Ito Calculus Research Articles

Covariance Control for Bilinear Stochastic Systems via Sliding Mode Control Concept

Based on the concept of sliding mode control, we study the problem of steady state covariance assignment for bilinear stochastic systems. We find that the invariance property of sliding mode control ensures nullity of the matched bilinear term in the system on the sliding mode. By suitably using Ito calculus, the controller u(t) can be designed to force the feedback gain matrix G to achieve the goal of steady state covariance assignment. We also compare our method with other approaches via simulations.

Exponential dissipativeness of the random-structure diffusion processes and problems of robust stabilization

Consideration was given to the class of systems described by a finite set of the controllable control-affine diffusion Ito processes with stepwise transitions defined by the evolution of the uniform Markov chain (Markov switchings). For these systems, the notion of exponential dissipativity was introduced, and its theory was developed and used to estimate the possible variations of the output feedback law under which the system retains its robust stability. For the set of linear systems with uncertain parameters, proposed was a two-step procedure for determination of the output feedback control based on comparison with the stochastic model and providing their simultaneous robust stabilization. At the first step, the robust stabilizing control is established by means of an iterative algorithm. Then, the possible variations of the feedback law for which the robust stability is retained are estimated by solving a system of matrix linear inequalities. An example was presented.

Dissipativity of diffusion Itô processes with Markovain switching and problems of robust stabilization

Consideration is given to a class of systems described by a finite set of controlled diffusion Ito processes that are control-affine, with jump transitions between them, and are defined by the evolution of a uniform Markovian chain (Markovian switching). Each state of this chain corresponds to a certain system mode. A stochastic version of the notion dissipativity by Willems is introduced, and properties of diffusion processes with Markovian switching are studied. The relationship between passivity and stabilizability in the process of output-feedback control is established. The obtained results are applied to the problem of robust simultaneous stabilization for the set of nonlinear systems with undetermined parameters. As a partial case, a problem of robust simultaneous stabilization for the set of linear systems where final results are obtained in terms of linear matrix inequalities.

A Framework Using Two-Factor Price Lattices for Generation Asset Valuation

In this paper, we use a real-options framework to value a power plant. The real option to commit or decommit a generating unit may be exercised on an hourly basis to maximize expected profit while subject to intertemporal operational constraints. The option-exercising process is modeled as a multistage stochastic problem. We develop a framework for generating discrete-time price lattices for two correlated Ito processes for electricity and fuel prices. We show that the proposed framework exceeds existing approaches in both lattice feasibility and computational efficiency. We prove that this framework guarantees existence of branching probabilities at all nodes and all stages of the lattice if the correlation between the two Ito processes is no greater than 4/√35 ≈ 0.676. With price evolution represented by a lattice, the valuation problem is solved using stochastic dynamic programming. We show how the obtained power plant value converges to the true expected value by refining the price lattice. Sensitivity analysis for the power plant value to changes of price parameters is also presented.

Dividends in the theory of derivative securities pricing

This paper develops the fundamental aspects of the theory of martingale pricing of derivative securites in a setting where the cumulative gains processes are Ito processes while the cumulative dividend processes of both the underliers and the derivative securities are general enough to cover all cases encountered in practical applications. A key ingredient is a general formula for how to change the unit of account of a cumulative dividend process. The formula is inconsistent with parts of the earlier literature. It obeys a unit-invariance rule for trading strategies, satisfies a consistency property when the unit is changed twice in a row, gives the correct results in well-known and uncontroversial special cases, and fits perfectly into a generalization of the martingale valuation theory. Using that generalized theory, we show that the value of a dividend process equals the value of a claim to the nominal amount of dividends yet to be accumulated plus the value of a flow of interest on the cumulative dividends at each point in time.

A Generalized Ito's Formula in Two-Dimensions and Stochastic Lebesgue-Stieltjes Integrals

In this paper, a generalized It${\hat {\rm o}}$'s formula for continuous functions of two-dimensional continuous semimartingales is proved. The formula uses the local time of each coordinate process of the semimartingale, the left space first derivatives and the second derivative $\nabla _1^- \nabla _2^-f$, and the stochastic Lebesgue-Stieltjes integrals of two parameters. The second derivative $\nabla _1^- \nabla _2^-f$ is only assumed to be of locally bounded variation in certain variables. Integration by parts formulae are asserted for the integrals of local times. The two-parameter integral is defined as a natural generalization of both the Ito integral and the Lebesgue-Stieltjes integral through a type of It${\hat {\rm o }}$ isometry formula.

An Introduction to Quantum Filtering

This paper provides an introduction to quantum filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation as a least squares estimate, and culminating in the construction of Wiener and Poisson processes on the Fock space. We describe the quantum Ito calculus and its use in the modeling of physical systems. We use both reference probability and innovations methods to obtain quantum filtering equations for system-probe models from quantum optics.

Representation and Weak Convergence of Stochastic Integrals with Fractional Integrator Processes

This paper considers the asymptotic distribution of the covariance of a nonstationary fractionally integrated process with the stationary increments of another such process - possibly, itself. Questions of interest include the relationship between the harmonic representation of these random variables, which we have analysed in a previous paper, and the construction derived from moving average representations in the time domain. The limiting integrals are shown to be expressible in terms of functionals of Ito integrals with respect to two distinct Brownian motions. Their mean is nonetheless shown to match that of the harmonic representation, and they satisfy the required integration by parts rule. The advantages of our approach over the harmonic analysis include the facts that our formulae are valid for the full range of the long memory parameters, and extend to non-Gaussian processes.

Representation theorems, set-valued and fuzzy set-valued Ito integral

In this paper, we shall present representation theorems of set-valued martingales and set-valued processes of finite variation with continuous time. We shall also obtain a representation theorem of a predictable set-valued stochastic process. We shall give a new definition of Ito integral of a set-valued stochastic process with respect to a Brownian motion based on the work [E.J. Jung, J.H. Kim, On set-valued stochastic integrals, Stochastic Anal. Appl. 21(2) (2003) 401–418.]. We shall also discuss some properties of set-valued Ito integral, especially the presentation theorem of set-valued Ito integral. Finally, we extend some of above results to the fuzzy set-valued case.

Moment equations and Hermite expansion for nonlinear stochastic differential equations with application to stock price models

Exact moment equations for nonlinear Ito processes are derived. Taylor expansion of the drift and diffusion coefficients around the first conditional moment gives a hierarchy of coupled moment equations which can be closed by truncation or a Gaussian assumption. The state transition density is expanded into a Hermite orthogonal series with leading Gaussian term and the Fourier coefficients are expressed in terms of the moments. The resulting approximate likelihood is maximized by using a quasi Newton algorithm with BFGS secant updates. A simulation study for the CEV stock price model compares the several approximate likelihood estimators with the Euler approximation and the exact ML estimator (Feller, in Ann Math 54: 173---182, 1951).

A diffusion model with cubic drift: statistical and computational aspects and application to modelling of the global CO2 emission in Spain

AbstractThe aim of this work is the study of a new stochastic diffusion model with a cubic‐type drift coefficient. The model is considered as the solution of an Ito stochastic differential equation. Using the Ito's stochastic calculus and properties of the Kummer function, the trend functions and steady‐state distribution for the process are obtained. Statistical estimation and corresponding computational methodology are established. Finally, the model is applied to modelling and prediction of the global CO2 emission in Spain. Copyright © 2006 John Wiley & Sons, Ltd.

Estimation of the drift function for Ito processes and a class of semimartingales via histogram sieve

A histogram sieve estimator of the drift function in Ito processes and some semimartingales is constructed. It is proved that the estimator is pointwise and $L^{1}$ consistent and its finite-dimensional distributions are asymptotically normal. Our approac

Integrating product and process design with optimal control: A case study of solvent recycling

This paper proposes the simultaneous integration of environmentally benign solvent selection (product design), solvent recycling (process design) and optimal control for the separation of azeotropic systems using batch distillation. The previous work performed by Kim et al. (2004. Entrainer selection and solvent recycling in complex batch distillation. Chemical Engineering Communications 191(12), 1606–1633) combines the chemical synthesis and process synthesis under uncertainty. For batch distillation, optimal operation is also important due to the unsteady state nature of the process and high operating costs. Optimal control allows us to optimize the column operating policy by selecting a trajectory for the reflux ratio. However, there are time-dependent uncertainties in thermodynamic models of batch distillation due to the assumption of constant relative volatility. In this paper, the uncertainties in relative volatility were modeled using Ito processes and the stochastic optimal control problem was solved by combined maximum principle and non-linear programming (NLP) techniques. Then the previous work of optimal solvent selection and recycling was coupled with optimal control. As a real world example for this integrated approach, a waste stream containing acetonitrile–water was studied. The optimal design parameters obtained by Kim et al. (2004. Entrainer selection and solvent recycling in complex batch distillation. Chemical Engineering Communications 191(12), 1606–1633), for this separation were used and the optimal control policy is computed first without considering uncertainties by variable transformation technique. The deterministic optimal control policy improves the product yield by 4.0% as compared to the base case, verified using a rigorous simulator for batch distillation. When the stochastic optimal control policy was computed representing the relative volatility as an Ito process, a similar recovery rate was obtained from simulations, but the batch time was reduced significantly, producing the most profitable operation.

On estimating stress in free-draining Kramers chain simulations using stochastic filtering

For free-draining Kramers chains, several possible algorithms for calculating the stress tensor are examined, when Liu’s algorithm is used for the dynamics. Previous methods assumed that Stratonovich calculus was necessary to interpret the stochastic differential equation and calculate the stress, which is incorrect. Here, we consider these previous methods and derive two new ones using Ito calculus. All methods considered give correct averages. However, differences are seen in the statistical error, computational costs, and simplicity. The explicit expression derived by Liu is the most expensive to calculate, and most complicated. A version using ‘stochastic filtering’, proposed by Grassia and Hinch [P. Grassia, E.J. Hinch, Computer simulations of polymer chain relaxation via Brownian motion, J. Fluid Mech. 308 (1996) 255–288], and independently by Doyle et al. [P.S. Doyle, E.S.G. Shaqfeh, A.P. Gast, Dynamic simulation of freely draining flexible polymers in steady linear flows, J. Fluid Mech. 334 (1997) 251–291] yields the same statistical error, but requires less computational cost than Liu’s expression. A new algorithm is proposed, which yields uncertainties identical to those found by Grassia and Hinch, but is simpler, and requires smaller memory storage. An even simpler expression is proposed which, unlike that using stochastic filtering, requires no additional information from that required for the Brownian dynamics simulations. This last expression yields a larger statistical error than the method proposed here, and is found not to be advantageous for computational costs. Hence, the simplified stress calculator introduced here is the recommended one.

Uncertainties in parameter estimation and optimal control in batch distillation

Optimal control problems in batch distillation involve finding a trajectory for the reflux ratio so as to maximize a performance index. Then the controller is asked to follow this trajectory in an open loop fashion. It is important to minimize the effect of uncertainties in thermodynamic models on the optimal control profiles to achieve a better operating performance. The non-linear parameter estimation problem in vapor–liquid equilibrium modeling involves determining the values of model parameters, which provide the best fit to experimental data. It was shown previously by Gau et al. [Fluid Phase Equilibria, 168, 1–8, (2000)] that, using a global optimization procedure based on interval-Newton technique combined with interval-branch-and-bound can significantly reduce the error between the predicted and experimental data. Using this method, it was also shown that for some of the data sets published in DECHEMA, the parameters estimated correspond to local minima. The effect of locally and globally optimal parameter estimates on batch distillation optimal control profiles is demonstrated in this work. Since batch distillation is a dynamic process, the static (parametric) uncertainties are translated into time-dependent uncertainties. The time-dependent changes in relative volatility for the two cases are analyzed and represented by Ito processes. Next, the optimal control problem is solved using the maximum principle and NLP approach. Numerical case studies show that using globally optimal parameter estimates versus locally optimal parameter estimates results in a better product yield and the minimum error between the specified purity and the purity that is achieved.

A note on Wick products and the fractional Black-Scholes model

In some recent papers (Elliott and van der Hoek 2003; Hu and Oksendal 2003) a fractional Black-Scholes model has been proposed as an improvement of the classical Black-Scholes model (see also Benth 2003; Biagini et al. 2002; Biagini and Oksendal 2004). Common to these fractional Black-Scholes models is that the driving Brownian motion is replaced by a fractional Brownian motion and that the Ito integral is replaced by the Wick integral, and proofs have been presented that these fractional Black-Scholes models are free of arbitrage. These results on absence of arbitrage complelety contradict a number of earlier results in the literature which prove that the fractional Black-Scholes model (and related models) will in fact admit arbitrage. The objective of the present paper is to resolve this contradiction by pointing out that the definition of the self-financing trading strategies and/or the definition of the value of a portfolio used in the above papers does not have a reasonable economic interpretation, and thus that the results in these papers are not economically meaningful. In particular we show that in the framework of Elliott and van der Hoek 2003, a naive buy-and-hold strategy does not in general qualify as “self-financing”. We also show that in Hu and Oksendal 2003, a portfolio consisting of a positive number of shares of a stock with a positive price may, with positive probability, have a negative “value”.

LARGE DEVIATION PRINCIPLE FOR DIFFUSION PROCESSES IN A CONUCLEAR SPACE

We consider a type of large deviation principle obtained by Freidlin and Wentzell for the solution of Stochastic differential equations in a conuclear space. We are using exponential tail estimates and exit probability of a Ito process. The nuclear structure of the state space is also used.

Classical solutions to reaction–diffusion systems for hedging problems with interacting Itô and point processes

We use probabilistic methods to study classical solutions for systems of interacting semilinear parabolic partial differential equations. In a modeling framework for a financial market with interacting Ito and point processes, such PDEs are shown to provide a natural description for the solution of hedging and valuation problems for contingent claims with a recursive payoff structure.

A Series Approach to Stochastic Differential Equations with Infinite Dimensional Noise

This paper considers semilinear stochastic differential equations in Hilbert spaces with Lipschitz nonlinearities and with the noise terms driven by sequences of independent scalar Wiener processes (Brownian motions). The interpretation of such equations requires a stochastic integral. By means of a series of Ito integrals, an elementary and direct construction of a Hilbert space valued stochastic integral with respect to a sequence of independent scalar Wiener processes is given. As an application, existence and strong and weak uniqueness for the stochastic differential equation are shown by exploiting the series construction of the integral.

Optimal Solution of Investment Problems Via Linear Parabolic Equations Generated by Kalman Filter

We consider optimal investment problems for a diffusion market model with non-observable random drifts that evolve as an Ito's process. Admissible strategies do not use direct observations of the market parameters, but rather use historical stock prices. For a non-linear problem with a general performance criterion, the optimal portfolio strategy is expressed via the solution of a scalar minimization problem and a linear parabolic equation with coefficients generated by the Kalman filter.