Stochastic Differential Equations and Diffusions in a Nutshell

Methods for investigating the sensitivity of joint resultants to body segment parameter variations

In this chapter we give a brief overview of the basic mathematical concepts used for pricing financial products. Modern pricing theory is mainly based on describing the ups and downs of market prices via stochastic processes. Therefore, we start with the definition and characteristics of stochastic processes in Section 2.1. Usually the price-changes of the most basic financial instruments are modelled in terms of a so-called stochastic differential equation (SDE). The price of a financial derivative is considered to be a function of the basic financial instruments’ prices. One of the central tools in stochastic calculus which can be applied to determine the stochastic differential equation for the prices of these derivatives, Itô’s lemma, is described in Section 2.4. This concept is closely related to the stochastic integral, which is defined in Section 2.3 to be the mean square limit of some random sums, carefully put together so that the resulting limit is a (local) martingale. And in fact, martingales are one of the most important elements needed for the evaluation of financial instruments. They are defined in Section 2.5 and are used to describe the prices of financial derivatives in terms of conditional expectations. The Feynman-Kac formula provides a partial differential equation (PDE) that corresponds to such conditional expectations as described in Section 2.6. The advantage of dealing with partial differential equations is that they can be solved numerically. Also, PDE methods come naturally to applied mathematicians and physicists. This approach led Black and Scholes [BS73] to their famous equation for pricing European options. Their model will be discussed in Section 3.5. By contrast, martingales and Itô (stochastic) calculus come naturally to probabilists, and this approach led Merton [Mer74] to his option pricing formula.KeywordsStochastic ProcessStochastic Differential EquationConditional ExpectationWiener ProcessLocal MartingaleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Stochastic differential equations (SDEs) and the Kolmogorov partial differential equations (PDEs) associated to them have been widely used in models from engineering, finance, and the natural sciences. In particular, SDEs and Kolmogorov PDEs, respectively, are highly employed in models for the approximative pricing of financial derivatives. Kolmogorov PDEs and SDEs, respectively, can typically not be solved explicitly and it has been and still is an active topic of research to design and analyze numerical methods which are able to approximately solve Kolmogorov PDEs and SDEs, respectively. Nearly all approximation methods for Kolmogorov PDEs in the literature suffer under the curse of dimensionality or only provide approximations of the solution of the PDE at a single fixed space-time point. In this paper we derive and propose a numerical approximation method which aims to overcome both of the above mentioned drawbacks and intends to deliver a numerical approximation of the Kolmogorov PDE on an entire region \([a,b]^d\) without suffering from the curse of dimensionality. Numerical results on examples including the heat equation, the Black–Scholes model, the stochastic Lorenz equation, and the Heston model suggest that the proposed approximation algorithm is quite effective in high dimensions in terms of both accuracy and speed.

Random dynamical systems can be generated by stochastic differential equations (sde) on the one hand, and by random differential equations (rde), i.e. randomly parametrized ordinary differential equations on the other hand. Due to conflicting concepts in stochastic calculus and ergodic theory, asymptotic problems for systems associated with sde are harder to treat. We show that both objects are basically identical, modulo a stationary coordinate change (cohomology) on the state space. This observation opens completely new opportunities for the treatment of asymptotic problems for systems related to sde: just study them for the conjugate rde, which is often possible by simple path-by-path classical arguments. This is exemplified for the problem of local linearization of random dynamical systems, the classical analogue of which leads to the Hartman–Grobman theorem.

Roughly speaking, stochastic differential equations are differential equations driven by Gaussian white noise. Here, we are using the term “stochastic differential equations” in a restricted sense and not merely to denote differential equations with some probabilistic aspects. The importance of. stochastic differential equations is largely due to the fact that the solution of such an equation is a sample-continuous Markov process, and conversely, a large and important class of sample-continuous Markov processes can be modeled by the solutions of stochastic differential equations. From the point of view of applications, this is a direct benefit of using white noise as a noise model, and this fact accounts for its popularity. After all, white noise is, at best, a tolerable abstraction and is never a completely faithful representation of a physical noise source. Its raison d’être is the simplicity of analysis that it brings about. We have seen this in connection with Kalman filtering, and we shall see it again in the Markovian nature of solutions to stochastic differential equations.

A Study on Uncertainty of Discharge in River Channel Using Stochastic Partial Differential Equation

This paper presents the cyber-physical system (CPS) of a numerical method (the widely-used Euler- Maruyama method) and establishes a foundational theory of the CPSs of numerical methods for stochastic differential equations (SDEs), which transforms the way we understand the relationship between the numerical method and the underlying dynamical system. Unlike in the literature where they are treated as separate systems linked by inequalities, the CPS is a seamless integration of the SDE and the numerical method and we construct a new and general class of stochastic impulsive differential equations (SiDEs) that can serve as a canonic form of the CPSs of numerical methods. By the CPS approach, we show the equivalence and intrinsic relationship between the stability of the SDE and the stability of the numerical method using the Lyapunov stability theory we develop for our class of SiDEs. Applying our established theory, we present the CPS Lyapunov inequality that is the necessary and sufficient condition for meansquare stability of the CPS of the Euler-Maruyama method for linear SDEs. The proposed CPS and theory initiate the study of systems numerics and provoke many open and interesting problems for future work.

In this chapter, we survey some of the concepts in the theory of stochastic processes which will be used in the sequel. The material is intended to be only a discussion of some of the main ideas which will be needed and to serve as a convenient reference. It is not intended to be complete in any sense. Additional details, motivation and background can be found in the references. The reader familiar with the basic facts concerning stochastic differential equations can skip the chapter and refer to the results as needed. Section 1 gives some definitions and facts concerning martingales. In Section 2, we define stochastic integrals with respect to a Wiener process and state Itô's Formula, the fundamental tool in the stochastic calculus. Section 3 uses the stochastic integrals to define stochastic differential equations (SDE) of the diffusion type, and obtains bounds on their solutions which will be used later in the study of properties of the solution processes, as well as in the proofs of tightness of sequences of solutions to such equations. In Section 4, we discuss three standard methods for obtaining existence and uniqueness of solutions to SDE's. These sections give some flavor of a few of the basic ideas. But they just touch the surface of the subject. Sections 5 and 6 concern an alternative and very useful method for verifying whether a process satisfies an SDE of the diffusion type. The so-called "martingale problem" method which is discussed in these sections is perhaps the most useful approach to showing that the limit of a weakly convergent sequence of processes is, indeed, a solution to a SDE.

Stochastic differential equations (SDEs) or diffusions are continuous-valued continuous-time stochastic processes widely used in the applied and mathematical sciences. Simulating paths from these processes is usually an intractable problem, and typically involves time-discretization approximations. We propose an exact Markov chain Monte Carlo sampling algorithm that involves no such time-discretization error. Our sampler is applicable to the problem of prior simulation from an SDE, posterior simulation conditioned on noisy observations, as well as parameter inference given noisy observations. Our work recasts an existing rejection sampling algorithm for a class of diffusions as a latent variable model, and then derives an auxiliary variable Gibbs sampling algorithm that targets the associated joint distribution. At a high level, the resulting algorithm involves two steps: simulating a random grid of times from an inhomogeneous Poisson process, and updating the SDE trajectory conditioned on this grid. Our work allows the vast literature of Monte Carlo sampling algorithms from the Gaussian process literature to be brought to bear to applications involving diffusions. We study our method on synthetic and real datasets, where we demonstrate superior performance over competing methods. Supplementary materials for this article are available online.

In a standard integration scheme for a measurable/integrable modification existence, a certain criterion is suggested. It is also shown, how a stochastic differential can be determined for a given stochastic function.

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.

This chapter contains an extensive introduction to fundamental concepts leading to the definition of ambit fields and processes. The theory of Levy bases is followed by concepts of stochastic integration in time and space necessary in order to state a mathematically rigorous definition of an ambit field. The integration theory with respect to Levy bases is not as well-developed as the temporal counterpart (i.e. the integration with respect to Levy process); however, the integration theories introduced by Rajput and Rosinski (Probab Theory Related Fields 82(3):451–487, 1989), Walsh (An introduction to stochastic partial differential equations. In Carmona R, Kesten H, Walsh J (eds) Lecture notes in mathematics 1180, Ecole d’Ete de Probabilites de Saint-Flour XIV (1984), Springer, 1986) and Bichteler and Jacod (Random measures and stochastic integration. In Kallianpur G (ed) Theory and application of random fields, vol 49. Lecture notes in control and information sciences. Springer, Berlin, pp 1–18, 1983) are available and fit our purposes. Following the mathematical definition of ambit fields, we give a series of basic properties focussing on the characteristic function and the second order properties of the fields. Moreover, we explore the link between stochastic partial differential equations and ambit fields. The final part of Chap. 5 is devoted to subordination of Levy bases, where we define the concepts of metatimes and chronometers. Chapter 5 contains a series of particular examples of ambit fields, highlighting the theory as well as serving for later analysis.

A numerical method for some stochastic differential equations with multiplicative noise

Stochastic differential equation models play an increasingly prominent role in a wide variety of application areas, particularly those characterized by complexity and random behavior. Examples of such areas include biology, epidemiology, mechanics, economics, and finance. This issue's Education section contains a paper, "An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations," by Desmond Higham, that makes this important material accessible to a wide range of readers. Higham's paper is written from the perspective that the best way to gain insight quickly into the topics of stochastic differential equations (SDEs) is to see examples of numerical methods in use. The author feels that it is possible to explain and understand how simple numerical methods in this area work, without requiring the reader to be familiar with SDE theory. Indeed, the paper takes the approach that simulations of the dynamic behavior of SDEs and experimentation with numerical examples can help develop an intuitive understanding of SDEs that lays the groundwork for subsequent theoretical treatments. Thus Higham's paper expects only that the reader's background includes Euler's method for deterministic ordinary differential equations and an intuitive understanding of random variables. It provides a beautifully written treatment that is aimed at upper-division undergraduates and beginning graduate students. The topics covered in Higham's paper include investigation of stochastic integration, linear stability, and strong and weak convergence. The paper is built around ten complete downloadable MATLAB programs that are used as examples and contain ample comment lines. The programs are linked to relevant discussion points in the body of the paper. This paper is suitable as a module in a numerical methods course, such as a course on numerical methods for ordinary differential equations. It could also find application in courses in the areas where SDEs arise, such as finance, epidemiology, or general mathematical modeling. And it can serve excellently as the starting point for a self-study project in the area of SDEs for advanced undergraduates or beginning graduate students. Finally, many professionals in the SIAM community will find this article useful as an introduction to an area of numerical computation that has grown in prominence since many of us were students!

We develop a stochastic integration theory with respect to creation, annihilation and gauge operators on the full Fock space. This is done by using a kernel representation for a large class of bounded operators on the full Fock space. It is shown that the kernels form a Banach algebra. Having established the definition of processes and stochastic integrals we go on to prove an Ito formula and use this for examining stochastic evolutions and constructing dilations of special completely positive semigroups. Explicit solutions of the corresponding stochastic differential equations are given.