. Already there is evidence that agent-based models yield more accurate approximations to the true or observed behavior in financial markets. Agent-based models exhibit emergent behaviors that have been linked to non-Gaussian interaction metrics and singularities in the time series they generate (Chan 2011). Farmer et al. constructed a agent-based model that is capable of exhibiting many of the statistical properties observed in real financial data simply due to the rules governing a double auction order book (Farmer et al. 2005).During the 20th century, economists relied heavily on Brownian motion to construct theoretical models of finance. The advantages of using Brownian motion stem from its foundation in the Gaussian distribution; and with advances in stochastic calculus and differential equations, it became a staple in models of empirical finance. The well-known Black-Scholes model for options pricing is a prime example of this application. Multifractal analysis has the ability to illuminate the underlying difference between a monofractal like Brownian motion and a considerably more diverse construction like the multifractal binomial measure. Thus, combining multifractal analysis with an agent-based model has the potential to provide insight into the underlying processes and behaviors of a dynamic economic system that is constantly changing and evolving.To analyze market-return time series exhibiting volatility clustering, long-range dependence, or heavy-tailed marginals, we exploit multifractal analysis and agent-based simulation. We develop a robust, automated software tool for extracting the multifractal spectrum of a time series based on MF-DFA. Guidelines are given for setting MF-DFA's parameters in practice. The software is tested on simulated data with closed-form monofractal and multifractal spectra as well as on observed data, and the results are analyzed. We also present a prototype agent-based financial market model based on the zero-intelligence model of Farmer et al. and analyze its output using MF-DFA. The figure below compares the multifractal spectrum of the output from the prototype model with that of the multifractal binomial measure. The ultimate objective is to expand this model to study the effects of microlevel agent behaviors on the macrolevel time series output as analyzed by MF-DFA. Finally we explore the potential for validating agent-based models using MF-DFA and thus being able to tune these models to the multifractal spectrum of empirical data.

Effective pulse dynamics in optical fibers with polarization mode dispersion

In this work, we deal with a backward stochastic differential equation driven by two mutually independent fractional Brownian motions (with Hurst parameter greater than 1/2). We establish the existence and uniqueness of the solution in the case of non-Lipschitz condition on the generator. The stochastic integral used throughout the paper is the divergence-type integral.

In this thesis we address the problem of data-driven coarse-graining, i.e. the process of inferring simplified models, which describe the evolution of the essential characteristics of a complex system, from available data (e.g. experimental observation or simulation data). Specifically, we consider the case where the coarse-grained model can be formulated as a stochastic differential equation. The main part of this work is concerned with data-driven coarse-graining when the underlying complex system is characterised by processes occurring across two widely separated time scales. It is known that in this setting commonly used statistical techniques fail to obtain reasonable estimators for parameters in the coarse-grained model, due to the multiscale structure of the data. To enable reliable data-driven coarse-graining techniques for diffusion processes with multiple time scales, we develop a novel estimation procedure which decisively relies on combining techniques from mathematical statistics and numerical analysis. We demonstrate, both rigorously and by means of extensive simulations, that this methodology yields accurate approximations of coarse-grained SDE models. In the final part of this work, we then discuss a systematic framework to analyse and predict complex systems using observations. Specifically, we use data-driven techniques to identify simple, yet adequate, coarse-grained models, which in turn allow to study statistical properties that cannot be investigated directly from the time series. The value of this generic framework is exemplified through two seemingly unrelated data sets of real world phenomena.

Realistic mathematical models of physical processes contain uncertainties. These models are often described by stochastic differential equations (SDEs) or stochastic partial differential equations (SPDEs) with multiplicative noise. The uncertainties in the right-hand side or the coefficients are represented as random fields. To solve a given SPDE numerically one has to discretise the deterministic operator as well as the stochastic fields. The total dimension of the SPDE is the product of the dimensions of the deterministic part and the stochastic part. To approximate random fields with as few random variables as possible, but still retaining the essential information, the Karhunen–Loève expansion (KLE) becomes important. The KLE of a random field requires the solution of a large eigenvalue problem. Usually it is solved by a Krylov subspace method with a sparse matrix approximation. We demonstrate the use of sparse hierarchical matrix techniques for this. A log-linear computational cost of the matrix-vector product and a log-linear storage requirement yield an efficient and fast discretisation of the random fields presented.

On solutions to backward stochastic partial differential equations for Lévy processes

SummaryIn this article, we discuss an infinite horizon optimal control of the stochastic system with partial information, where the state is governed by a mean‐field stochastic differential delay equation driven by Teugels martingales associated with Lévy processes and an independent Brownian motion. First, we show the existence and uniqueness theorem for an infinite horizon mean‐field anticipated backward stochastic differential equation driven by Teugels martingales. Then applying different approaches for the underlying system, we establish two classes of stochastic maximum principles, which include two necessary conditions and two sufficient conditions for optimality, under a convex control domain. Moreover, compared with the finite horizon optimal control, we add the transversality conditions to the two kinds of stochastic maximum principles. Finally, using the stochastic maximum principle II, we settle an infinite horizon optimal consumption problem driven by Teugels martingales associated with Gamma processes.

Two-dimensional stochastic dynamics as model for time evolution of the financial market

Background. The study of the asymptotic behavior of solutions of stochastic differential equations is one of the main places in many sections of insurance and financial mathematics, economics, management theory since stochastic differential equations, as an effective model of random process is the basis for the study of random phenomena. Objective. In this paper we consider the almost sure asymptotic behavior of the solution of the nonautonomous stochastic differential equation. Methods. We proposed a method to study the y-asymptotic properties of a solution of a stochastic differential equation by comparison with a solution of an ordinary differential equations obtained by dropping the stochastic part. We also use of the theory of pseudo-regularly varying functions. Results. We investigate the asymptotic behavior of solutions stochastic differential equations and establish sufficient conditions that provide different types of asymptotic behavior of a random process. Conclusions. Stochastic models approximate the real processes much better than deterministic ones, however, deterministic modelling has been preferred to stochastic one because of much greater ease of computability. The presented result enabled comparing properties of solution a stochastic differential equation with a solution of an ordinary differential equation.

This thesis is centered on the theory of stochastic processes and their applications in biological systems characterized by a complex environment. Three case studies have been modeled by the use of the three fundamental tools of stochastic processes: the master equation (ME), the stochastic differential equation (SDE) and the partial differential equation (PDE). The principal approach here applied to deal with complexity is the characterization of the system by means of probability distributions describing each a parameter of the model or the introduction of fractional order derivatives to include non-local and memory effects maintaining the linearity in the equations. In Chapter 1 we briefly review the theory of stochastic processes. In Chapter 2 we derive a birth-death process master equation to test if Long Interspersed Elements (LINEs) can be modeled according to the neutral theory of biodiversity. In Chapter 3 we derive a model of anomalous diffusion based on a Langevin approach in which anomalous behavior arise in the asymptotic intermediate state as a consequence of the heterogeneity of the system, from the superposition of Ornstein-Uhlenback processes. In Chapter 4 we propose an extension of the cable equation, used to describe anomalous diffusion phenomena as the signal conduction in spiny dendrites, by introducing a Caputo time fractional derivative.

Validation of synthetic daily Landsat NDVI time series data generated by the improved spatial and temporal data fusion approach

Event Abstract Back to Event Stochastic modeling of neuronal signaling Jukka Intosalmi1*, Tiina Manninen1, Keijo Ruohonen1 and Marja-Leena Linne1 1 Tampere University of Technology, Finland The time evolution of biochemical systems in neurons is traditionally modeled using deterministic ordinary differential equations (ODEs). Chemical reactions, however, are random in nature, and the deterministic approach is valid only for a restricted class of systems. Stochastic models take random fluctuations into account and are thus more realistic. Biochemical reactions can be modeled stochastically using numerous different methods. An ideal model would have the following three important properties. First, the model should be as realistic as possible, second, the mathematical method should be easily implementable as a computer algorithm, and third, the algorithm should be computationally effective. Naturally, all these three conditions cannot be fulfilled at the same time. Some realistic modeling approaches can be derived directly from chemical kinetics without making any approximations. Such approaches are called exact. A good example of an exact modeling approach is the stochastic simulation algorithm (SSA) developed by Gillespie. The SSA is applicable when the molecular populations in the system are small, but it becomes computationally inefficient when the numbers of molecules increase. In order to construct stochastic models that can be effectively simulated, new mathematical approaches have to be explored. As an approximate method also stochastic differential equations (SDEs) have been considered a promising way to model biochemical reactions stochastically. The SDE approach is attractive especially if we consider a system for which the SSA is computationally inefficient and the traditional deterministic ODE approach cannot be used as a good approximation. SDE models treat the chemical populations as real numbers and the construction of the model is based on the law of mass action, similarly as the construction of the traditional deterministic modeling approach. The SDE model, however, takes the random fluctuations into account by describing the time evolution of the system with the stochastic Itô process instead of the deterministic set of ODEs. The Itô process is basically a set of coupled stochastic differential equations. Although SDE modeling usually leads to equations that cannot be solved analytically, the solutions can be approximated numerically using different numerical integration methods. Thus, SDEs provide a mathematically rigorous way to construct models that can be simulated effectively. In order to investigate how well the SDE approach models biochemical systems, the results have to be compared with experimental data or with the simulation results from some exact simulation procedure. In this study, we use the methodology of spectral analysis to compare the simulation results of the SDE model and the SSA. As case studies, we consider e.g. calcium binding and protein kinase C (PKC) signal transduction pathway. We investigate the nature of noise in different modeling approaches and try to interpret the meaning of the noise in real biological systems. The main goal of our study is to find the most essential components and parameters in the SDE model and to adjust them so that the model is capable of giving similar results as the SSA. The simulations are carried out using the distributed computing resources (GRID) provided by Techila Technologies Ltd. Conference: Neuroinformatics 2008, Stockholm, Sweden, 7 Sep - 9 Sep, 2008. Presentation Type: Poster Presentation Topic: Computational Neuroscience Citation: Intosalmi J, Manninen T, Ruohonen K and Linne M (2008). Stochastic modeling of neuronal signaling. Front. Neuroinform. Conference Abstract: Neuroinformatics 2008. doi: 10.3389/conf.neuro.11.2008.01.023 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 28 Jul 2008; Published Online: 28 Jul 2008. * Correspondence: Jukka Intosalmi, Tampere University of Technology, Tampere, Finland, jukka.intosalmi@tut.fi Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Google Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Google Scholar Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne PubMed Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

In this paper, a stochastic SIRS epidemic model with general awareness-induced and four independent Brownian Motions is established. We verify the global existence of a unique positive solution and find out the noise modified reproduction number R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</sup> which is a sharp threshold for the dynamics: If R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</sup> <; 1, the disease will die out; if R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</sup> > 1, the disease persists and there exists a global asymptotically stable stationary distribution under parameter restrictive conditions. Numerical simulations are presented to illustrate the theoretical results.