Stochastic Calculus Research Articles

Stochastic integration and differential equations for typical paths

The goal of this paper is to define stochastic integrals and to solve stochastic differential equations for typical paths taking values in a possibly infinite dimensional separable Hilbert space without imposing any probabilistic structure. In the spirit of [33, 37] and motivated by the pricing duality result obtained in [4] we introduce an outer measure as a variant of the pathwise minimal superhedging price where agents are allowed to trade not only in $\omega $ but also in $\int \omega \,d\omega :=\omega ^{2} -\langle \omega \rangle $ and where they are allowed to include beliefs in future paths of the price process expressed by a prediction set. We then call a property to hold true on typical paths if the set of paths where the property fails is null with respect to our outer measure. It turns out that adding the second term $\omega ^{2} -\langle \omega \rangle $ in the definition of the outer measure enables to directly construct stochastic integrals which are continuous, even for typical paths taking values in an infinite dimensional separable Hilbert space. Moreover, when restricting to continuous paths whose quadratic variation is absolutely continuous with uniformly bounded derivative, a second construction of model-free stochastic integrals for typical paths is presented, which then allows to solve in a model-free way stochastic differential equations for typical paths.

A Valuation Model for Callable Eurobonds

This paper models the value of callable Eurobonds, using stochastic calculus, by assuming that the exchange rate follows a geometric Brownian motion process and the arrival time of an early redemption of the bond by the issuer conforms to a negative exponential distribution. The solution to the stochastic model shows that there is a relationship between the call premium and the expected time to the call which is consistent with traditional Black-Scholes pricing formulae. The magnitude of the call premium can be viewed as a signal to the market on a Government treasury’s or company’s expectations about the future level of interest rates and possible refinancing strategies. This paper is unique because as of July 2019 there exists no attempt at valuing Callable Eurobonds in the research literature.

Nonlinear Cointegrating Power Function Regression with Endogeneity

This paper develops an asymptotic theory for nonlinear cointegrating power function regression. The framework extends earlier work on the deterministic trend case and allows for both endogeneity and heteroskedasticity, which makes the models and inferential methods relevant to many empirical economic and financial applications, including predictive regression. Accompanying the asymptotic theory of nonlinear regression, the paper establishes some new results on weak convergence to stochastic integrals that go beyond the usual semi-martingale structure and considerably extend existing limit theory, complementing other recent findings on stochastic integral asymptotics. The paper also provides a general framework for extremum estimation limit theory that encompasses stochastically nonstationary time series and should be of wide applicability.

A remark on global solutions to random 3D vorticity equations for small initial data

<p style='text-indent:20px;'>In this paper, we prove that the solution constructed in [<xref ref-type="bibr" rid="b2">2</xref>] satisfies the stochastic vorticity equations with the stochastic integration being understood in the sense of the integration of controlled rough path introduced in [<xref ref-type="bibr" rid="b8">8</xref>]. As a result, we obtain the existence and uniqueness of the global solutions to the stochastic vorticity equations in 3D case for the small initial data independent of time, which can be viewed as a stochastic version of the Kato-Fujita result (see [<xref ref-type="bibr" rid="b10">10</xref>]).

Expansion of iterated Stratonovich stochastic integrals based on generalized multiple Fourier series

The article is devoted to the expansions of multiple Stratonovich stochastic integrals of multiplicities 1-4 on the basis of the method of generalized multiple Fourier series. Mean-square convergence of the expansions for the case of Legendre polynomials and for the case of trigonometric functions is proven. The considered expansions contain only one operation of the limit transition in contrast to its existing analogues. This property is comfortable for the mean-square approximation of multiple stochastic integrals. The results of the article can be applied to numerical integration of Ito stochastic differential equations.

Physical Layer Authentication in Mission-Critical MTC Networks: A Security and Delay Performance Analysis

We study the detection and delay performance impacts of a feature-based physical layer authentication (PLA) protocol in mission-critical machine-type communication (MTC) networks. The PLA protocol uses generalized likelihood-ratio testing based on the line-of-sight (LOS), single-input multiple-output channel-state information in order to mitigate impersonation attempts from an adversary node. We study the detection performance, develop a queueing model that captures the delay impacts of erroneous decisions in the PLA (i.e., the false alarms and missed detections), and model three different adversary strategies: data injection, disassociation, and Sybil attacks. Our main contribution is the derivation of analytical delay performance bounds that allow us to quantify the delay introduced by PLA that potentially can degrade the performance in mission-critical MTC networks. For the delay analysis, we utilize tools from stochastic network calculus. Our results show that with a sufficient number of receive antennas (approximately 4–8) and sufficiently strong LOS components from legitimate devices, PLA is a viable option for securing mission-critical MTC systems, despite the low-latency requirements associated to corresponding use cases. Furthermore, we find that PLA can be very effective in detecting the considered attacks, and in particular, it can significantly reduce the delay impacts of disassociation and Sybil attacks.

A Bi-Level Capacity Optimization of an Isolated Microgrid With Load Demand Management Considering Load and Renewable Generation Uncertainties

Aiming at capacity optimization of an isolated microgrid, this paper establishes a bi-level capacity optimization model that considers load demand management (LDM) while comprehensively considering load and renewable generation uncertainties. The uncertainties in this paper are brought by the source and load on the same timescale, as well as by the different characteristics of uncertainty presented over different timescales. For long timescales, the problem of source/load random uncertainty is solved using the stochastic network calculus theory to meet the energy balance constraints. For short timescales, we primarily aim to resolve the problem of power balance at the operation level, considering the uncertainty of source/load prediction errors and the impact of LDM. Particularly, by controlling the interruptible and shiftable loads, the LDM can optimize load characteristics, reduce operation costs, and increase system stability. The bi-level optimization model established in this paper is analyzed with regard to energy and power balance constraints, and the proposed mixed integer linear programming (MILP) model is solved by utilizing the CPLEX solver to minimize the investment cost. A typical microgrid, comprising a wind turbine (WT), a photovoltaic panel (PV), a controllable micro generator (CMG), and an energy storage system (ESS), is taken as an example to study capacity optimization problems. The simulation results verify the rationality and effectiveness of the proposed model and method.

Performance Analysis and Uplink Scheduling for QoS-Aware NB-IoT Networks in Mobile Computing

Low-power wide-area (LPWA) communication has gained increasing attention in recent years with the rapid growth of fifth generation evolution (5G), the Internet of Things (IoT), and mobile computing. Narrowband Internet of Things (NB-IoT) is one kind of LPWA technology based on cellular IoT, which supports massive connections, wide area coverage, ultra-low power consumption, and ultra-low cost. Research on NB-IoT communication is increasingly attractive. Network calculus theory facilitates the performance analysis and network optimization of the NB-IoT system. We aim to analyze and optimize the NB-IoT networks. In this paper, we construct a random access traffic model including the NB-IoT user equipment (UE) arrival process and eNB service process. Then, we utilize the stochastic network calculus (SNC) to analyze the network delay in NB-IoT traffic model. Random latency bounds in different arrival processes are derived. Simulations show that SNC can evaluate the system delay under different distributions effectively. For the condition that numerous UEs access simultaneously following the Beta distribution, we first propose an improved K-means algorithm to cluster the NB-IoT terminals. Then, we raise the scheduling strategy on the basis of priority. It consists of the priority generation algorithm IPGNTQ and the NB-IoT task scheduling algorithm SANTQ. The extensive experiment results verify that our proposed optimized strategy can alleviate the network congestion effectually. Moreover, we compare our proposed optimized scheme with four existing uplink traffic scheduling schemes, showing that ours outperforms all of them.

Delay Performance of the Multiuser MISO Downlink Under Imperfect CSI and Finite-Length Coding

We use stochastic network calculus to investigate the delay performance of a multiuser MISO system with zero-forcing beamforming. First, we consider ideal assumptions with long codewords and perfect CSI at the transmitter, where we observe a strong channel hardening effect that results in very high reliability with respect to the maximum delay of the application. We then study the system under more realistic assumptions with imperfect CSI and finite blocklength channel coding. These effects lead to interference and to transmission errors, and we derive closed-form approximations for the resulting error probability. Compared to the ideal case, imperfect CSI and finite length coding cause massive degradations in the average transmission rate. Surprisingly, the system nevertheless maintains the same qualitative behavior as in the ideal case: as long as the average transmission rate is higher than the arrival rate, the system can still achieve very high reliability with respect to the maximum delay.

On the stability of matrix-valued Riccati diffusions

The stability properties of matrix-valued Riccati diffusions are investigated. The matrix-valued Riccati diffusion processes considered in this work are of interest in their own right, as a rather prototypical model of a matrix-valued quadratic stochastic process. Under rather natural observability and controllability conditions, we derive time-uniform moment and fluctuation estimates and exponential contraction inequalities. Our approach combines spectral theory with nonlinear semigroup methods and stochastic matrix calculus. This analysis seem to be the first of its kind for this class of matrix-valued stochastic differential equation. This class of stochastic models arise in signal processing and data assimilation, and more particularly in ensemble Kalman-Bucy filtering theory. In this context, the Riccati diffusion represents the flow of the sample covariance matrices associated with McKean-Vlasov-type interacting Kalman-Bucy filters. The analysis developed here applies to filtering problems with unstable signals.

A Nonlocal Approach to The Quantum Kolmogorov Backward Equation and Links to Noncommutative Geometry

The Accardi-Boukas quantum Black-Scholes equation (Luigi Accardi, Andreas Boukas: The Quantum Black-Scholes Equation, Global Journal of Pure and Applied Mathematics, (2006) vol.2, no.2, pp. 155-170.) can be used as an alternative to the classical approach to finance, and has been found to have a number of useful benefits. The quantum Kolmogorov backward equations, and associated quantum Fokker-Planck equations, that arise from this general framework, are derived using the Hudson-Parthasarathy quantum stochastic calculus (Hudson, R.L.; Parthasarathy, K.R: Quantum Ito's Formula and Stochastic Evolutions. Commun Math. Phys. 1984, 93, 301--323.). In this paper we show how these equations can be derived using a nonlocal approach to quantum mechanics. We show how nonlocal diffusions, and quantum stochastic processes can be linked, and discuss how moment matching can be used for deriving solutions.

Numerical Methods for First Order Uncertain Stochastic Differential Equations

Uncertain stochastic calculus is a relatively new sub discipline of mathematics. This branch of mathematical sciences seeks to develop models that capture aleatory and epistemic features of generic uncertainty in dynamical systems. The growth of uncertain stochastic theory has given birth to a novel class of differential equations called uncertain stochastic differential equations (USDEs). Exact and analytic solutions to this family of differential equations are not always available. In such cases, numerical analysis provides a gateway to approximate solutions. This paper examines a Runge-Kutta method for solving USDEs. Before examining and applying the Runge-Kutta method, the paper states and proves the existence and uniqueness theorem. The Runge-Kutta method is then applied to solve an American call option pricing problem. This numerical algorithm proves to be effective and efficient because it produces almost the same results as compared to Chen's analytic formula and the classical Black-Scholes model.

Closed Quantum Black-Scholes: Quantum Drift and the Heisenberg Equation of Motion

In this article we model a financial derivative price as an observable on the market state function. We apply geometric techniques to integrating the Heisenberg Equation of Motion. We illustrate how the non-commutative nature of the model introduces quantum interference effects that can act as either a drag or a boost on the resulting return. The ultimate objective is to investigate the nature of quantum drift in the Accardi-Boukas quantum Black-Scholes framework which involves modelling the financial market as a quantum observable, and introduces randomness through the Hudson-Parthasarathy quantum stochastic calculus. In particular we aim to differentiate randomness that is introduced through external noise (quantum stochastic calculus) and randomness that is fundamental to a quantum system (Heisenberg Equation of Motion).

Maximal Inequalities and Exponential Estimates for Stochastic Convolutions Driven by Lévy-type Processes in Banach Spaces with Application to Stochastic Quasi-Geostrophic Equations

We present remarkably simple proofs of Burkholder-Davis-Gundy inequalities for stochastic integrals and maximal inequalities for stochastic convolutions in Banach spaces driven by L\'{e}vy-type processes. Exponential estimates for stochastic convolutions are obtained and two versions of It\^{o}'s formula in Banach spaces are also derived. Based on the obtained maximal inequality, the existence and uniqueness of mild solutions of stochastic quasi-geostrophic equation with L\'{e}vy noise is established.

Numerical Homogenization of Elliptic PDEs with Similar Coefficients

We consider a sequence of elliptic partial differential equations (PDEs) with different but similar rapidly varying coefficients. Such sequences appear, for example, in splitting schemes for time-dependent problems (with one coefficient per time step) and in sample based stochastic integration of outputs from an elliptic PDE (with one coefficient per sample member). We propose a parallelizable algorithm based on Petrov--Galerkin localized orthogonal decomposition that adaptively (using computable and theoretically derived error indicators) recomputes the local corrector problems only where it improves accuracy. The method is illustrated in detail by an example of a time-dependent two-pase Darcy flow problem in three dimensions.

Rescaled Whittaker driven stochastic differential equations converge to the additive stochastic heat equation

We study some linear SDEs arising from the two-dimensional $q$-Whittaker driven particle system on the torus as $q\to 1$. The main result proves that the SDEs along certain characteristics converge to the additive stochastic heat equation. Extensions for the SDEs with generalized coefficients and in other spatial dimensions are also obtained. Our proof views the limiting process after recentering as a process of the convolution of a space-time white noise and the Fourier transform of the heat kernel. Accordingly we turn to similar space-time stochastic integrals defined by the SDEs, but now the convolution and the Fourier transform are broken. To obtain tightness of these induced integrals, we bound the oscillations of complex exponentials arising from divergence of the characteristics, with two methods of different nature.

Convergence of random oscillatory integrals in the presence of long-range dependence and application to homogenization

This paper deals with the asymptotic behavior of random oscillatory integrals in the presence of long-range dependence. As a byproduct, we solve the corrector problem in random homogenization of onedimensional elliptic equations with highly oscillatory random coefficients displaying long-range dependence, by proving convergence to stochastic integrals with respect to Hermite processes.

Stochastic equivalence for performance analysis of concurrent systems in dtsiPBC

We propose an extension with immediate multiactions of discrete time stochastic Petri box calculus (dtsPBC), pre- sented by I.V. Tarasyuk. The resulting algebra dtsiPBC is a discrete time analogue of stochastic Petri box calculus (sPBC) with immediate multiactions, proposed by H. MaciV. Valero and others within a continuous time domain. The step operational semantics is constructed via labeled probabilistic transition systems. The denotational seman- tics is defined on the basis of a subclass of labeled discrete time stochastic Petri nets with immediate transitions. A consistency of the both semantics is demonstrated. In order to evaluate performance, the corresponding semi-Markov chains and (reduced) discrete time Markov chains are analyzed. We define step stochastic bisimulation equivalence of expressions and prove that it can be applied to reduce their transition systems and underlying semi-Markov chains while preserving the functionality and performance characteristics. We explain how this equivalence may help to sim- plify performance analysis of the algebraic processes. In a case study, a method of modeling, performance evaluation and behaviour preserving reduction of concurrent systems is outlined and applied to the shared memory system.

Strong sequential completeness of the natural domain of a conditional expectation operator in Riesz spaces

Strong convergence and convergence in probability were generalized to the setting of a Riesz space with conditional expectation operator, $T$, in [{{\sc Y. Azouzi, W.-C. Kuo, K. Ramdane, B. A. Watson}, {Convergence in Riesz spaces with conditional expectation operators}, {\em Positivity}, {\bf 19} {(2015), 647-657}}] as $T$-strong convergence and convergence in $T$-conditional probability, respectively. Generalized $L^{p}$ spaces for the cases of $p=1,2,\infty$, were discussed in the setting of Riesz spaces as $\mathcal{L}^{p}(T)$ spaces in [{{\sc C. C. A. Labuschagne, B. A. Watson}, {Discrete stochastic integration in Riesz spaces}, {\em Positivity}, {\bf 14} {(2010), 859-875}}]. An $R(T)$ valued norm, for the cases of $p=1,\infty,$ was introduced on these spaces in [{{\sc W. Kuo, M. Rogans, B.A. Watson}, {Mixing processes in Riesz spaces}, {\em Journal of Mathematical Analysis and Application}, {\bf 456} {(2017), 992-1004}}] where it was also shown that $R(T)$ is a universally complete $f$-algebra and that these spaces are $R(T)$-modules. In [{{\sc Y. Azouzi, M. Trabelsi}, {$L^p$-spaces with respect to conditional expectation on Riesz spaces}, {\em Journal of Mathematical Analysis and Application}, {\bf 447} {(2017), 798-816}}] functional calculus was used to consider $\mathcal{L}^{p}(T)$ for $p\in (1,\infty)$. In this paper we prove the strong sequential completeness of the space $\mathcal{L}^{1}(T)$, the natural domain of the conditional expectation operator $T$, and the strong completeness of $\mathcal{L}^{\infty}(T)$.

Mean square stability of linear stochastic neutral‐type time‐delay systems with multiple delays

SummaryThis paper studies mean square exponential stability of linear stochastic neutral‐type time‐delay systems with multiple point delays by using an augmented Lyapunov‐Krasovskii functional (LKF) approach. To build a suitable augmented LKF, a method is proposed to find an augmented state vector whose elements are linearly independent. With the help of the linearly independent augmented state vector, the constructed LKF, and properties of the stochastic integral, sufficient delay‐dependent stability conditions expressed by linear matrix inequalities are established to guarantee the mean square exponential stability of the system. Differently from previous results where the difference operator associated with the system needs to satisfy a condition in terms of matrix norms, in the current paper, the difference operator only needs to satisfy a less restrictive condition in terms of matrix spectral radius. The effectiveness of the proposed approach is illustrated by two numerical examples.