International Journal of Stochastic Analysis The latest articles from Hindawi © 2018 , Hindawi Limited . All rights reserved. Option Price Decomposition in Spot-Dependent Volatility Models and Some Applications Mon, 31 Jul 2017 07:16:43 +0000 We obtain a Hull and White type option price decomposition for a general local volatility model. We apply the obtained formula to CEV model. As an application we give an approximated closed formula for the call option price under a CEV model and an approximated short term implied volatility surface. These approximated formulas are used to estimate model parameters. Numerical comparison is performed for our new method with exact and approximated formulas existing in the literature. Raúl Merino and Josep Vives Copyright © 2017 Raúl Merino and Josep Vives. All rights reserved. Malliavin Differentiability of Solutions of SPDEs with Lévy White Noise Sun, 12 Mar 2017 00:00:00 +0000 We consider a stochastic partial differential equation (SPDE) driven by a Lévy white noise, with Lipschitz multiplicative term . We prove that, under some conditions, this equation has a unique random field solution. These conditions are verified by the stochastic heat and wave equations. We introduce the basic elements of Malliavin calculus with respect to the compensated Poisson random measure associated with the Lévy white noise. If is affine, we prove that the solution is Malliavin differentiable and its Malliavin derivative satisfies a stochastic integral equation. Raluca M. Balan and Cheikh B. Ndongo Copyright © 2017 Raluca M. Balan and Cheikh B. Ndongo. All rights reserved. Semigroup Solution of Path-Dependent Second-Order Parabolic Partial Differential Equations Mon, 27 Feb 2017 00:00:00 +0000 We apply a new series representation of martingales, developed by Malliavin calculus, to characterize the solution of the second-order path-dependent partial differential equations (PDEs) of parabolic type. For instance, we show that the generator of the semigroup characterizing the solution of the path-dependent heat equation is equal to one-half times the second-order Malliavin derivative evaluated along the frozen path. Sixian Jin and Henry Schellhorn Copyright © 2017 Sixian Jin and Henry Schellhorn. All rights reserved. Global Stability of Nonlinear Stochastic SEI Epidemic Model with Fluctuations in Transmission Rate of Disease Mon, 23 Jan 2017 00:00:00 +0000 We derive and analyze the dynamic of a stochastic SEI epidemic model for disease spread. Fluctuations in the transmission rate of the disease bring about stochasticity in model. We discuss the asymptotic stability of the infection-free equilibrium by first deriving the closed form deterministic () and stochastic () basic reproductive number. Contrary to some author’s remark that different diffusion rates have no effect on the stability of the disease-free equilibrium, we showed that even if no epidemic invasion occurs with respect to the deterministic version of the SEI model (i.e., ), epidemic can still grow initially (if ) because of the presence of noise in the stochastic version of the model. That is, diffusion rates can have effect on the stability by causing a transient epidemic advance. A threshold criterion for epidemic invasion was derived in the presence of external noise. Olusegun Michael Otunuga Copyright © 2017 Olusegun Michael Otunuga. All rights reserved. Generalisation of Hajek’s Stochastic Comparison Results to Stochastic Sums Mon, 05 Sep 2016 16:18:57 +0000 Hajek’s univariate stochastic comparison result is generalised to multivariate stochastic sum processes with univariate convex data functions and for univariate monotonic nondecreasing convex data functions for processes with and without drift, respectively. As a consequence strategies for a class of multivariate optimal control problems can be determined by maximizing variance. An example is passport options written on multivariate traded accounts. The argument describes a narrow path between impossibilities of generalisations to jump processes or impossibilities of more general data functions. Jörg Kampen Copyright © 2016 Jörg Kampen. All rights reserved. Asymptotic Time Averages and Frequency Distributions Mon, 05 Sep 2016 06:27:26 +0000 Consider an arbitrary nonnegative deterministic process (in a stochastic setting is a fixed realization, i.e., sample-path of the underlying stochastic process) with state space . Using a sample-path approach, we give necessary and sufficient conditions for the long-run time average of a measurable function of process to be equal to the expectation taken with respect to the same measurable function of its long-run frequency distribution. The results are further extended to allow unrestricted parameter (time) space. Examples are provided to show that our condition is not superfluous and that it is weaker than uniform integrability. The case of discrete-time processes is also considered. The relationship to previously known sufficient conditions, usually given in stochastic settings, will also be discussed. Our approach is applied to regenerative processes and an extension of a well-known result is given. For researchers interested in sample-path analysis, our results will give them the choice to work with the time average of a process or its frequency distribution function and go back and forth between the two under a mild condition. Muhammad El-Taha Copyright © 2016 Muhammad El-Taha. All rights reserved. A BSDE with Delayed Generator Approach to Pricing under Counterparty Risk and Collateralization Tue, 02 Aug 2016 06:36:12 +0000 We consider a nonlinear pricing problem that takes into account credit risk and funding issues. The aforementioned problem is formulated as a stochastic forward-backward system with delay, both in the forward and in the backward component, whose solution is characterized in terms of viscosity solution to a suitable type of path-dependent PDE. Francesco Cordoni and Luca Di Persio Copyright © 2016 Francesco Cordoni and Luca Di Persio. All rights reserved. Stochastic Analysis of Gaussian Processes via Fredholm Representation Sun, 31 Jul 2016 13:12:25 +0000 We show that every separable Gaussian process with integrable variance function admits a Fredholm representation with respect to a Brownian motion. We extend the Fredholm representation to a transfer principle and develop stochastic analysis by using it. We show the convenience of the Fredholm representation by giving applications to equivalence in law, bridges, series expansions, stochastic differential equations, and maximum likelihood estimations. Tommi Sottinen and Lauri Viitasaari Copyright © 2016 Tommi Sottinen and Lauri Viitasaari. All rights reserved. Optimal Bounds for the Variance of Self-Intersection Local Times Wed, 20 Jul 2016 08:55:32 +0000 For a -valued random walk , let be its local time at the site . For , define the -fold self-intersection local time as . Also let be the corresponding quantities for the simple random walk in . Without imposing any moment conditions, we show that the variance of the self-intersection local time of any genuinely -dimensional random walk is bounded above by the corresponding quantity for the simple symmetric random walk; that is, . In particular, for any genuinely -dimensional random walk, with , we have . On the other hand, in dimensions we show that if the behaviour resembles that of simple random walk, in the sense that , then the increments of the random walk must have zero mean and finite second moment. George Deligiannidis and Sergey Utev Copyright © 2016 George Deligiannidis and Sergey Utev. All rights reserved. Analysis of a Priority Queue with Phase-Type Service and Failures Sun, 17 Jul 2016 12:25:23 +0000 We consider a single server queue with two types of customers. We propose a discipline of flexible priority in access that combines the features of randomization and the threshold type control. We introduce a new class of distributions, phase-type with failures () distribution, that generalizes the well-known phase-type () distribution to the case when failures can occur during service of a customer. The arrival flow is described by the marked Markovian arrival process. The service time distribution is of type with the parameters depending on the type of a customer. Customers of both types can be impatient. Behavior of the system is described by the multidimensional Markov chain. Problem of existence and computation of the stationary distribution of this Markov chain is discussed in brief as well as the problem of computation of the key performance measures of the system. Numerical examples are presented that give some insight into behavior of the system performance measures under different values of the parameters defining the strategy of customers access to service. Alexander Dudin and Sergei Dudin Copyright © 2016 Alexander Dudin and Sergei Dudin. All rights reserved. Multiserver Queue with Guard Channel for Priority and Retrial Customers Thu, 03 Mar 2016 15:16:49 +0000 This paper considers a retrial queueing model where a group of guard channels is reserved for priority and retrial customers. Priority and normal customers arrive at the system according to two distinct Poisson processes. Priority customers are accepted if there is an idle channel upon arrival while normal customers are accepted if and only if the number of idle channels is larger than the number of guard channels. Blocked customers (priority or normal) join a virtual orbit and repeat their attempts in a later time. Customers from the orbit (retrial customers) are accepted if there is an idle channel available upon arrival. We formulate the queueing system using a level dependent quasi-birth-and-death (QBD) process. We obtain a Taylor series expansion for the nonzero elements of the rate matrices of the level dependent QBD process. Using the expansion results, we obtain an asymptotic upper bound for the joint stationary distribution of the number of busy channels and that of customers in the orbit. Furthermore, we develop an efficient numerical algorithm to calculate the joint stationary distribution. Kazuki Kajiwara and Tuan Phung-Duc Copyright © 2016 Kazuki Kajiwara and Tuan Phung-Duc. All rights reserved. Large Deviation Analysis of a Droplet Model Having a Poisson Equilibrium Distribution Wed, 07 Oct 2015 07:04:19 +0000 In this paper we use large deviation theory to determine the equilibrium distribution of a basic droplet model that underlies a number of important models in material science and statistical mechanics. Given and , distinguishable particles are placed, each with equal probability , onto the sites of a lattice, where equals . We focus on configurations for which each site is occupied by a minimum of particles. The main result is the large deviation principle (LDP), in the limit and with , for a sequence of random, number-density measures, which are the empirical measures of dependent random variables that count the droplet sizes. The rate function in the LDP is the relative entropy , where is a possible asymptotic configuration of the number-density measures and is a Poisson distribution with mean , restricted to the set of positive integers satisfying . This LDP implies that is the equilibrium distribution of the number-density measures, which in turn implies that is the equilibrium distribution of the random variables that count the droplet sizes. Richard S. Ellis and Shlomo Ta’asan Copyright © 2015 Richard S. Ellis and Shlomo Ta’asan. All rights reserved. On Continuous Selection Sets of Non-Lipschitzian Quantum Stochastic Evolution Inclusions Tue, 28 Jul 2015 08:43:57 +0000 We establish existence of a continuous selection of multifunctions associated with quantum stochastic evolution inclusions under a general Lipschitz condition. The coefficients here are multifunctions but not necessarily Lipschitz. Sheila Bishop Copyright © 2015 Sheila Bishop. All rights reserved. Pricing FX Options in the Heston/CIR Jump-Diffusion Model with Log-Normal and Log-Uniform Jump Amplitudes Sun, 26 Jul 2015 06:17:28 +0000 We examine foreign exchange options in the jump-diffusion version of the Heston stochastic volatility model for the exchange rate with log-normal jump amplitudes and the volatility model with log-uniformly distributed jump amplitudes. We assume that the domestic and foreign stochastic interest rates are governed by the CIR dynamics. The instantaneous volatility is correlated with the dynamics of the exchange rate return, whereas the domestic and foreign short-term rates are assumed to be independent of the dynamics of the exchange rate and its volatility. The main result furnishes a semianalytical formula for the price of the foreign exchange European call option. Rehez Ahlip and Ante Prodan Copyright © 2015 Rehez Ahlip and Ante Prodan. All rights reserved. A Generic Decomposition Formula for Pricing Vanilla Options under Stochastic Volatility Models Wed, 17 Jun 2015 08:39:25 +0000 We obtain a decomposition of the call option price for a very general stochastic volatility diffusion model, extending a previous decomposition formula for the Heston model. We realize that a new term arises when the stock price does not follow an exponential model. The techniques used for this purpose are nonanticipative. In particular, we also see that equivalent results can be obtained by using Functional Itô Calculus. Using the same generalizing ideas, we also extend to nonexponential models the alternative call option price decomposition formula written in terms of the Malliavin derivative of the volatility process. Finally, we give a general expression for the derivative of the implied volatility under both the anticipative and the nonanticipative cases. Raúl Merino and Josep Vives Copyright © 2015 Raúl Merino and Josep Vives. All rights reserved. Stochastic Nonlinear Equations Describing the Mesoscopic Voltage-Gated Ion Channels Sun, 05 Apr 2015 16:01:57 +0000 We propose a stochastic nonlinear system to model the gating activity coupled with the membrane potential for a typical neuron. It distinguishes two different levels: a macroscopic one, for the membrane potential, and a mesoscopic one, for the gating process through the movement of its voltage sensors. Such a nonlinear system can be handled to form a Hodgkin-Huxley-like model, which links those two levels unlike the original deterministic Hodgkin-Huxley model which is positioned at a macroscopic scale only. Also, we show that an interacting particle system can be used to approximate our model, which is an approximation technique similar to the jump Markov processes, used to approximate the original Hodgkin-Huxley model. Mauricio Tejo Copyright © 2015 Mauricio Tejo. All rights reserved. A Comparative Numerical Study of the Spectral Theory Approach of Nishimura and the Roots Method Based on the Analysis of BDMMAP/G/1 Queue Tue, 17 Feb 2015 09:51:28 +0000 This paper considers an infinite-buffer queuing system with birth-death modulated Markovian arrival process (BDMMAP) with arbitrary service time distribution. BDMMAP is an excellent representation of the arrival process, where the fractal behavior such as burstiness, correlation, and self-similarity is observed, for example, in ethernet LAN traffic systems. This model was first apprised by Nishimura (2003), and to analyze it, he proposed a twofold spectral theory approach. It seems from the investigations that Nishimura’s approach is tedious and difficult to employ for practical purposes. The objective of this paper is to analyze the same model with an alternative methodology proposed by Chaudhry et al. (2013) (to be referred to as CGG method). The CGG method appears to be rather simple, mathematically tractable, and easy to implement as compared to Nishimura’s approach. The Achilles tendon of the CGG method is the roots of the characteristic equation associated with the probability generating function (pgf) of the queue length distribution, which absolves any eigenvalue algebra and iterative analysis. Both the methods are presented in stepwise manner for easy accessibility, followed by some illustrative examples in accordance with the context. Arunava Maity and U. C. Gupta Copyright © 2015 Arunava Maity and U. C. Gupta. All rights reserved. Asymptotic Stabilizability of a Class of Stochastic Nonlinear Hybrid Systems Wed, 11 Feb 2015 07:54:05 +0000 The problem of the asymptotic stabilizability in probability of a class of stochastic nonlinear control hybrid systems (with a linear dependence of the control) with state dependent, Markovian, and any switching rule is considered in the paper. To solve the issue, the Lyapunov technique, including a common, single, and multiple Lyapunov function, the hybrid control theory, and some results for stochastic nonhybrid systems are used. Sufficient conditions for the asymptotic stabilizability in probability for a considered class of hybrid systems are formulated. Also the stabilizing control in a feedback form is considered. Furthermore, in the case of hybrid systems with the state dependent switching rule, a method for a construction of stabilizing switching rules is proposed. Obtained results are illustrated by examples and numerical simulations. Ewelina Seroka Copyright © 2015 Ewelina Seroka. All rights reserved. A General Multidimensional Monte Carlo Approach for Dynamic Hedging under Stochastic Volatility Sun, 08 Feb 2015 14:05:28 +0000 We propose a feasible and constructive methodology which allows us to compute pure hedging strategies with respect to arbitrary square-integrable claims in incomplete markets. In contrast to previous works based on PDE and BSDE methods, the main merit of our approach is the flexibility of quadratic hedging in full generality without a priori smoothness assumptions on the payoff. In particular, the methodology can be applied to multidimensional quadratic hedging-type strategies for fully path-dependent options with stochastic volatility and discontinuous payoffs. In order to demonstrate that our methodology is indeed applicable, we provide a Monte Carlo study on generalized Föllmer-Schweizer decompositions, locally risk minimizing, and mean variance hedging strategies for vanilla and path-dependent options written on local volatility and stochastic volatility models. Daniel Bonetti, Dorival Leão, Alberto Ohashi, and Vinícius Siqueira Copyright © 2015 Daniel Bonetti et al. All rights reserved. A Stochastic Flows Approach for Asset Allocation with Hidden Economic Environment Tue, 27 Jan 2015 11:15:54 +0000 An optimal asset allocation problem for a quite general class of utility functions is discussed in a simple two-state Markovian regime-switching model, where the appreciation rate of a risky share changes over time according to the state of a hidden economy. As usual, standard filtering theory is used to transform a financial model with hidden information into one with complete information, where a martingale approach is applied to discuss the optimal asset allocation problem. Using a martingale representation coupled with stochastic flows of diffeomorphisms for the filtering equation, the integrand in the martingale representation is identified which gives rise to an optimal portfolio strategy under some differentiability conditions. Tak Kuen Siu Copyright © 2015 Tak Kuen Siu. All rights reserved. Yamada-Watanabe Results for Stochastic Differential Equations with Jumps Thu, 01 Jan 2015 09:34:03 +0000 Recently, Kurtz (2007, 2014) obtained a general version of the Yamada-Watanabe and Engelbert theorems relating existence and uniqueness of weak and strong solutions of stochastic equations covering also the case of stochastic differential equations with jumps. Following the original method of Yamada and Watanabe (1971), we give alternative proofs for the following two statements: pathwise uniqueness implies uniqueness in the sense of probability law, and weak existence together with pathwise uniqueness implies strong existence for stochastic differential equations with jumps. Mátyás Barczy, Zenghu Li, and Gyula Pap Copyright © 2015 Mátyás Barczy et al. All rights reserved. Adaptive Algorithm for Estimation of Two-Dimensional Autoregressive Fields from Noisy Observations Thu, 25 Dec 2014 00:10:03 +0000 This paper deals with the problem of two-dimensional autoregressive (AR) estimation from noisy observations. The Yule-Walker equations are solved using adaptive steepest descent (SD) algorithm. Performance comparisons are made with other existing methods to demonstrate merits of the proposed method. Alimorad Mahmoudi Copyright © 2014 Alimorad Mahmoudi. All rights reserved. On Henstock Method to Stratonovich Integral with respect to Continuous Semimartingale Thu, 18 Dec 2014 00:10:15 +0000 We will use the Henstock (or generalized Riemann) approach to define the Stratonovich integral with respect to continuous semimartingale in space. Our definition of Stratonovich integral encompasses the classical definition of Stratonovich integral. Haifeng Yang and Tin Lam Toh Copyright © 2014 Haifeng Yang and Tin Lam Toh. All rights reserved. Strong Law of Large Numbers for Hidden Markov Chains Indexed by an Infinite Tree with Uniformly Bounded Degrees Tue, 09 Dec 2014 06:59:15 +0000 We study strong limit theorems for hidden Markov chains fields indexed by an infinite tree with uniformly bounded degrees. We mainly establish the strong law of large numbers for hidden Markov chains fields indexed by an infinite tree with uniformly bounded degrees and give the strong limit law of the conditional sample entropy rate. Huilin Huang Copyright © 2014 Huilin Huang. All rights reserved. Optimal Foreign Exchange Rate Intervention in Lévy Markets Wed, 26 Nov 2014 00:10:03 +0000 This paper considers an exchange rate problem in Lévy markets, where the Central Bank has to intervene. We assume that, in the absence of control, the exchange rate evolves according to Brownian motion with a jump component. The Central Bank is allowed to intervene in order to keep the exchange rate as close as possible to a prespecified target value. The interventions by the Central Bank are associated with costs. We present the situation as an impulse control problem, where the objective of the bank is to minimize the intervention costs. In particular, the paper extends the model by Huang, 2009, to incorporate a jump component. We formulate and prove an optimal verification theorem for the impulse control. We then propose an impulse control and construct a value function and then verify that they solve the quasivariational inequalities. Our results suggest that if the expected number of jumps is high the Central Bank will intervene more frequently and with large intervention amounts hence the intervention costs will be high. Masimba Aspinas Mutakaya, Eriyoti Chikodza, and Edward T. Chiyaka Copyright © 2014 Masimba Aspinas Mutakaya et al. All rights reserved. A Note on the Distribution of Multivariate Brownian Extrema Sun, 16 Nov 2014 11:47:53 +0000 This paper presents a closed-form solution for the joint probability of the endpoints and minimums of a multidimensional Wiener process for some correlation matrices. This is the only explicit expressions found in the literature for this joint probability. The analysis can only be carried out for special correlation structures as it is related to the fundamentals regions of irreducible spherical simplexes generated by reflections and the link to the method of images. This joint distribution can be used in financial mathematics to obtain prices of credit or market related products in high dimension. The solution could be generalized to account for stochastic volatility and other stylized features of the financial markets. Marcos Escobar and Julio Hernandez Copyright © 2014 Marcos Escobar and Julio Hernandez. All rights reserved. A Discrete-Time Queue with Balking, Reneging, and Working Vacations Wed, 29 Oct 2014 00:00:00 +0000 This paper presents an analysis of balking and reneging in finite-buffer discrete-time single server queue with single and multiple working vacations. An arriving customer may balk with a probability or renege after joining according to a geometric distribution. The server works with different service rates rather than completely stopping the service during a vacation period. The service times during a busy period, vacation period, and vacation times are assumed to be geometrically distributed. We find the explicit expressions for the stationary state probabilities. Various system performance measures and a cost model to determine the optimal service rates are presented. Moreover, some queueing models presented in the literature are derived as special cases of our model. Finally, the influence of various parameters on the performance characteristics is shown numerically. Veena Goswami Copyright © 2014 Veena Goswami. All rights reserved. Limit Properties of Transition Functions of Continuous-Time Markov Branching Processes Sun, 19 Oct 2014 00:00:00 +0000 Consider the Markov Branching Process with continuous time. Our focus is on the limit properties of transition functions of this process. Using differential analogue of the Basic Lemma we prove local limit theorems for all cases and observe invariant properties of considering process. Azam A. Imomov Copyright © 2014 Azam A. Imomov. All rights reserved. A Semigroup Expansion for Pricing Barrier Options Sun, 14 Sep 2014 13:08:59 +0000 This paper presents a new asymptotic expansion method for pricing continuously monitoring barrier options. In particular, we develop a semigroup expansion scheme for the Cauchy-Dirichlet problem in the second-order parabolic partial differential equations (PDEs) arising in barrier option pricing. As an application, we propose a concrete approximation formula under a stochastic volatility model and demonstrate its validity by some numerical experiments. Takashi Kato, Akihiko Takahashi, and Toshihiro Yamada Copyright © 2014 Takashi Kato et al. All rights reserved. Backward Stochastic Differential Equations Approach to Hedging, Option Pricing, and Insurance Problems Thu, 11 Sep 2014 06:36:48 +0000 In the present work we give a self-contained introduction to financial mathematical models characterized by noise of Lévy type in the framework of the backward stochastic differential equations theory. Such techniques will be then used to analyse an innovative model related to insurance and death processes setting. Francesco Cordoni and Luca Di Persio Copyright © 2014 Francesco Cordoni and Luca Di Persio. All rights reserved.