Showing papers in "Stochastic Analysis and Applications in 2012"
TL;DR: In this paper, the authors discuss an SIR model with stochastic perturbation and deduce the globally asymptotical stability and exponential meansquare stability of the disease-free equilibrium under some conditions, which means the disease will die out.
Abstract: In this article,we discuss an SIR model with stochastic perturbation. We show that there is a nonnegative solution that belongs to a positively invariant set. Then,by stochastic Lyapunov functional methods,we deduce the globally asymptotical stability and exponential meansquare stability of the disease-free equilibrium under some conditions,which means the disease will die out. Comparing with the deterministic model,there is no endemic equilibrium. To show when the disease will prevail,we investigate the asymptotic behavior of the solution around the endemic equilibrium of the deterministic model. Last,we illustrate the dynamic behavior of the model and their approximations via a range of numerical experiments.
102 citations
TL;DR: In this article, a new proof of existence of weak solutions to stochastic differential equations with continuous coefficients based on ideas from infinite-dimensional (i.e., stochastically analytically stable) analysis is presented.
Abstract: A new proof of existence of weak solutions to stochastic differential equations with continuous coefficients based on ideas from infinite-dimensional stochastic analysis is presented. The proof is fairly elementary, in particular, neither theorems on representation of martingales by stochastic integrals nor results on almost sure representation for tight sequences of random variables are needed.
58 citations
TL;DR: In this paper, the spectral decomposition of tensor random fields on the sphere was studied based on the theory of representations of the rotation group and the conditions of weak isotropy and mean square continuity.
Abstract: We study the representations of tensor random fields on the sphere basing on the theory of representations of the rotation group. Introducing specific components of a tensor field and imposing the conditions of weak isotropy and mean square continuity, we derive their spectral decompositions in terms of generalized spherical functions. The properties of random coefficients of the decompositions are characterized, including such an important question as conditions of Gaussianity.
40 citations
TL;DR: The Milstein scheme is the simplest nontrivial numerical scheme for stochastic differential equations with a strong order of convergence as mentioned in this paper, which has been extended to the Stochastic Delay Dierential Equation (SLDE).
Abstract: The Milstein scheme is the simplest nontrivial numerical scheme for stochastic differential
equations with a strong order of convergence one. The scheme has been extended to the
stochastic delay dierential equations but the analysis of the convergence is technically complicated
due to anticipative integrals in the remainder terms. This paper employs an elementary method
to derive the Milstein scheme and its rst order strong rate of convergence for stochastic delay
dierential equations.
40 citations
TL;DR: In this paper, the authors apply the Esscher transform on power spot models based on mean-reverting processes driven by independent increment (time-inhomogeneous Levy) processes, and show that the effect of the transform on the short end of the forward curve can explain the occurrence of a positive risk premium in the long end.
Abstract: In power markets one frequently encounters a risk premium being positive in the short end of the forward curve and negative in the long end. Economically it has been argued that the positive premium is reflecting retailers aversion for spike risk, wheras in the long end of the forward curve, the hedging pressure kicks in as in other commodity markets. Mathematically, forward prices are expressed as risk-neutral expectations of the spot at delivery. We apply the Esscher transform on power spot models based on mean-reverting processes driven by independent increment (time-inhomogeneous Levy) processes. It is shown that the Esscher transform is yielding a change of mean-reversion level. Moreover, we show that an Esscher transform together with jumps occuring seasonally may explain the occurence of a positive risk premium in the short end. This is demonstrated both mathematically and by a numerical example for a two-factor spot model being relevant for electricity markets.
32 citations
TL;DR: In this article, finite sample stability properties of sequential Monte Carlo methods for approximating sequences of probability distributions are studied. But the results presented in this paper are applicable in the scenario where the start and end distributions in the sequence are fixed and the number of intermediate steps is a parameter of the algorithm.
Abstract: This article addresses finite sample stability properties of sequential Monte Carlo methods for approximating sequences of probability distributions. The results presented herein are applicable in the scenario where the start and end distributions in the sequence are fixed and the number of intermediate steps is a parameter of the algorithm. Under assumptions which hold on noncompact spaces, it is shown that the effect of the initial distribution decays exponentially fast in the number of intermediate steps and the corresponding stochastic error is stable in 𝕃 p norm.
29 citations
TL;DR: The hyperbolic vector random field is introduced, which is formulated as a scale mixture of Gaussian random fields and is thus an elliptically contoured (or spherically invariant) random field.
Abstract: This article introduces the hyperbolic vector random field whose finite-dimensional distributions are the generalized hyperbolic one, which is formulated as a scale mixture of Gaussian random fields and is thus an elliptically contoured (or spherically invariant) random field. Such a vector random field may or may not have second-order moments, while a second-order one is characterized by its mean function and its covariance matrix function, just as in a Gaussian case. Some covariance matrix structures of hyperbolic type are constructed in this paper for second-order hyperbolic vector random fields.
25 citations
TL;DR: In this paper, a risk-sensitive stochastic optimal control problem motivated by a kind of optimal portfolio choice problem in the financial market is considered, which is similar in form to its risk-neutral counterpart.
Abstract: This article is concerned with a risk-sensitive stochastic optimal control problem motivated by a kind of optimal portfolio choice problem in the financial market. The maximum principle for this kind of problem is obtained, which is similar in form to its risk-neutral counterpart. But the adjoint equations and maximum condition heavily depend on the risk-sensitive parameter. This result is used to solve a kind of optimal portfolio choice problem and the optimal portfolio choice strategy is obtained. Computational results and figures explicitly illustrate the optimal solution and the sensitivity to the volatility rate parameter.
24 citations
TL;DR: In this paper, the authors fuse two pieces of theory to make a tractable model for asset pricing using a stochastic discounting function (SDF) and a Markov chain.
Abstract: This article fuses two pieces of theory to make a tractable model for asset pricing. The first is the theory of asset pricing using a stochastic discounting function (SDF). This will be reviewed. The second is to model uncertainty in an economy using a Markov chain. Using the semi-martingale dynamics for the chain these models can be calibrated and asset valuations derived. Interest rate models, stock price models, futures pricing, exchange rates can all be introduced endogenously in this framework.
22 citations
TL;DR: In this article, the authors compare several notions of almost periodicity for continuous processes defined on the time interval I = ℝ or I = ǫ or I Ã = Ã + Ã (0, Ã ) with values in a separable Banach space 𝔼.
Abstract: We compare several notions of almost periodicity for continuous processes defined on the time interval I = ℝ or I = [0, + ∞) with values in a separable Banach space 𝔼 (or more generally a separable completely regular topological space): almost periodicity in distribution, in probability, in quadratic mean, almost sure almost periodicity, almost equi-almost periodicity. In the deterministic case, all these notions reduce to Bochner-almost periodicity, which is equivalent to Bohr-almost periodicity when I = ℝ, and to asymptotic Bohr-almost periodicity when I = [0, + ∞).
21 citations
TL;DR: In this article, the authors considered an infinite capacity queueing system with a single server to which customers arrive according to a Poisson process and the service time follows an exponential distribution.
Abstract: Server induced interruptions such as server break downs, server attending a high priority customer, and server taking a vacation in queues have been extensively studied in the literature. However, customer-induced interruptions such as customers leaving in the middle of a service due to not having enough information for completing a service and customer breakdowns have not been studied so far. The purpose of this work is to introduce customer interruptions in queueing systems. We consider an infinite capacity queueing system with a single server to which customers arrive according to a Poisson process and the service time follows an exponential distribution. The customer interruption while in service occurs according to a Poisson process and the interruption duration follows an exponential distribution. The self-interrupted customers will enter into a finite buffer of size K. Any interrupted customer, finding the buffer full, is considered lost. Those interrupted customers who complete their interruptions...
TL;DR: In this paper, the authors deal with the almost-sure stability of stochastic differential delay equations and obtain improved theorems while conditions imposed on the Lyapunov function are much weaker.
Abstract: The main aim of this article is to deal with the almost-sure stability of stochastic differential delay equations. Our improved theorems give better results while conditions imposed on the Lyapunov function are much weaker, thus, it is easier to find a right Lyapunov function in application.
TL;DR: In this article, a characterization of all possible affine diffusions with polyhedral and quadratic state spaces is presented and necessary and sufficient conditions on the behavior of drift and diffusion on the boundary of the state space in order to obtain invariance and to prove strong existence and uniqueness.
Abstract: Multidimensional affine diffusions have been studied in detail for the case of a canonical state space. We present results for general state spaces and provide a complete characterization of all possible affine diffusions with polyhedral and quadratic state space. We give necessary and sufficient conditions on the behavior of drift and diffusion on the boundary of the state space in order to obtain invariance and to prove strong existence and uniqueness.
TL;DR: In this paper, a model with the dependence through fractal activity time is described, which is implemented via superpositions of Ornstein-Uhlenbeck type processes driven by Levy noise.
Abstract: Risky asset models with the dependence through fractal activity time are described. The construction of the fractal activity time is implemented via superpositions of Ornstein-Uhlenbeck type processes driven by Levy noise. The model features both tractable dependence structure and desired marginal distributions of the returns from the generalized hyperbolic class: the Variance Gamma and normal inverse Gaussian. These distributions provide good fit to real financial data. Pricing formulae for the proposed models are derived.
TL;DR: In this paper, the authors extend the complete convergence theorem for row sums from an array of rowwise independent random variables obtained by Sung, Volodin, and Hu [8] to a set of rowwise independent random elements taking values in a real separable Rademacher type p Banach space.
Abstract: We extend in several directions a complete convergence theorem for row sums from an array of rowwise independent random variables obtained by Sung, Volodin, and Hu [8] to an array of rowwise independent random elements taking values in a real separable Rademacher type p Banach space. An example is presented which illustrates that our result extends the Sung, Volodin, and Hu result even for the random variable case.
TL;DR: In this article, the authors consider a system with one massive particle coupling to an ideal random wave field, evolved according to classical mechanical principles, and prove the almost sure existence and uniqueness of the solution of the considered dynamics.
Abstract: We consider the problem of deriving Brownian motions from classical mechanical systems. Specifically, we consider a system with one massive particle coupling to an ideal random wave field, evolved according to classical mechanical principles. We prove the almost sure existence and uniqueness of the solution of the considered dynamics, prove the convergence of the solution under a certain scaling limit and give the precise expression of the limiting process, a diffusion process.
TL;DR: In this paper, it was shown that spaces of predictable processes with values in a Banach space are isomorphic to spaces of progressive resp. adapted, measurable processes, which is a straightforward extension of the It integral in infinite dimensions.
Abstract: The goal of this article is to give an easy proof that spaces of predictable processes with values in a Banach space are isomorphic to spaces of progressive resp. adapted, measurable processes. This provides a straightforward extension of the It integral in infinite dimensions. We also outline an application to stochastic partial differential equations.
TL;DR: In this paper, the authors considered a two commodity continuous review inventory system with Markovian demands, where the two commodities are assumed to be substitutable, and the limiting probability distribution for the joint inventory levels is computed, and various operational characteristics are derived.
Abstract: This article considers a two commodity continuous review inventory system with Markovian demands. The two commodities are assumed to be substitutable, that is, if the inventory level of one commodity reaches zero, then a demand for this commodity will be satisfied by an item of the other commodity. Reordering for supply is initiated as soon as the sum of the on-hand inventory levels of the two commodities reaches a certain level s, and there is a lead time until the reorder arrives. For this model, an associated Markovian model is derived which is then investigated in detail. The limiting probability distribution for the joint inventory levels is computed, and various operational characteristics are derived. The results are illustrated with numerical examples.
TL;DR: In this paper, the robustness of options prices to model variation in a multidimensional jump-diffusion framework was studied for both European and Exotic options and their deltas using two approaches: the Malliavin method and the Fourier method.
Abstract: We study the robustness of options prices to model variation in a multidimensional jump-diffusion framework. In particular, we consider price dynamics in which small variations are modeled either by a Poisson random measure with infinite activity or by a Brownian motion. We consider both European and Exotic options and we study their deltas using two approaches: the Malliavin method and the Fourier method. We prove robustness of the deltas to model variation. We apply these results to the study of stochastic volatility models for the underlying and the corresponding options.
TL;DR: In this article, guaranteed minimum death benefits (GMDBs) are considered as a form of portfolio insurance, and quantile hedges are developed to cover the contingent claims in incomplete markets when perfect hedging is not possible.
Abstract: The efficient hedging minimizes the average of the shortfall risk weighted by a loss function, where the hedging efficiency refers to the effectiveness of a hedge to accomplish the desired goal of risk management. Quantile hedging refers to the percentage of the hedge that can cover the contingent claim, which plays a key role for contingent claims in incomplete markets when perfect hedging is not possible. As observed in [8], the concept of quantile hedging can be considered as a dynamic version of the familiar value at risk concept (VaR). Treating regime switching diffusion models, this article focuses on guaranteed minimum death benefits (GMDBs), which are present in many variableannuity contracts, and act as a form of portfolio insurance. The GMDBs cannot be perfectly hedged due to the mortality component and incompleteness resulting from the regime switching, and as a result, quantile hedges are developed. Numerical examples are also presented to illustrate our results.
TL;DR: In this paper, the stochastic stabilization problems of partial differential equations by Levy noise are studied and sufficient conditions under which the perturbed systems decay with general rate functions are provided and some examples are constructed to demonstrate the applications of their theory.
Abstract: In this article, we focus on the stochastic stabilization problems of partial differential equations by Levy noise. Sufficient conditions under which the perturbed systems decay with general rate functions are provided and some examples are constructed to demonstrate the applications of our theory.
TL;DR: In this article, Liptser and Shiryaev provide the absolute continuity of the law of the solution of one equation with respect to the other one by means of Girsanov transform; then they consider the equivalence of these laws.
Abstract: Given two stochastic equations with different drift terms, under very weak assumptions Liptser and Shiryaev provide the absolute continuity of the law of the solution of one equation with respect to the other one by means of Girsanov transform; then they consider the equivalence of these laws. Their assumptions involve both the drift terms. We are interested in the same results but with the main assumption involving only the difference of the drift terms. Applications of our result will be presented in the finite as well as in the infinite-dimensional setting.
TL;DR: In this article, it was shown that the most important (H, δ)-dilatively stable limit processes (e.g., the LISOU and the LISCBI processes) almost surely have a local Holder exponent.
Abstract: First, we present some results about the Holder continuity of the sample paths of so-called dilatively stable processes which are certain infinitely divisible processes having a more general scaling property than self-similarity. As a corollary, we obtain that the most important (H, δ)-dilatively stable limit processes (e.g., the LISOU and the LISCBI processes, see [4]) almost surely have a local Holder exponent H. Next we prove that, under some slight regularity assumptions, any two dilatively stable processes with stationary increments are singular (in the sense that their distributions have disjoint supports) if their parameters H are different. We also study the more general case of not having stationary increments. Throughout the article, we specialize our results to some basic dilatively stable processes such as the above-mentioned limit processes and the fractional Levy process.
TL;DR: In this article, a backward stochastic differential equation (BSDE) approach is used to evaluate convex risk measures for unhedged positions of derivative securities in a continuous-time economy.
Abstract: A backward stochastic differential equation (BSDE) approach is used to evaluate convex risk measures for unhedged positions of derivative securities in a continuous-time economy. The convex risk measure is represented as the solution of a BSDE. We use the Clark-Ocone representation result together with Malliavin calculus to identify the integrand in the martingale representation associated with the BSDE. In the Markov case, we relate the BSDE solution to a partial differential equation solution for convex risk measure evaluation.
TL;DR: In this article, the authors derived the instantaneous defaultable forward rate f d (t, T) defined by in the real world probability in a market of defaultable bonds with jumps.
Abstract: We construct a model for the term structure in a market of defaultable bonds with jumps {p d (t, T); t ≤ T}, T ∈ (0, T*]. We derive the instantaneous defaultable forward rate f d (t, T) defined by in the real world probability. We are also given default-free bonds {p(t, T); t ≤ T}, T ∈ (0, T*] and we establish the market consisting of both the defaultable and the nondefaultable bonds. In this market, we study the common equivalent martingale measure and in this arbitrage free market we derive the relationship between the forward rates f(t, T) and f d (t, T) associated with the two sorts of bonds. Especially, it is proved that in a parameterized market with common equivalent martingale measure where f(t, T) can be described by (1.1) the defaultable forward rate f d (t, T) can be reconstructed from the special form of the default-free forward rate f(t, T) if a certain system of BSDEs has a solution. Finally, we extend the results to a market with recovery rate and give examples where the system of BSDEs has...
TL;DR: Theorem 3.4 of as discussed by the authors states that there cannot exist a global solution and the local solution will blow up at a finite time in the mean L p ǫ−ǫ norm for p ≥ 1.
Abstract: This article is concerned with explosive solutions of the initial-boundary problem for a class of nonlinear stochastic wave equations in a domain 𝒟 ⊂ ℝ d . Under appropriate conditions on the initial data, the nonlinear term and the noise intensity, it is proved in Theorem 3.4 that there cannot exist a global solution and the local solution will blow up at a finite time in the mean L p − norm for p ≥ 1. An example is given to show the application of this theorem.
TL;DR: In this paper, the problem of pricing defaultable bonds in a Markov modulated market is addressed using Merton's structural approach, where various types of defaultable bond are combination of European type contingent claims.
Abstract: We address the problem of pricing defaultable bonds in a Markov modulated market. Using Merton's structural approach we show that various types of defaultable bonds are combination of European type contingent claims. Thus pricing a defaultable bond is tantamount to pricing a contingent claim in a Markov modulated market. Since the market is incomplete, we use the method of quadratic hedging and minimal martingale measure to derive locally risk minimizing derivative prices, hedging strategies and the corresponding residual risks. The price of defaultable bonds are obtained as solutions to a system of PDEs with weak coupling subject to appropriate terminal and boundary conditions. We solve the system of PDEs numerically and carry out a numerical investigation for the defaultable bond prices. We compare their credit spreads with some of the existing models. We observe higher spreads in the Markov modulated market. We show how business cycles can be easily incorporated in the proposed framework. We demonstrate the impact on spreads of the inclusion of rare states that attempt to capture a tight liquidity situation. These states are characterized by low risk-free interest rate, high payout rate and high volatility.
TL;DR: In this article, the average number of real zeros of a class of trigonometric polynomials of the form where the a k's are independent standard normally distributed random variables and the b k's are binomial coefficients is estimated.
Abstract: In this article, we estimate the average number of real zeros of a class of trigonometric polynomials of the form where the a k 's are independent standard normally distributed random variables and the b k 's are binomial coefficients .
TL;DR: In this article, the problem of delay-dependent stability for Markovian jumping stochastic systems with interval time-varying delays and nonlinear perturbations is addressed.
Abstract: This article addresses the problem of delay-dependent stability for Markovian jumping stochastic systems with interval time-varying delays and nonlinear perturbations. The delay is assumed to be time-varying and belongs to a given interval. By resorting to Lyapunov–Krasovskii functionals and stochastic stability theory, a new delay interval-dependent stability criterion for the system is obtained. It is shown that the addressed problem can be solved if a set of linear matrix inequalities (LMIs) are feasible. Finally, a numerical example is employed to illustrate the effectiveness and less conservativeness of the developed techniques.
TL;DR: In this paper, the authors established the almost-sure central limit theorem (ASCLT) for a quasi-left continuous vector martingale with explosive and mixed (regular and explosive) growth, and proved a quadratic extension and established several new central limit theorems associated with the obtained ASCLT.
Abstract: In this article, we establish the almost-sure central limit theorem (ASCLT) for a quasi-left continuous vector martingale with explosive and mixed (regular and explosive) growth. We also prove a quadratic extension and establish several new central limit theorems associated with the obtained ASCLT. Finally, we study the problem of parameter estimation in the particular case of multidimensional diffusion processes, which illustrates in a concrete manner the use of our results.