International Journal of Stochastic Analysis
[SJR: 0.234] [H-I: 19] [4 followers] Follow
Open Access journal
ISSN (Print) 2090-3332 - ISSN (Online) 2090-3340
Published by Hindawi [333 journals]
- Malliavin Differentiability of Solutions of SPDEs with Lévy White
Abstract: 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.
PubDate: Sun, 12 Mar 2017 00:00:00 +000
- Semigroup Solution of Path-Dependent Second-Order Parabolic Partial
Abstract: 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.
PubDate: Mon, 27 Feb 2017 00:00:00 +000
- Global Stability of Nonlinear Stochastic SEI Epidemic Model with
Fluctuations in Transmission Rate of Disease
Abstract: 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.
PubDate: Mon, 23 Jan 2017 00:00:00 +000
- Generalisation of Hajek’s Stochastic Comparison Results to
Abstract: 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.
PubDate: Mon, 05 Sep 2016 16:18:57 +000
- Asymptotic Time Averages and Frequency Distributions
Abstract: 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.
PubDate: Mon, 05 Sep 2016 06:27:26 +000
- A BSDE with Delayed Generator Approach to Pricing under Counterparty Risk
Abstract: 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.
PubDate: Tue, 02 Aug 2016 06:36:12 +000
- Stochastic Analysis of Gaussian Processes via Fredholm Representation
Abstract: 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.
PubDate: Sun, 31 Jul 2016 13:12:25 +000
- Optimal Bounds for the Variance of Self-Intersection Local Times
Abstract: 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.
PubDate: Wed, 20 Jul 2016 08:55:32 +000
- Analysis of a Priority Queue with Phase-Type Service and Failures
Abstract: 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.
PubDate: Sun, 17 Jul 2016 12:25:23 +000
- Multiserver Queue with Guard Channel for Priority and Retrial Customers
Abstract: 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.
PubDate: Thu, 03 Mar 2016 15:16:49 +000
- Large Deviation Analysis of a Droplet Model Having a Poisson Equilibrium
Abstract: 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.
PubDate: Wed, 07 Oct 2015 07:04:19 +000
- On Continuous Selection Sets of Non-Lipschitzian Quantum Stochastic
Abstract: 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.
PubDate: Tue, 28 Jul 2015 08:43:57 +000
- Pricing FX Options in the Heston/CIR Jump-Diffusion Model with Log-Normal
and Log-Uniform Jump Amplitudes
Abstract: We examine foreign exchange options in the jump-diffusion version of the Heston stochastic volatilitymodel 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.
PubDate: Sun, 26 Jul 2015 06:17:28 +000
- A Generic Decomposition Formula for Pricing Vanilla Options under
Stochastic Volatility Models
Abstract: 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.
PubDate: Wed, 17 Jun 2015 08:39:25 +000
- Stochastic Nonlinear Equations Describing the Mesoscopic Voltage-Gated Ion
Abstract: 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.
PubDate: Sun, 05 Apr 2015 16:01:57 +000
- A Comparative Numerical Study of the Spectral Theory Approach of Nishimura
and the Roots Method Based on the Analysis of BDMMAP/G/1 Queue
Abstract: 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.
PubDate: Tue, 17 Feb 2015 09:51:28 +000
- Asymptotic Stabilizability of a Class of Stochastic Nonlinear Hybrid
Abstract: 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.
PubDate: Wed, 11 Feb 2015 07:54:05 +000
- A General Multidimensional Monte Carlo Approach for Dynamic Hedging under
Abstract: 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.
PubDate: Sun, 08 Feb 2015 14:05:28 +000
- A Stochastic Flows Approach for Asset Allocation with Hidden Economic
Abstract: 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.
PubDate: Tue, 27 Jan 2015 11:15:54 +000
- Yamada-Watanabe Results for Stochastic Differential Equations with Jumps
Abstract: 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.
PubDate: Thu, 01 Jan 2015 09:34:03 +000
- Adaptive Algorithm for Estimation of Two-Dimensional Autoregressive Fields
from Noisy Observations
Abstract: 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.
PubDate: Thu, 25 Dec 2014 00:10:03 +000
- On Henstock Method to Stratonovich Integral with respect to Continuous
Abstract: 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.
PubDate: Thu, 18 Dec 2014 00:10:15 +000
- Strong Law of Large Numbers for Hidden Markov Chains Indexed by an
Infinite Tree with Uniformly Bounded Degrees
Abstract: 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.
PubDate: Tue, 09 Dec 2014 06:59:15 +000
- Optimal Foreign Exchange Rate Intervention in Lévy Markets
Abstract: 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.
PubDate: Wed, 26 Nov 2014 00:10:03 +000
- A Note on the Distribution of Multivariate Brownian Extrema
Abstract: 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 byreflections 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.
PubDate: Sun, 16 Nov 2014 11:47:53 +000
- A Discrete-Time Queue with Balking, Reneging, and Working Vacations
Abstract: 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.
PubDate: Wed, 29 Oct 2014 00:00:00 +000
- Limit Properties of Transition Functions of Continuous-Time Markov
Abstract: 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.
PubDate: Sun, 19 Oct 2014 00:00:00 +000
- A Semigroup Expansion for Pricing Barrier Options
Abstract: 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.
PubDate: Sun, 14 Sep 2014 13:08:59 +000
- Backward Stochastic Differential Equations Approach to Hedging, Option
Pricing, and Insurance Problems
Abstract: 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.
PubDate: Thu, 11 Sep 2014 06:36:48 +000
- Efficient Variable Step Size Approximations for Strong Solutions of
Stochastic Differential Equations with Additive Noise and Time Singularity
Abstract: We consider stochastic differential equations with additive noise and conditions on the coefficients in those equations that allow a time singularity in the drift coefficient. Given a maximum step size, , we specify variable (adaptive) step sizes relative to which decrease as the time node points approach the singularity. We use an Euler-type numerical scheme to produce an approximate solution and estimate the error in the approximation. When the solution is restricted to a fixed closed time interval excluding the singularity, we obtain a global pointwise error of order . An order of error for any is obtained when the approximation is run up to a time within of the singularity for an appropriate choice of exponent . We apply this scheme to Brownian bridge, which is defined as the nonanticipating solution of a stochastic differential equation of the type under consideration. In this special case, we show that the global pointwise error is of order , independent of how close to the singularity the approximation is considered.
PubDate: Wed, 02 Jul 2014 10:48:33 +000