Ito Formula Research Articles

Comments on “Solving nonlinear stochastic differential equations with fractional Brownian motion using reducibility approach” [Nonlinear Dyn. 67, 2719–2726 (2012)

In this note, some comments are pointed out to the paper (Zeng et al. in Nonlinear Dyn 67:2719–2726, 2012). It is shown that the authors in the mentioned paper have wrongly utilized the fractional Ito formula to derive the reducibility conditions of nonlinear fractional stochastic differential equations (SDEs) to linear fractional SDEs. This incorrect use of the fractional Ito formula has led to fundamental flaws in the proposed theorems.

A Feynman–Kac–Itô formula for magnetic Schrödinger operators on graphs

In this paper we prove a Feynman–Kac–Ito formula for magnetic Schrodinger operators on arbitrary weighted graphs. To do so, we have to provide a natural and general framework both on the operator theoretic and the probabilistic side of the equation. On the operator side we identify a very general class of potentials that allows the definition of magnetic Schrodinger operators. On the probabilistic side, we introduce an appropriate notion of stochastic line integrals with respect to magnetic potentials. Apart from linking the world of discrete magnetic operators with the probabilistic world through the Feynman–Kac–Ito formula, the insights from this paper gained on both sides should be of an independent interest. As applications of the Feynman–Kac–Ito formula, we prove a Kato inequality, a Golden–Thompson inequality and an explicit representation of the quadratic form domains corresponding to a large class of potentials.

Semi-Compatible Quantum Finance Models

Two observables in a quantum probability space may not be compatible unless they commute. In this note, each basic asset in a market is assumed to commute with the history of trading strategies. A modified quantum finance model, in contrast to non-commuting ones, termed semi-compatible quantum finance model in this note, can be defined with this additional feature. The fundamental asset pricing theorem resumes its original form. It is shown that a self-financing strategy is again self-financing after numeraire change in the Boson Fock space quantum finance model described in Su H., A Quantum Finance Model, Preprint, (2014) and in the Free Brownian motion quantum finance model constructed in Su H., Quantum Finance Models, Preprint, (2015). The proofs are applications of respective Ito formula which dictates the corresponding quantum stochastic calculus. The two very different Ito correction terms which acknowledge the intrinsic distinctions between tensor independence and the free independence in quantum probability lead to individual calculations.

Reflected BSDEs when the obstacle is not right-continuous and optimal stopping

In the first part of the paper, we study reflected backward stochastic differential equations (RBSDEs) with lower obstacle which is assumed to be right upper-semicontinuous but not necessarily right-continuous. We prove existence and uniqueness of the solutions to such RBSDEs in appropriate Banach spaces. The result is established by using some tools from the general theory of processes such as Mertens decomposition of optional strong (but not necessarily right-continuous) supermartingales, some tools from optimal stopping theory, as well as an appropriate generalization of Ito's formula due to Gal'chouk and Lenglart. In the second part of the paper, we provide some links between the RBSDE studied in the first part and an optimal stopping problem in which the risk of a financial position $\xi$ is assessed by an $f$-conditional expectation $\mathcal{E}^f(\cdot)$ (where $f$ is a Lipschitz driver). We characterize the value function of the problem in terms of the solution to our RBSDE. Under an additional assumption of left upper-semicontinuity on $\xi$, we show the existence of an optimal stopping time. We also provide a generalization of Mertens decomposition to the case of strong $\mathcal{E}^f$-supermartingales.

The Ito Formula for the Ito Processes Driven by the Wiener Processes in a Banach Space

Using traditional methods it is possible to prove the Ito formula in a Hilbert space and some Banach spaces with special geometrical properties. The class of such Banach spaces is very narrow-they are subclass of reflexive Banach spaces. Using the definition of a generalized stochastic integral, early we proved the Ito formula in an arbitrary Banach space for the case, when as initial Ito process was the Wiener process. For an arbitrary Banach space and an arbitrary Ito process it is impossible to find the sequence of corresponding step functions with the desired convergence. We consider the space of generalized random processes, introduce general Ito process there and prove in it the Ito formula. Afterward, from the main Ito process in a Banach space we receive the generalized Ito process in the space of generalized random processes and we get the Ito formula in this space. Then we check decompasibilility of the members of the received equality and as they turn out Banach space valued, we get the Ito formula in an arbitrary Banach space. We implemented this approach when the stochastic integral in the Ito process was taken from a Banach space valued non-anticipating random process by the one dimensional Wiener process. In this paper we consider the case, when the stochastic integral is taken from an operator- valued non-anticipating random process by the Wiener process with values in a Banach space.

The obstacle problem for quasilinear stochastic PDEs with non-homogeneous operator

We prove the existence and uniqueness of solution of the obstacle problem for quasilinear Stochastic PDEs with non-homogeneous second order operator. Our method is based on analytical technics coming from the parabolic potential theory. The solution is expressed as a pair $(u,\nu)$ where $u$ is a predictable continuous process which takes values in a proper Sobolev space and $\nu$ is a random regular measure satisfying minimal Skohorod condition. Moreover, we establish a maximum principle for local solutions of such class of stochastic PDEs. The proofs are based on a version of Ito's formula and estimates for the positive part of a local solution which is non-positive on the lateral boundary.

Itô Formula for Integral Processes Related to Space-Time Lévy Noise

In this article, we give a new proof of the Ito formula for some integral processes related to the space-time Levy noise introduced in [1] [2] as an alternative for the Gaussian white noise perturbing an SPDE. We discuss two applications of this result, which are useful in the study of SPDEs driven by a space-time Levy noise with finite variance: a maximal inequality for the p-th moment of the stochastic integral, and the Ito representation theorem leading to a chaos expansion similar to the Gaussian case.

Robust stability analysis of stochastic neural networks with Markovian jumping parameters and probabilistic time‐varying delays

This article discusses the issue of robust stability analysis for a class of Markovian jumping stochastic neural networks (NNs) with probabilistic time‐varying delays. The jumping parameters are represented as a continuous‐time discrete‐state Markov chain. Using the stochastic stability theory, properties of Brownian motion, the information of probabilistic time‐varying delay, the generalized Ito's formula, and linear matrix inequality (LMI) technique, some novel sufficient conditions are obtained to guarantee the stochastical stability of the given NNs. In particular, the activation functions considered in this article are reasonably general in view of the fact that they may depend on Markovian jump parameters and they are more general than those usual Lipschitz conditions. The main features of this article are described in the following: first one is that, based on generalized Finsler lemma, some improved delay‐dependent stability criteria are established and the second one is that the nonlinear stochastic perturbation acting on the system satisfies a class of Lipschitz linear growth conditions. By resorting to the Lyapunov–Krasovskii stability theory and the stochastic analysis tools, sufficient stability conditions are established using an efficient LMI approach. Finally, two numerical examples and its simulations are given to demonstrate the usefulness and effectiveness of the proposed results. © 2014 Wiley Periodicals, Inc. Complexity 21: 59–72, 2016

Functional solution about stochastic differential equation driven by $G$-Brownian motion

Peng introduced the notions of $G$-expectation and $G$-Brownian motion as well as $G$-Ito formula in 2006. The $G$-Brownian motion has many rich and new properties comparing to classical Brownian motion. In this paper, we present a method to solve stochastic differential equation driven by $G$-Brownian motion without using $G$-Ito formula. Our method is mainly depending on Frobenius's Theorem. Many classical models in mathematical finance are investigated to illustrate the method. As a by-product, this financial models are extended to the case of $G$-Brownian motion.

On integration with respect to the [formula omitted]-Brownian motion

For a parameter 0<q<1, we use the Jackson q-integral to define integration with respect to the so called q-Brownian motion. Our main results are the q-analogs of the L2-isometry and of the Ito formula for polynomial integrands. We also indicate how the L2-isometry extends the integral to more general functions.

Stochastic quasi-synchronization for uncertain chaotic delayed neural networks

The stochastic quasi-synchronization issue for uncertain chaotic delayed neural networks (DNNs) is investigated. Stochastic perturbation and three uncertain elements, including the discontinuous activation functions, mismatched connection weight parameters and unknown connection weight parameters, are considered in the chaotic DNNs. According to the Ito formula and the inequality techniques, the parameters update laws and the control laws are given to realize the synchronization. And a stochastic quasi-synchronization criterion is established. Furthermore, sufficient conditions are proposed for the control of the synchronization error bound by choosing appropriate control laws. Some numerical simulations are presented to demonstrate the effectiveness of the theoretical results.

Delay-Dependent exponential stability for nonlinear reaction-diffusion uncertain Cohen-Grossberg neural networks with partially known transition rates via Hardy-Poincaré inequality

In this paper, stochastic global exponential stability criteria for delayed impulsive Markovian jumping reaction-diffusion Cohen-Grossberg neural networks (CGNNs for short) are obtained by using a novel Lyapunov-Krasovskii functional approach, linear matrix inequalities (LMIs for short) technique, Ito formula, Poincare inequality and Hardy-Poincare inequality, where the CGNNs involve uncertain parameters, partially unknown Markovian transition rates, and even nonlinear p-Laplace diffusion (p > 1). It is worth mentioning that ellipsoid domains in ℝ m (m ≥ 3) can be considered in numerical simulations for the first time owing to the synthetic applications of Poincare inequality and Hardy-Poincare inequality. Moreover, the simulation numerical results show that even the corollaries of the obtained results are more feasible and effective than the main results of some recent related literatures in view of significant improvement in the allowable upper bounds of delays.

Generalized stochastic evolution equations

An approach to generalized stochastic evolution equations is presented which is based on a generalized Ito formula. This allows the consideration of interesting examples which are stochastic generalizations of evolution equations of mixed type or second order in time hyperbolic equations. It includes more standard material involving a Gelfand triple of spaces as a special case. Several examples are given which illustrate the use of the abstract theory presented.

Optimal control for infinite dimensional stochastic differential equations with infinite Markov jumps and multiplicative noise

In this paper we solve an infinite-horizon linear quadratic control problem for a class of differential equations with countably infinite Markov jumps and multiplicative noise. The global solvability of the associated differential Riccati-type equations is studied under detectability hypotheses. A nonstochastic, operatorial approach is used. Some properties of the linear stochastic systems, such as stability, stabilizability and detectability, are also discussed on the basis of a new solution representation result. A generalized Ito's formula which applies to infinite dimensional stochastic differential equations with countably infinite Markov jumps is also provided.

The Pricing of a Supply Contract under Uncertainty with Long-Range Dependence

This paper aims to address a contracting problem between upstream and downstream agents in a supply chain using a stochastic demand process with autocorrelation properties. For example, the quarterly global sales volumes of Apple's iPhone are highly autocorrelated over time although the time lag is as long as 10 quarters. Based on such empirical evidence, an autocorrelated demand process referred to as fractional Brownian motion is adopted in this paper. It is assumed that there are two echelons in the supply chain: business and consumer markets. The information flows fall into four categories: demand flow, marketing info flow, uncertainty flow, and premium charge flow. The downstream agent can transfer demand uncertainty to the upstream firm (uncertainty flow) by signing a supply contract (contracting agent). The demand in the consumer market is assumed to follow a fractional Brownian motion. Based on the fractional Ito formula for the real option model, the result demonstrates that the real option value can be an increasing or decreasing function of the degree of autocorrelation in which the real option value reaches its maximum at the critical point. As a consequence, the trading price determined in the supply contract without considering the autocorrelation of demand could be significantly undervalued or overvalued. In other words, to ensure a fair game in a contracting activity, the upstream agent should charge more for the trading price depending on the degree of autocorrelation in demand.

Asymptotic Error Distributions of the Crank–Nicholson Scheme for SDEs Driven by Fractional Brownian Motion

We investigate the difference between the solution to a stochastic differential equation driven by a fractional Brownian motion and the approximation by the Crank–Nicholson scheme associated with the equation. In preceding results, researchers deal with the errors of the Euler scheme and the Crank–Nicholson scheme for some fixed time as real-valued random variables and study the convergence rates and the limit distributions. In the present paper, we consider the error as stochastic processes and determine the convergence rate of the error and the limit distribution in the Skorohod topology. The limit distribution is expressed in terms of the solution to the equation and the Ito integral with respect to a standard Brownian motion independent of the driving process of the equation. This result extends those contained in J Theor Probab 20(4):871–899, 2007. The key ingredients in our proof are asymptotic behavior of weighted Hermite variations as stochastic processes. We also give the Ito formula for fractional Brownian motion.

Maximum principle for quasilinear stochastic PDEs with obstacle

We prove a maximum principle for local solutions of quasi linear stochastic PDEs with obstacle (in short OSPDE). The proofs are based on a version of Ito's formula and estimates for the positive part of a local solution which is non-positive on the lateral boundary.

疲労き裂の不規則成長に対するPoisson型ノイズを用いた確率モデルの新たな提案

A probabilistic model describing random fatigue crack growth is newly proposed, in which a stochastic differential equation driven by a temporally inhomogeneous compound Poisson process is applied. The new model has several outstanding features such as (i) we can perfectly remove the probability that a crack length decreases in the process of fatigue failure and (ii) an analytical approach is possible for obtaining probabilistic properties associated with the crack growth process. The former feature gives a quite effective solution over a critical disadvantage point inherent in the so-called diffusive model which has been widely used for describing the random fatigue crack growth.First, the well-known Paris-Erdogan law is extended to a stochastic differential equation by introducing a temporally inhomogeneous compound Poisson process expressing the random behavior of crack propagation resistance. Next, its solution process is analytically derived by the use of the generalized Ito formula and probabilistic properties associated with the random fatigue crack growth process are also clarified. Finally, some numerical examples are given so that we can quantitatively clarify sample behavior of the crack growth and temporal variation of a probability distribution of a crack length.

High-order discretization schemes for stochastic volatility models

In usual stochastic volatility models, the process driving the volatility of the asset price evolves according to an autonomous one-dimensional stochastic differential equation. We assume that the coefficients of this equation are smooth. Using Ito's formula, we get rid, in the asset price dynamics, of the stochastic integral with respect to the Brownian motion driving this SDE. Taking advantage of this structure, we propose - a scheme, based on the Milstein discretization of this SDE, with order one of weak trajectorial convergence for the asset price, - a scheme, based on the Ninomiya-Victoir discretization of this SDE, with order two of weak convergence for the asset price. We also propose a specific scheme with improved convergence properties when the volatility of the asset price is driven by an Orstein-Uhlenbeck process. We confirm the theoretical rates of convergence by numerical experiments and show that our schemes are well adapted to the multilevel Monte Carlo method introduced by Giles [2008a, 2008b].

The Radial Part of Brownian Motion with Respect to $$\mathcal L $$ L -Distance Under Ricci Flow

Let \(\{g_t\}_{t\in [0,T)}\) be a family of complete time-depending Riemannian metrics on a manifold, which evolves under backwards Ricci flow. The Ito formula is established for the \(\mathcal L \)-distance of the \(g_t\)-Brownian motion to a fixed reference point (\(\mathcal L \)-base). Furthermore, as an application, we construct a coupling by parallel displacement, which yields a new proof of some results of Topping.