Generalized moment estimation of stochastic differential equations

We study the semiparametric estimation of stochastic differential equations employing methods based on moment conditions, comparing the finite sample and robustness properties of generalized method of moments, empirical likelihood and minimum contrast methods using unconditional and conditional formulations of moment conditions. The results obtained indicate that the estimators proposed, particularly, the estimators based on exponential tilting, obtain better results than those of the generalized methods of moments normally used to estimate stochastic differential equations. This conclusion is mainly derived from the robustness properties of this method in the presence of problems of incorrect specification.     

Transition Specifications for Dynamic Abstract Data Types

Transition specifications extend algebraic specifications by a notion of states and state transitions, introducing a second dynamic layer on abstract data types. This makes possible a reduction of complex specifications by the introduction of an explicit system state, a formal comparison of algebraic specifications and imperative algorithms, and the specification of input/output and other communication features. States are modelled as partial algebras that extend a given algebra by an environment, that is, a partial function that assigns contents to references. State transitions are specified by conditional parallel assignments, analoguous to the conditional existence equations of partial equational specifications. A framework of transition specifications is developed, including initial model semantics, parameterization and general composition mechanisms, and a notion of model correctness. Examples from programming languages show the applicability of the approach.     

Nonlinear filtering theory for coral/starfish and plant/herbivore interactions †

Several nonlinear filtering problems associated with specific 4 dimensional differential equation models of coral/starfish or chemically mediated plant/herbivore population dynamics are studied. Extensive use is made of H. Kunita's backward Stratonovich calculus and stochastic partial differential equations theory to obtain exact solution measures of the Zakai and Kushner equations. The hypoellipticity problem is solved positively, so that these measures all possess c^∞-densities. Thus, explicit formulas are obtained for the estimation of signal processes conditional on observational data. For example, biomass production/consumption processes are least squares estimated conditional on observations on the population dynamics of the producing and consuming units themselves.     

Generalized conditional symmetries of evolution equations

We analyze the relationship of generalized conditional symmetries of evolution equations to the formal compatibility and passivity of systems of differential equations as well as to systems of vector fields in involution. Earlier results on the connection between generalized conditional invariance and generalized reduction of evolution equations are revisited. This leads to a no-go theorem on determining equations for operators of generalized conditional symmetry. It is also shown that up to certain equivalences there exists a one-to-one correspondence between generalized conditional symmetries of an evolution equation and parametric families of its solutions.     

A Coherent Aggregation Framework for Stress Testing and Scenario Analysis

Abstract

We present a methodology to aggregate in a coherent manner conditional stress losses in a trading or banking book. The approach bypasses the specification of unconditional probabilities of the individual stress events and ensures by a linear programming approach so that the (subjective or frequentist) conditional probabilities chosen by the risk manager are internally consistent. The admissibility requirement greatly reduces the degree of arbitrariness in the conditional probability matrix if this is assigned subjectively. The approach can be used to address the requirements of the regulators on the Instantaneous Risk Charge.     

Invariant subspaces and conditional Lie-Bäcklund symmetries of inhomogeneous nonlinear diffusion equations

The inhomogeneous nonlinear difusion equation is studied by invariant subspace and conditional Lie-Bcklund symmetry methods.It is shown that the equations admit a class of invariant subspaces governed by the nonlinear ordinary diferential equations,which is equivalent to a kind of higher-order conditional Lie-Bcklund symmetries of the equations.As a consequence,a number of new solutions to the inhomogeneous nonlinear difusion equations are constructed explicitly or reduced to solving fnite-dimensional dynamical systems.     

Scheduling of production and logistics operations of steam production systems in food industries: a case study of the tomato processing industry

This study presents an optimization approach by mathematical modelling to support some of the main operational decisions in steam production systems with multiple industrial boilers. Decisions are related to boiler operations scheduling (start-up, warm-up and shutdown time), fuel replenishment (transportation and inventory management) and fuel composition consumed by each piece of equipment. These decisions are often taken based on practical experience of people involved, instead of any decision support tool using optimization techniques; as a consequence, unnecessary costs are likely to be incurred. The optimization approach is based on mixed integer programming and parameters experimental adjustment procedures. A case study of a large tomato processing plant in Brazil was carried out along 1 year using a 3-year database. Owing to the reasonably good outcomes achieved (annually potential savings around 10%), we consider the proposed approach as a suitable tool to support some of the key decisions in boiler scheduling and fuel logistics in steam production systems for tomato processing and other similar industries.     

Paradoxes in conditional probability

It is shown that paradoxes arise in conditional probability calculations, due to incomplete specification of the problem at hand. This is illustrated with the Borel and the Kac-Slepian type paradoxes. These are significant in applications including Bayesian inference. Also Rényi's axiomatic setup does not resolve them. An open problem on calculation of conditional probabilities in the continuous case is noted.     

非线性反应扩散方程的广义条件对称及其精确解

利用广义条件对称,考虑非线性反应扩散方程的精确解,对应于不同的参数讨论,得到相应的方程及其允许的广义条件对称,进而得到方程的精确解.     

On the geometry of the Clairin theory of conditional symmetries for higher-order systems of PDEs with applications

This work presents a geometrical formulation of the Clairin theory of conditional symmetries for higher-order systems of partial differential equations (PDEs). We devise methods for obtaining Lie algebras of conditional symmetries from known conditional symmetries, and unnecessary previous assumptions of the theory are removed. As a consequence, new insights into other types of conditional symmetries arise. We then apply the so-called PDE Lie systems to the derivation and analysis of Lie algebras of conditional symmetries. In particular, we develop a method for obtaining solutions of a higher-order system of PDEs via the solutions and geometric properties of a PDE Lie system, whose form gives a Lie algebra of conditional symmetries of the Clairin type. Our methods are illustrated with physically relevant examples such as nonlinear wave equations, the Gauss–Codazzi equations for minimal soliton surfaces, and generalised Liouville equations.     

A multiplicative model for volume and volatility

We first present prima facie evidence for the predictions generated by the mixture of distributions hypothesis, using daily German stock returns and their corresponding daily trading volumes and number of trades. These last two variables are used as proxies for the stochastic rate of information arrival when one wishes to explain GARCH effects by adhering to the mixture of distributions hypothesis. We show that there is no need for these proxies when the stochastic rate of information arrival follows an inverted gamma distribution. Daily trading volume and the daily number of trades, however, empirically provide an explanation for the occurrence of conditional heteroskedasticity of the GARCH form. We estimate several specifications where daily trading volume is included in the conditional variance equation additively and multiplicatively. The new multiplicative specification clearly outperforms the additive specification.     

State estimation for partially observed jump processes

In this paper partially observed jump processes are considered and optimal filtering equations are given for the conditional expectation of a functional on the past of the process.Rudemo [6] derived filtering equations for a partially observed jump Markov process. Snyder [3] gives equations for the conditional characteristic function of a jump process. Segall et al. [2] discuss filtering for processes with counting observations. Their work carries over to processes with counting observations the martingale methods that Fujisaki et al. [1] had used to derive nonlinear filtering equations for processes governed by Ito equations. Many further references to filtering for processes with discrete state measurements are given in the references cited.The objective of this paper is to show that by making use of the concept of a representation of a functional the idea of Rudemo's proof of [6, pp. 595–599] can be carried over to jump processes. The author feels that this is a very interesting proof because of its simplicity. It involves only calculations with conditional expectations and the rule for differentiation of a quotient.     

多孔介质方程的广义条件对称和精确解

运用广义条件对称方法对径向对称的多孔介质方程进行了对称约化.确定了允许二阶广义条件对称的方程形式,并给出了方程相应的不变解.     

Implied Filtering Densities on the Hidden State of Stochastic Volatility

Abstract

We formulate and analyse an inverse problem using derivative prices to obtain an implied filtering density on volatility’s hidden state. Stochastic volatility is the unobserved state in a hidden Markov model (HMM) and can be tracked using Bayesian filtering. However, derivative data can be considered as conditional expectations that are already observed in the market, and which can be used as input to an inverse problem whose solution is an implied conditional density on volatility. Our analysis relies on a specification of the martingale change of measure, which we refer to as separability. This specification has a multiplicative component that behaves like a risk premium on volatility uncertainty in the market. When applied to SPX options data, the estimated model and implied densities produce variance-swap rates that are consistent with the VIX volatility index. The implied densities are relatively stable over time and pick up some of the monthly effects that occur due to the options’ expiration, indicating that the volatility-uncertainty premium could experience cyclic effects due to the maturity date of the options.     

On the Regularity of Optimal Transportation Potentials on Round Spheres

In this paper the regularity of optimal transportation potentials defined on round spheres is investigated. Specifically, this research generalises the calculations done by Loeper, where he showed that the strong (A3) condition of Trudinger and Wang is satisfied on the round sphere, when the cost-function is the geodesic distance squared. In order to generalise Loeper’s calculation to a broader class of cost-functions, the (A3) condition is reformulated via a stereographic projection that maps charts of the sphere into Euclidean space. This reformulation subsequently allows one to verify the (A3) condition for any case where the cost-function of the associated optimal transportation problem can be expressed as a function of the geodesic distance between points on a round sphere. With this, several examples of such cost-functions are then analysed to see whether or not they satisfy this (A3) condition.     

Modeling the dynamics of interest rate volatility with skewed fat-tailed distributions

This paper proposes generalized parametric models of the short-term interest rate that nest one-factor CEV and discrete time GARCH models. The paper estimates the generalized and nested models with skewed fat-tailed distributions to determine the correct specification of the conditional distribution of interest rates. The results indicate that the discrete time models that incorporate the level and GARCH effects into the diffusion function and that accommodate the tail-thickness of the interest rate distribution perform much better than the CEV model in forecasting the future volatility of interest rates. The results also show that the significance of nonlinearity in the drift function relies crucially on the specification of the volatility function.     

Conditional Lie–Bäcklund Symmetries and Invariant Subspaces to Nonlinear Diffusion Equations with Convection and Source

The conditional Lie–Bäcklund symmetry method is used to study the invariant subspace of the nonlinear diffusion equations with convection and source terms. We obtain a complete list of canonical forms for such equations which admit higher order conditional Lie–Bäcklund symmetries and multidimensional invariant subspaces. The functionally generalized separable solutions to the resulting equations are constructed due to the corresponding symmetry reductions. For most of the cases, they are reduced to solving finite‐dimensional dynamical systems.     

Bivariate distributions via a Pareto conditional distribution and a regression function

Uniqueness of specification of a bivariate distribution by a Pareto conditional and a consistent regression function is investigated. New characterizations of the Mardia bivariate Pareto distribution and the bivariate Pareto conditionals distribution are obtained.     

State estimation for partially observed Markov chains

Recursive equations are derived for the conditional distribution of the state of a Markov chain, given observations of a function of the state. Mainly continuous time chains are considered. The equations for the conditional distribution are given in matrix form and in differential equation form. The conditional distribution itself forms a Markov process. Special cases considered are doubly stochastic Poisson processes with a Markovian intensity, Markov chains with a random time, and Markovian approximations of semi-Markov processes. Further the results are used to compute the Radon-Nikodym derivative for two probability measures for a Markov chain, when a function of the state is observed.