Observable Stochastic Process Research Articles

Indifference pricing of pure endowments via BSDEs under partial information

In this paper, we investigate the pricing problem of a pure endowment contract when the insurance company has a limited information on the mortality intensity of the policyholder. The payoff of this kind of policies depends on the residual life time of the insured as well as the trend of a portfolio traded in the financial market, where investments in a riskless asset, a risky asset and a longevity bond are allowed. We propose a modeling framework that takes into account mutual dependence between the financial and the insurance markets via an observable stochastic process, which affects the risky asset and the mortality index dynamics. Since the market is incomplete due to the presence of basis risk, in alternative to arbitrage pricing we use expected utility maximization under exponential preferences as evaluation approach, which leads to the so-called indifference price. Under partial information this methodology requires filtering techniques that can reduce the original control problem to an equivalent problem in complete information. Using stochastic dynamics techniques, we characterize the indifference price of the insurance derivative in terms of the solutions of two backward stochastic differential equations. Finally, we discuss two special cases where we get a more explicit representation of the indifference price process.

Indifference Pricing of Pure Endowments Via BSDEs under Partial Information

In this paper we investigate the pricing problem of a pure endowment contract when the insurer has a limited information on the mortality intensity of the policyholder. The payoff of this kind of policies depends on the residual life time of the insured as well as the trend of a portfolio traded in the financial market, where investments in a riskless asset, a risky asset and a longevity bond are allowed. We propose a modeling framework that takes into account mutual dependence between the financial and the insurance markets via an observable stochastic process, which affects the risky asset and the mortality index dynamics. Since the market is incomplete due to the presence of basis risk, in alternative to arbitrage pricing we use expected utility maximization under exponential preferences as evaluation approach, which leads to the so-called indifference price. Under partial information this methodology requires filtering techniques that can reduce the original control problem to an equivalent problem in complete information. Using stochastic dynamics techniques, we characterize the indifference price of the insurance derivative via the solutions of suitable backward stochastic differential equations.

A discrete regularization method for hidden Markov models embedded into reproducing kernel Hilbert space

Hidden Markov models are a well-known probabilistic graphical model for time series of discrete, partially observable stochastic processes. We consider the method to extend the application of hidden Markov models to non-Gaussian continuous distributions by embedding a priori probability distribution of the state space into reproducing kernel Hilbert space. Corresponding regularization techniques are proposed to reduce the tendency to overfitting and computational complexity of the algorithm, i.e. Nystr¨om subsampling and the general regularization family for inversion of feature and kernel matrices. This method may be applied to various statistical inference and learning problems, including classification, prediction, identification, segmentation, and as an online algorithm it may be used for dynamic data mining and data stream mining. We investigate, both theoretically and empirically, the regularization and approximation bounds of the discrete regularization method. Furthermore, we discuss applications of the method to real-world problems, comparing the approach to several state-of-the-art algorithms.

On the optimal stopping with incomplete data

The Kalman–Bucy continuous model of partially observable stochastic processes is considered. The problem of optimal stopping of a stochastic process with incomplete data is reduced to the problem of optimal stopping with complete data. The convergence of payoffs is proved when ε1→0,ε2→0, where ε1, and ε2 are small perturbation parameters of the non observable and observable processes respectively.

Prediction of annual rainfall pattern using Hidden Markov Model (HMM) in Jos, Plateau State, Nigeria

A Hidden Markov Model (HMM) is a double stochastic process in which one of the stochastic processes is an underlying Markov chain, the other stochastic process is an observable stochastic process. Hidden Markov model is very influential in stochastic world because of its uniqueness, double stochastic nature and independence assumption between consecutive observations. A hidden Markov model to predict annual rainfall pattern has been presented in this paper. The model is developed to provide necessary information for the farmers, agronomists, water resource management scientists and policy makers to enable them plan for the uncertainty of annual rainfall. The model classified annual rainfall amount into three states, each with eight possible observations. The parameters of the model were estimated from the annual rainfall data of Jos, Plateau state, Nigeria for the period of 39 years (1977-2015). After which, the model was trained using Baum-Welch algorithm to attend maximum likelihood. The model is designed such that, if given any of the three rainfall states and its observation in the present year, it is possible to make quantitative prediction on how rainfall will be in the following year and in the subsequent years. The test HMM1 was able to make prediction with 75% accuracy in state and 50% accuracy in observations. The accuracy level of the model shows that, it is dependable and therefore, information from the model could be used as a guide to the farmers, agronomists, water resources management scientists and the government to plan strategies for crop production in the region. Keywords: Markov model, Hidden Markov model, Transition probability, Observation probability, Crop Production, Annual Rainfall

Comparative Analysis of Data Mining Tools and Techniques for Information Retrieval

Background/Objectives: There are a lot of information retrieval techniques available for getting information from different kinds of sources. Main aim of this paper is to improve information retrieval activities to a higher level. Different methods for information retrieval have been studied and discussed. It involves use of Fuzzy Ontology Generation framework (FOGA) framework along with Formal Concept Analysis (FCA) based clustering and keyword matching approach. Hidden Markov Model has been used for retrieval of data from search engines in an intelligent and efficient way for correct identification and retrieval from the database. Classification algorithm has been used for detection of community and conversion of large community graph to sub community graph for its better study and usage. Findings: It has been found that fuzzy ontology generation framework can automatically generate the fuzzy ontology which is very hard and time consuming task otherwise. Clustering of data can be done using the technique of formal concept analysis along with keyword matching method. There is a large amount of data available under similar words but with different meanings. So there has been a lot of problem in retrieval of exact data as required in a very short amount of time. In this case Hidden Markov Model can be used which can find the non-observable or hidden stochastic process from the observable stochastic process. Application/Improvements: Generalized Expectation-Maximization algorithm used with Hidden Markov Model can find unknown parameters. By adding frequency tracking algorithm along with Hidden Markov Model, we can also track audible data from a large database. Community detection algorithm along with Informap and Bigclam algorithms used with Hidden Markov Model will increase the modularity of data. Its applications include use of information retrieval of different types of data in an extremely faster and efficient way.

Supply Contracts with Financial Hedging

We study the performance of a stylized supply chain where two firms, a retailer and a producer, compete in a Stackelberg game. The retailer purchases a single product from the producer and afterward sells it in the retail market at a stochastic clearance price. The retailer, however, is budget constrained and is therefore limited in the number of units that he may purchase from the producer. We also assume that the retailer's profit depends in part on the realized path or terminal value of some observable stochastic process. We interpret this process as a financial process such as a foreign exchange rate or interest rate. More generally, the process can be interpreted as any relevant economic index. We consider a variation (the flexible contract) of the traditional wholesale price contract that is offered by the producer to the retailer. Under this flexible contract, at t = 0 the producer offers a menu of wholesale prices to the retailer, one for each realization of the financial process up to a future time τ. The retailer then commits to purchasing at time τ a variable number of units, with the specific quantity depending on the realization of the process up to time τ. Because of the retailer's budget constraint, the supply chain might be more profitable if the retailer was able to shift some of the budget from states where the constraint is not binding to states where it is binding. We therefore consider a variation of the flexible contract, where we assume that the retailer is able to trade dynamically between zero and τ in the financial market. We refer to this variation as the flexible contract with hedging. We compare the decentralized competitive solution for the two contracts with the solutions obtained by a central planner. We also compare the supply chain's performance across the two contracts. We find, for example, that the producer always prefers the flexible contract with hedging to the flexible contract without hedging. Depending on model parameters, however, the retailer might or might not prefer the flexible contract with hedging.

Particle filtering approximations for a Gaussian-generalized inverse Gaussian model

We consider the filtering problem for a class of discrete-time partially observable stochastic processes. Under strong conditions on the parameters involved and on the initial condition, we are able to prove that it admits a finite dimensional filter. Relaxing these assumptions, we use a Rao Blackwellization procedure to perform a Particle filtering approximation of the filtering distribution, then we prove its convergence and extend this study to a jump Markov model.

A Gaussian-generalized inverse Gaussian finite-dimensional filter

We consider the filtering problem for a partially observable stochastic process {X n,Z n,Y n} n∈ N , solution to a nonlinear system of stochastic difference equations, which provides a stochastic modellization for both the mean and the variance of the Gaussian observation distribution. The noises in the equations are given by two sequences of independent Gaussian random variables and a sequence of independent gamma random variables. We are able to prove that there exists a finite-dimensional filter system for this model, since, for each n, the conditional distribution of ( X n , Z n ) given ( Y 0,…, Y n ) is that of a suitable bivariate Gaussian-generalized inverse Gaussian random variable.

Finite dimensional filters for nonlinear stochastic difference equations with multiplicative noises

We consider the filtering problem for partially observable stochastic processes {X n,Y n} n∈ N , solutions to systems of stochastic difference equations. In the first part of the paper we shall present a simple constructive method to obtain finite dimensional filters in discrete time. Then, applying some well-known results, mainly on the product of independent positive random variables, we shall present new finite dimensional filters and interpret some known results in a more general setting.

A Forward Interpolation Equation of a Semimartingale by Observations over a Point Process

Let $(\Omega ,\mathcal{F}_\infty ,{\bf P})$ be a complete probability space, and let $(\mathcal{F}_t )$, $t \in {\bf R}_ + $, be a nondecreasing right-continuous family of sub-$\sigma $-algebras of $\mathcal{F}_\infty $ completed by sets from $\mathcal{F}_\infty $ of zero probability. A two-dimensional partially observable stochastic process is given on the probability space $(\Omega ,\mathcal{F}_\infty ,{\bf P})$, where $\theta _t $ is an $(\mathcal{F}_t )$-adapted, $0 \leq t < \infty $, unobservable component and $(T_n ,X_n )$, $n \geq 1$, is an observable one. We consider the problem of optimal interpolation, which consists of finding an optimal mean square estimate $\theta _s $ from the observations of the process $(T_n ,X_n )$ on $[0,t]$, $t \geq s$. This paper contains a deduction of an equation of optimal nonlinear interpolation on the basis of an equation of optimal nonlinear filtering.

On the existence of finite-dimensional filters in discrete time

In the present paper we shall study the filtering problem for a partially observable stochastic process with the unobservable component defined by a linear model of the form x n+1=A n x n + B n v n where (v n ) n is a sequence of independent identically distributed random vectors. We obtain necessary conditions for the existence of finite dimensional filters expressed in terms of conditions on the characteristic function of the filtering- and prediction distribution, as well as on the distribution of the noise sequence (v n ) n . Additional conditions, including conditions on (X 0) n , are derived for the existence of Gaussian Finite Dimensional Filters

On the convergence of costs in the problem of optimal stopping of stochastic processes in the scheme of kalman-bucy

We obtain the bound of order ∊ for the difference between the costs of optimal stopping problems for completely and partially observable cases when the partially observable stochastic process contains a small ∊ parmeter. The key step in teh analysis is an introduction of the auxiliary standard normal variablen and the subsequent change of time.

Determining the form of the mean of a stochastic process

Let μ be the mean function of an observable stochastic process whose sample paths fall in some Banach space with a basis and assume μ is also in this space. A procedure like Cover's ( Ann. Statist. 1, 862–871, 1973) is given which has the property that if the last nonzero coordinate of μ is the mth then with probability one this is discovered after at most a finite number of erros. If μ has an infinite number of nonzero coordinates, then with probability one this is discovered after at most a finite number of errors except for a set of μ of prior probability zero.