On Stochastic Integration and Differentiation

A numerical method for some stochastic differential equations with multiplicative noise

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

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.

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.

Little seems to be known about evaluating the stochastic stability of stochastic differential equations (SDEs) driven by fractional Brownian motion (fBm) via stochastic Lyapunov technique. The objective of this paper is to work with stochastic stability criterions for such systems. By defining a new derivative operator and constructing some suitable stochastic Lyapunov function, we establish some sufficient conditions for two types of stability, that is, stability in probability and moment exponential stability of a class of nonlinear SDEs driven by fBm. We will also give an example to illustrate our theory. Specifically, the obtained results open a possible way to stochastic stabilization and destabilization problem associated with nonlinear SDEs driven by fBm.

Stochastic differential equations in a Banach space driven by the cylindrical Wiener process

Stochastic differential equations (SDEs) are a powerful and natural tool for the modelling of complex systems that change continuously in time. This chapter provides a short introduction to SDEs and their solutions, which under regularity conditions agree with the class of diffusion processes. In particular, it covers the motivation and introduction of stochastic integrals as opposed to the classical Lebesgue-Stieltjes integral, the definition of diffusion processes, key properties and formulas from stochastic calculus, and finally numerical approximation and exact sampling methods. The chapter serves as a basis for the remaining parts of this book and offers a quick access to stochastic calculus.

For system identification, the ordinary differential equations (ODEs) model is popular for its accuracy and effectiveness. Consequently, the ODEs model is extended to the stochastic differential equations (SDEs) model to tackle the stochastic case intuitively. But the existence of stochastic integral is a rigid barrier. We simply transform the SDEs to their corresponding stochastic difference equations (SDCEs) to eliminate stochastic integrals and propose an easy but effective solution to stochastic system identification. In this solution, the maximum likelihood estimation can be applied and the evolutionary algorithms are used to determine structures and parameters of the unknown system.

This paper presents geometry-preserving weak solution techniques for Ito stochastic differential equations (SDEs) whose solutions are constrained to evolve on a geometrical space. Weak solutions are required where estimation of the average behaviour of the solution paths of an SDE is useful. The weak approximation to the SDE on the manifold is developed by exploiting the relation between the manifold and its Lie algebra through exponential mapping. The work demonstrates three weak integration schemes up to order 3.0 based on the Ito–Taylor expansion of the SDE on Lie algebra with associated stochastic integrals. Detailed formulation of the geometric operators associated with the higher-order schemes is shown along with convergence analysis. Effects of additive and multiplicative noise in dynamical systems on the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>S</mml:mtext> <mml:mn>2</mml:mn> </mml:msup> </mml:math> manifold are explored. The accuracy of the proposed integration schemes is shown to be consistent with the order of weak accuracy and it is shown that higher-order weak schemes provide better solution estimates and accurate description of the stable regions for the nonlinear dynamical systems on the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>S</mml:mtext> <mml:mn>2</mml:mn> </mml:msup> </mml:math> manifold even at coarser time steps. The proposed method accommodates a wide range of stochastic dynamical systems on the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>S</mml:mtext> <mml:mn>2</mml:mn> </mml:msup> </mml:math> manifold whose dynamics can be represented by an Ito SDE.

In this thesis, stochastic processes closely related to a prominent process called Brownian motion is introduced. The processes are called local Brownian motions. Two rich families are constructed, which can be used to model stochastic dynamical systems in such fields as finance, biology, and physics. Stochastic calculus with respect to local Brownian motion is developed, and strong solutions are presented for some stochastic differential equations driven by the noise generated by local Brownian motions.

We apply the methodology of our recent paper ‘The Dynamics of the Hubbard Model through Stochastic Calculus and Girsanov Transformation’ [1] to thermodynamic correlation functions in the Fermi-Hubbard model. They can be obtained from a stochastic differential equation (SDE) system. To this SDE system, a Girsanov transformation can be applied. This has the effect that the usual determinant or pfaffian which shows up in a pfaffian quantum Monte Carlo (PfQMC) representation [2] basically gets absorbed into the new integration variables and information from that pfaffian moves into the drift part of the transformed SDE system. While the PfQMC representation depends heavily on the choice of how the quartic interaction has been factorized into quadratic quantities in the beginning, the Girsanov transformed formula has the very remarkable property that it is nearly independent of that choice, the drift part of the transformed SDE system as well as a remaining exponential which has the obvious meaning of energy are always the same and do not depend on the Hubbard-Stratonovich details. The resulting formula may serve as a starting point for further theoretical or numerical investigations. Here we consider the spin-spin correlation at half-filling on a bipartite lattice and obtain an analytical proof that the signs of these correlations have to be of antiferromagnetic type, at arbitrary temperatures. Also, by checking against available benchmark data [3], we find that approximate ground state energies can be obtained from an ODE system. This may even hold for exact ground state energies, but future work would be required to prove or disprove this. As in [1], the methodology is generic and can be applied to arbitrary quantum many body or quantum field theoretical models.

Part 1 Stochastic processes: generated theta-algebras stochastic processes stopping times convergence in Lp and uniform integrability. Part 2 Martingales: martingale, submartingale and supermartingale fundamental submartingale inequalities convergence of submartingales uniformly integrable submartingales regularity of sample functions of submartingales increasing processes. Part 3 Stochastic integrals: L2-martingales and quadratic variation processes stochastic integrals with respect to martingales Ft-Brownian motions local martingales and extensions of the stochastic integral Ito's formula Ito's stochastic calculus. Part 4 Stochastic differential equations: the space of continuous functions on R++ definition and function space representation of solutions existence and uniqueness of solutions strong solutions.

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.