Ito Calculus Research Articles

Properties of fuzzy Ito integrals with respect to fuzzy Brownian motion

Integral of a stochastic process with respect to Brownian motion is called Ito integral. Here the stochastic process and Brownian motion are random as well as fuzzy. Hence the Ito integral is fuzzy Ito integral. This paper deals with the properties of fuzzy Ito integral for simple adapted process with respect to fuzzy Brownian motion. The quadratic variance and covariance of FII are discussed. The concept of fuzzy simple adapted process, fuzzy martingale, fuzzy functions are used to derive the properties of fuzzy Ito integrals.

Option pricing: the reduced-form SDE model

We use partial differential equations (PDEs) to describe the pricing process of options in an illiquid market. These equations are derived from stochastic differential equations built on the Ito process. With the help of Lie symmetry analysis, this paper focuses on the pricing of a model that incorporates the effect of large traders in an illiquid market. The nonlinear PDE representing this model incorporates a nonzero risk-neutral interest rate. This PDE is singularly perturbed and quadratic in the highest derivative. Using the method of Lie symmetry analysis, we obtain symmetries in the mathematical package Program Lie, and these symmetries are used to analyse the equation and to reduce the PDE to ordinary differential equations. When the equations are solved, they yield group invariant solutions to the PDE. We give a graphical representation of the obtained solutions. These invariant solutions are new to the field and can be used in place of simulations.

Optical solitons in the Sasa�Satsuma model with multiplicative noise via Ito calculus

Optical solitons in the Sasa�Satsuma model with multiplicative noise via Ito calculus

Optimization of the mean-square approximation procedures for iterated Stratonovich stochastic integrals of multiplicities 1 to 3 with respect to components of the multi-dimensional Wiener process based on Multiple Fourier–Legendre series

The article is devoted to approximation of iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 3 by the method of multiple Fourier–Legendre series. The mentioned stochastic integrals are part of strong numerical methods with convergence order 1.5 for Ito stochastic differential equations with multidimensional noncommutative noise. These numerical methods are based on the so-called Taylor–Ito and Taylor–Stratonovich expansions. We calculate the exact lengths of sequences of independent standard Gaussian random variables required for the mean-square approximation of iterated Stratonovich stochastic integrals. Thus, the computational cost for the implementation of numerical methods can be significantly reduced.

Leaving your tailings behind: Environmental bonds, bankruptcy and waste cleanup

This paper studies the impacts of an environmental bond, which fully covers waste cleanup costs, on a mining firm's optimal actions over the full life cycle of a mine, when bankruptcy may shift cleanup costs to the government. A firm's stochastic optimal control problem is described by an HJB equation with the resource price modelled as an Ito process. A theoretical result shows that under certain assumptions, an environmental bond can ensure that the option to declare bankruptcy has no impact on the firm's optimal actions. A numerical solution is implemented for a hypothetical copper mine using two different models of bankruptcy risk. Numerical results show that when bankruptcy is an option and no bond is required, the firm produces too much waste relative to a benchmark case, resulting in an efficiency loss and a cleanup liability imposed on government. In the presence of bankruptcy risk, a bond ensures that the firm acts optimally and no efficiency loss is imposed on society.

Brownian Motion at the Speed of Light: A New Lorentz Invariant Family of Processes

We consider here a new family of processes which describe particles which only can move at the speed of light c in the ordinary 3D physical space. The velocity, which randomly changes direction, can be represented as a point on the surface of a sphere of radius c and its trajectories only may connect points of this variety. A process can be constructed both by considering jumps from one point to another (velocity changes discontinuously) and by continuous velocity trajectories on the surface. We recently proposed to follow this second strategy assuming that the velocity is described by a Wiener process (which is isotropic only in the ’rest frame’) on the surface of the sphere. Using both Ito calculus and Lorentz boost rules, we succeed here in characterizing the entire Lorentz-invariant family of processes. Moreover, we highlight and describe the short-term ballistic behavior versus the long-term diffusive behavior of the particles in the 3D physical space.

Equity-linked securities option pricing by fractional Brownian motion

Abstract In this paper, we use the Monte Carlo simulation (MCS) method to price the Equity-Linked Securities (ELS) option by fractional Brownian motion (fBm). We use the fBm that satisfy the Ito process instead of the geometric Brownian motion, and we call the proposed model ELS-fBm model. We first do the convergence test with the number of MCS. Then, we price the one-, two-, and three-asset ELS by employing the classical ELS model and the ELS-fBm model, respectively. We see that the price of ELS varies with different Hurst exponent, which indicates that the ELS pricing model considering the long-range dependence is necessary and more suitable for the financial market under uncertain environment. In addition, we prove the superiority of our proposed model with an actual three-assets ELS product which is issued by Future Asset company. We find the ELS price calculated by the proposed ELS-fBm model almost the same as the ELS price in the ELS product contract issued by Future Asset company, while the classical ELS model produces a great deviation.

Method of Construction of the Stochastic Integral with Respect to Fractional Brownian Motion

Since the pioneering work of Hurst, and Mandelbrot, the fractional brownian motions have played and increasingly important role in many fields of application such as hydrology, economics and telecommunications. For every value of the Hurst index H ∈ (0,1) we define a stochastic integral with respect to fractional Brownian motion of index H. This process is called a (standard) fractional Brownian motion with Hurst parameter H. To simplify the presentation, it is always assumed that the fractional Brownian motion is 0 at t=0. If H = 1/2, then the corresponding fractional Brownian motion is the usual standard Brownian motion. If 1/2 < <I>H</I> < 1, Fractional Brownian motion (FBM) is neither a finite variation nor a semi-martingale. Consequently, the standard Ito calculus is not available for stochastic integrals with respect to FBM as an integrator if 1/2 <<i> H</i> < 1. The classic methods (It<i>ô</i> and Stiliege) are excluted. The most studied case is that where H is between 0 and 1/2. Several attempts to define the stochastic integral are made. But so far some difficulties subjust. We give in this paper, several construction methods. So for the construction, we will use other tools to deal with such situations.

On copulas of self-similar Ito processes

Abstract We characterize the cumulative distribution functions and copulas of two-dimensional self-similar Ito processes, with randomly correlated Wiener margins, as solutions of certain elliptic partial differential equations.

Equilibrium models with heterogeneous agents under rational expectations and its numerical solution

In this work we assume rational expectations to pose general equilibrium models with heterogeneous firms that can enter or exit the industry. More precisely, we assume a general Ito process for the dynamics of the agents productivity, including the main dynamics in the literature. A Hamilton-Jacobi-Bellman (HJB) formulation models the endogenous decision of firms to remain or exit the industry. All firms that exit are immediately replaced by a group of new ones, so that the probability density function of firms satisfies an appropriate Kolmogorov-Fokker-Plank (KFP) equation with source term. Equilibrium models are completed with the household problem formulation and the feasibility conditions. In the evolutive and general stationary settings, analytical or semi-analytical formulas are not available, so that appropriate numerical methods are required. We propose a Crank-Nicolson scheme for the time discretization of the evolutive problems. Moreover, we use an augmented Lagrangian active set (ALAS) method combined with a finite difference discretization for the HJB formulation and a suitable finite differences discretization for the KFP problem. For the global equilibrium problem we propose a Steffensen algorithm. Numerical examples illustrate the performance of the proposed numerical methodologies as well as the expected behaviours of the computed economic variables.

Robert C. Merton and the Science of Finance

Starting with his 1970 doctoral dissertation and continuing to today, Robert C. Merton has revolutionized the theory and practice of finance. In 1997, Merton shared a Nobel Prize in Economics “for a new method to determine the value of derivatives.” His contributions to the science of finance, however, go far beyond that. In this article I describe Merton's main contributions. They include the following: 1. The introduction of continuous-time stochastic models (the Ito calculus) to the theory of household consumption and investment decisions. Merton's technique of dynamic hedging in continuous time provided a bridge between the theoretical complete-markets equilibrium model of Kenneth Arrow and the real world of personal financial planning and management. 2. The derivation of the multifactor Intertemporal Capital Asset Pricing Model (ICAPM). The ICAPM generalizes the single-factor CAPM and explains why that model might fail to properly account for observed market excess returns. It also provides a theory to identify potential forward-looking risk premia for use in factor-based investment strategies. It is therefore both a positive and normative theory. 3. The invention of Contingent Claims Analysis (CCA) as a generalization of option pricing theory. CCA applies the technique of dynamic replication to the valuation and risk management of a wide range of corporate and government liabilities. Merton's CCA model for the valuation and analysis of risky debt is known among scholars and practitioners alike as the Merton Model. 4. The development of financial engineering, which employs CCA to design and produce new financial products. Merton was the first to apply CCA to analyze government guaranty programs such as deposit insurance, and to suggest improvements in the way those programs are managed. He and his students have applied his insights at both the micro and macro policy levels. 5. And finally, the development of a theory of financial intermediation that explains and predicts how financial systems differ across countries and change over time. Merton has applied that theory, called functional and structural finance, to guide the design and regulation of financial systems at the levels of the firm, the industry, and the nation. He has also used it to propose reforms in pensions, sovereign wealth funds, and macrostabilization policy. This article is one of a pair of articles published in this volume about Robert C. Merton's contributions to the science of financial economics. This article was originally published in Volume 11 of the Annual Review of Financial Economics. The other article in this pair is “ Robert C. Merton: The First Financial Engineer ” by Andrew W. Lo.

P‐48: Study of ITO Electrode on the White Color Uniformity of TFT‐LCD

This paper makes a research on the relationship between TFT transmittance and white color uniformity, which based on 1st ITO process optimization. The results show that the better TFT transmittance uniformity leads to better white color uniformity, and the ITO oxidation is an important factor for TFT transmittance uniformity improvement. Annealing and increase oxygen flow during ITO sputtering when annealing is skipped are both the effective methods to obtain an excellent white color uniformity.

Leaving Your Tailings Behind: Environmental Bonds, Bankruptcy and Waste Cleanup

This paper studies the impacts of an environmental bond, which fully covers waste cleanup costs, on a mining firm's optimal actions over the full life cycle of a mine, when bankruptcy may shift cleanup costs to the government. A firm's stochastic optimal control problem is described by an HJB equation with the resource price modelled as an Ito process. A theoretical result shows that under certain assumptions, an environmental bond can ensure that the option to declare bankruptcy has no impact on the firm's optimal actions. A numerical solution is implemented for a hypothetical copper mine using two different models of bankruptcy risk. Numerical results show that when bankruptcy is an option and no bond is required, the firm produces too much waste relative to a benchmark case, resulting in an efficiency loss and a cleanup liability imposed on government. In the presence of bankruptcy risk, a bond ensures that the firm acts optimally and no efficiency loss is imposed on society.

Volatility and variance swaps and options in the fractional SABR model

Appropriate capturing the nature of financial market volatility is a significant factor for the pricing of volatility derivatives. A recent study by Gatheral, Jaisson and Rosenbaum [2018. “Volatility is Rough.” Quantitative Finance 18 (6): 933–949] has found that log-volatility behaves as a fractional Brownian motion with a small Hurst exponent at any reasonable time scale. Also, there are several empirical works showing that a stochastic volatility model driven by the fractional Brownian motion well approximates at-the-money volatility skew near expiration. In this paper, we choose the log-normal SABR model with fractional stochastic volatility to valuate variance and volatility swaps. We derive a closed-form exact solution for the fair strike price of the variance swap by using fractional Ito calculus, while we obtain an approximate solution for the fair strike price of the volatility swap by exploiting the shifted log-normal approximation. Also, solution formulas for the variance and volatility option prices are derived. Their accuracy is confirmed through numerical studies. Calibration to market variance swap rates demonstrates the strength of fractional SABR model compared to the Heston and SABR models.

Carleman linearisation based estimation technique for a noise influenced electronic circuit

Noise is an inherent characteristic of electronic circuits. In this Letter, the authors propose a novel estimation technique for a noise influenced electronic circuit. Stochasticity is accounted for by utilising Ito's stochastic calculus. This novel estimation technique unifies the Carleman linearisation with the Fokker–Planck equation for the Ito stochastic differential equations. To show the effectiveness of the proposed method, numerical simulation results are presented for a benchmark tunnel diode oscillator circuit. This method is beneficial to other electronic circuits for estimating the state trajectories under the influence of randomness when observations are not available.

Sliding mode control for nonlinear stochastic systems with Markovian jumping parameters and mode-dependent time-varying delays

This paper reports on the sliding mode control (SMC) problem for nonlinear stochastic systems with one features: time-delays are not only varied with time but also characterized by random delays changed in line with a set of Markov chains (namely, time-delays are mode-dependent time-varying delays). Based on given systems, an integral switching surface is introduced. In particular, such a switching surface with an Ito process is given so that the traditional assumption imposed on systems is removed. And by applying the Ito formula, the linear matrix inequalities method and the lemma provided, more relaxed and indeed delay-dependent criteria for the second moment exponential stability are given. Then, the sliding mode controller is constructed to guarantee the reachability of the switching surface and the existence of the sliding mode. Finally, the validity and the application for the presented SMC method are illustrated by the DC motor system.

Introduction to Stochastic Calculus and Its Applications

Stochastic Calculus deals with the rate of change of functions with respect to the randomness. It has been applied in the field of financial mathematics. When we look at the graph of stock prices, there are many irregular price fluctuations. The main tool for modeling the small fluctuations is the Brownian motion. This paper will mainly introduce the concept of Brownian motion, the concept of random walks, and the main flavors of stochastic calculus, Ito calculus. I will also discuss various applications, especially in the finance area.

From ODE to Open Markov Chains, via SDE: an application to models for infections in individuals and populations

Abstract We present a methodology to connect an ordinary differential equation (ODE) model of interacting entities at the individual level, to an open Markov chain (OMC) model of a population of such individuals, via a stochastic differential equation (SDE) intermediate model. The ODE model here presented is formulated as a dynamic change between two regimes; one regime is of mean reverting type and the other is of inverse logistic type. For the general purpose of defining an OMC model for a population of individuals, we associate an Ito processes, in the form of SDE to ODE system of equations, by means of the addition of Gaussian noise terms which may be thought to model non essential characteristics of the phenomena with small and undifferentiated influences. The next step consists on discretizing the SDE and using the discretized trajectories computed by simulation to define transitions of a finite valued Markov chain; for that, the state space of the Ito processes is partitioned according to some rule. For the example proposed for illustration, the state space of the ODE system referred – corresponding to a model of a viral infection – is partitioned into six infection classes determined by some of the critical points of the ODE system; we detail the evolution of some infected population in these infection classes.

Robert C. Merton and the Science of Finance

Starting with his 1970 doctoral dissertation and continuing to today, Robert C. Merton has revolutionized the theory and practice of finance. In 1997, Merton shared a Nobel Prize in Economics “for a new method to determine the value of derivatives.” His contributions to the science of finance, however, go far beyond that. In this article I describe Merton's main contributions. They include the following: 1. The introduction of continuous-time stochastic models (the Ito calculus) to the theory of household consumption and investment decisions. Merton's technique of dynamic hedging in continuous time provided a bridge between the theoretical complete-markets equilibrium model of Kenneth Arrow and the real world of personal financial planning and management. 2. The derivation of the multifactor Intertemporal Capital Asset Pricing Model (ICAPM). The ICAPM generalizes the single-factor CAPM and explains why that model might fail to properly account for observed market excess returns. It also provides a theory to identify potential forward-looking risk premia for use in factor-based investment strategies. It is therefore both a positive and normative theory. 3. The invention of Contingent Claims Analysis (CCA) as a generalization of option pricing theory. CCA applies the technique of dynamic replication to the valuation and risk management of a wide range of corporate and government liabilities. Merton's CCA model for the valuation and analysis of risky debt is known among scholars and practitioners alike as the Merton Model. 4. The development of financial engineering, which employs CCA to design and produce new financial products. Merton was the first to apply CCA to analyze government guaranty programs such as deposit insurance, and to suggest improvements in the way those programs are managed. He and his students have applied his insights at both the micro and macro policy levels. 5. And finally, the development of a theory of financial intermediation that explains and predicts how financial systems differ across countries and change over time. Merton has applied that theory, called functional and structural finance, to guide the design and regulation of financial systems at the levels of the firm, the industry, and the nation. He has also used it to propose reforms in pensions, sovereign wealth funds, and macrostabilization policy.

Semi-Transparent Perovskite Solar Cells with ITO Directly Sputtered on Spiro-OMeTAD for Tandem Applications.

Perovskite silicon tandem solar cells have the potential to overcome the efficiency limit of single-junction solar cells. For both monolithic and mechanically stacked tandem devices, a semi-transparent perovskite top solar cell, including a transparent contact, is required. Usually, this contact consists of a metal oxide buffer layer and a sputtered transparent conductive oxide. In this work, semi-transparent perovskite solar cells in the regular n-i-p structure are presented with tin-doped indium oxide (ITO) directly sputtered on the hole conducting material Spiro-OMeTAD. ITO process parameters such as sputter power, temperature, and pressure in the chamber are systematically varied. While a low temperature of 50 °C is crucial for good device performance, a low sputter power has only a slight effect, and an increased chamber pressure has no influence on device performance. For the 5 × 5 mm2 perovskite cell with a planar front side, a 105 nm thick ITO layer with a sheet resistance of 44 Ω sq-1 allowing for the omission of grid fingers and a MgF2 antireflection coating are used to improve transmission into the solar cells. The best device achieved an efficiency of 14.8%, which would result in 24.2% in a four-terminal tandem configuration.