Change Of Probability Measure Research Articles

Optimal investment–consumption–insurance strategy with inflation risk and stochastic income in an Itô–Lévy setting

This paper’s focus is on finding the optimal strategies for a trader who invests in stock, a money market account and an inflation-linked index bond. The stock follows a jump diffusion process and the bond is linked to inflation making the two risky. The optimal strategies are determined on two generations of the life of an investor, that is before the investor dies and after the investor dies. We applied the concept of change of probability measures considering Girsanov’s and the Radon–Nikodym theorems. We found the generator of the Backward Stochastic differential equations defined and employed the Hamilton–Jacobi–Bellman (HJB) dynamic programming in finding the stochastic optimal controls of interest.

Parallel assessment of the tropical cyclone wind hazard at multiple locations using the probability density evolution method integrated with the change of probability measure

Parallel assessment of the tropical cyclone wind hazard at multiple locations using the probability density evolution method integrated with the change of probability measure

Learning-based importance sampling via stochastic optimal control for stochastic reaction networks

We explore efficient estimation of statistical quantities, particularly rare event probabilities, for stochastic reaction networks. Consequently, we propose an importance sampling (IS) approach to improve the Monte Carlo (MC) estimator efficiency based on an approximate tau-leap scheme. The crucial step in the IS framework is choosing an appropriate change of probability measure to achieve substantial variance reduction. This task is typically challenging and often requires insights into the underlying problem. Therefore, we propose an automated approach to obtain a highly efficient path-dependent measure change based on an original connection in the stochastic reaction network context between finding optimal IS parameters within a class of probability measures and a stochastic optimal control formulation. Optimal IS parameters are obtained by solving a variance minimization problem. First, we derive an associated dynamic programming equation. Analytically solving this backward equation is challenging, hence we propose an approximate dynamic programming formulation to find near-optimal control parameters. To mitigate the curse of dimensionality, we propose a learning-based method to approximate the value function using a neural network, where the parameters are determined via a stochastic optimization algorithm. Our analysis and numerical experiments verify that the proposed learning-based IS approach substantially reduces MC estimator variance, resulting in a lower computational complexity in the rare event regime, compared with standard tau-leap MC estimators.

Improvements to the probability density evolution method integrated with the change of probability measure on quantifying hybrid uncertainties

The probability density evolution method (PDEM) synthesizing the change of probability measure (COM) provides a compatible framework for quantifying simultaneous aleatory and epistemic uncertainties. Nonetheless, when the effective support for the change of measure becomes narrow, the quantification error of the PDEM-COM may increase, and the PDEM-COM may even be invalid if the effective support is an empty set. To address this issue, in the present paper, the error analysis of the PDEM-COM is firstly explored, and the generalized F-discrepancy via the COM is utilized to measure the similarity of two different distributions. Then, an augmenting approach is proposed to improve the performance of the PDEM-COM by adding a number of computational simulations. The results indicate that, the PDEM-COM can still be relatively accurate as long as the effective support is sufficiently large, even when there is a significant shift in the probability measure. However, if the effective support turns to be extremely narrow, the error of hybrid uncertainty quantification via the PDEM-COM may be unsatisfied, but it can be significantly reduced by the suggested augmenting approach. Numerical examples are investigated to demonstrate the accuracy and efficiency of the proposed method. Some issues to be further studied are also outlined.

A unified approach for time-invariant and time-variant reliability-based design optimization with multiple most probable points

Reliability-based design optimization (RBDO) can fulfill performance-based design of static and dynamic structures considering various uncertainties. For time-invariant and time-variant RBDO problems with multiple most probable points (MPPs, also termed as design points), the existing approaches encounter some difficulties in searching for all MPPs and performing time-variant RBDO which requires complicated time-variant reliability analysis. In this paper, the direct probability integral method (DPIM) combining with the change of probability measure (COM), i.e., DPIM-COM approach, is proposed to attack time-invariant and time-variant RBDO problems with multiple MPPs in a unified framework. Firstly, the DPIM in conjunction with Heaviside function is developed to estimate time-invariant and time-variant probabilistic constraints directly and explicitly. Then, an efficient formula for calculating the sensitivity of probabilistic constraints with respect to random design variables is derived based on the DPIM and COM strategy, where the representative points for reliability computation keep unchanged and only the corresponding assigned probabilities need to be updated. Accordingly, the corresponding structural responses of representative points at the current design variables can be reused to evaluate the sensitivity of probabilistic constraints, which saves much computational cost, especially for time-variant RBDO problems. Finally, three numerical examples, including the ten-story building with tuned mass damper under deterministic earthquake ground motion and near-fault stochastic impulsive ground motions, demonstrate the effectiveness and versatility of the proposed DPIM-COM approach for addressing time-invariant and time-variant RBDO problems with multiple MPPs.

Reducing Bias in Event Time Simulations via Measure Changes

Stochastic point process models of event timing are common in many areas, including finance, insurance, and reliability. Monte Carlo simulation is often used to perform computations for these models. The standard sampling algorithm, which is based on a time-change argument, is widely applicable but generates biased simulation estimators. This article develops and analyzes a change of probability measure that can reduce or even eliminate the bias without restricting the scope of the algorithm. A result of independent interest offers new conditions that guarantee the existence of a broad class of point process martingales inducing changes of measure. Numerical results illustrate our approach.

Valuing guaranteed minimum accumulation benefits by a change of numéraire approach

Three correlated risk factors, namely, financial, mortality and lapse risks, are modelled in an integrated way to support the valuation of guaranteed minimum accumulation benefits (GMAB). The dynamics of such risk factors under the risk-adjusted measure needed for valuation are derived. A sequence of reference probability measure changes is employed to obtain a compact and implementable price representations. We demonstrate how a change of numéraire approach facilitates the valuation of a GMAB rider in a variable annuity. A numerical illustration is presented to confirm the computational efficiency and accuracy of our proposed valuation methodology. In particular, our proposed approach on average takes only 0.07% of the computing time for the Monte-Carlo (MC) simulation technique to complete the valuation calculation. Furthermore, the standard errors of our approach's results are lower than those obtained from the MC-based computations. A sensitivity analysis is also performed with respect to various parameters. When there are no renewal options in a GMAB contract, we get the special case of a guaranteed minimum maturity benefit for which a closed-form valuation solution is established.

A PDEM-COM framework for uncertainty quantification of backward issues involving both aleatory and epistemic uncertainties

Uncertainties that exist in nature or due to lack of knowledge have been widely recognized by researchers and engineering practitioners throughout engineering design and analysis for decades. Though great efforts have been devoted to the issues of uncertainty quantification (UQ) in various aspects, the methodologies on the quantification of aleatory uncertainty and epistemic uncertainty are usually logically inconsistent. For instance, the aleatory uncertainty is usually quantified in the framework of probability theory, whereas the epistemic uncertainty is quantified mostly by non-probabilistic methods. In the present paper, a probabilistically consistent framework for the quantification of both aleatory and epistemic uncertainty by synthesizing the probability density evolution method (PDEM) and the change of probability measure (COM) is outlined. The framework is then applied to the backward issues of uncertainty quantification. In particular, the uncertainty model updating issue is discussed in this paper. A numerical example is presented, and the results indicate the flexibility and efficiency of the proposed PDEM-COM framework.

A global sensitivity index based on Fréchet derivative and its efficient numerical analysis

Sensitivity analysis plays an important role in reliability evaluation, structural optimization and structural design, etc. The local sensitivity, i.e., the partial derivative of the quantity of interest in terms of parameters or basic variables, is inadequate when the basic variables are random in nature. Therefore, global sensitivity such as the Sobol’ indices based on the decomposition of variance and the moment-independent importance measure, among others, have been extensively studied. However, these indices are usually computationally expensive, and the information provided by them has some limitations for decision making. Specifically, all these indices are positive, and therefore they cannot reveal whether the effects of a basic variable on the quantity of interest are positive or adverse. In the present paper, a novel global sensitivity index is proposed when randomness is involved in structural parameters. Specifically, a functional perspective is firstly advocated, where the probability density function (PDF) of the output quantity of interest is regarded as the output of an operator on the PDF of the source basic random variables. The Fréchet derivative is then naturally taken as a measure for the global sensitivity. In some sense such functional perspective provides a unified perspective on the concepts of global sensitivity and local sensitivity. In the case the change of the PDF of a basic random variable is due to the change of parameters of the PDF of the basic random variable, the computation of the Fréchet-derivative-based global sensitivity index can be implemented with high efficiency by incorporating the probability density evolution method (PDEM) and change of probability measure (COM). The numerical algorithms are elaborated. Several examples are illustrated, demonstrating the effectiveness of the proposed method.

Structural optimization considering dynamic reliability constraints via probability density evolution method and change of probability measure

The present paper presents a framework for solving dynamic reliability–based design optimization (DRBDO) problems. The proposed approach is based on a sequential approximate programming technique, where the solution of the original design problem is obtained by converting it into a sequence of sub-optimization problems with a simple explicit algebraic structure. The technique is combined with the probability density evolution method (PDEM) for the purpose of estimating the reliability and/or the failure probability at the different designs in an efficient manner. The change of probability measure (COM), a strategy based on the Radon-Nikodym derivative, is employed to estimate the sensitivity information required by the optimization scheme. Based on this strategy, the sensitivity information can be obtained without any additional function evaluation. Thus, the numerical efforts associated with the reliability assessment and sensitivity analysis can be considerably reduced. The results of the numerical examples indicate that the proposed method is an effective and efficient tool for solving a class of DRBDO problems.

Solving Nonlinear Filtering Problems in Real Time by Legendre Galerkin Spectral Method

It is well known that the nonlinear filtering (NLF) problem has important applications in both military and civil industries. The central question is to solve the posterior conditional density function of the states, which satisfies the Kushner or the Duncan–Mortensen–Zakai (DMZ) equation after suitable change of probability measure. In this article, we shall follow the so-called Yau–Yau's algorithm to split the solution of the DMZ equation into on- and off-line part, where the off-line part is to solve the forward Kolmogorov equation (FKE) with the initial conditions to be the orthonormal bases in some suitable function space. Instead of the generalized Hermite function investigated by the second and the third author of this article, we shall explore the generalized Legendre polynomials. The Legendre spectral method (LSM) is used to numerically solve the FKE. Under certain conditions, the convergence rate of LSM is twice faster than that of the Hermite spectral method. Two two-dimensional numerical experiments of NLF problems (time-invariant and time-varying cases) have been numerically solved to illustrate the feasibility of our algorithm. Our algorithm outperforms the extended Kalman Filter and particle filter in both real-time manner and accuracy.

Unified formalism for entropy production and fluctuation relations.

Stochastic entropy production, which quantifies the difference between the probabilities of trajectories of a stochastic dynamics and its time reversals, has a central role in nonequilibrium thermodynamics. In the theory of probability, the change in the statistical properties of observables due to reversals can be represented by a change in the probability measure. We consider operators on the space of probability measures that induce changes in the statistical properties of a process, and we formulate entropy production in terms of these change-of-probability-measure (CPM) operators. This mathematical underpinning of the origin of entropy production allows us to achieve an organization of various forms of fluctuation relations: All entropy production has a nonnegative mean value, admit the integral fluctuation theorem, and satisfy a rather general fluctuation relation. Other results such as the transient fluctuation theorem and detailed fluctuation theorems then are derived from the general fluctuation relation with more constraints on the operator of entropy production. We use a discrete-time, discrete-state-space Markov process to draw the contradistinction among three reversals of a process: time reversal, protocol reversal, and the dual process. The properties of their corresponding CPM operators are examined, and the domains of validity of various fluctuation relations for entropy production in physics and chemistry are revealed. We also show that our CPM operator formalism can help us rather easily extend other fluctuation relations for excess work and heat, discuss the martingale properties of entropy production, and derive the stochastic integral formulas for entropy production in constant-noise diffusion process with Girsanov theorem. Our formalism provides a general and concise way to study the properties of entropy-related quantities in stochastic thermodynamics and information theory.

Bayesian credibility premium with GB2 copulas

Abstract For observations over a period of time, Bayesian credibility premium may be used to predict the value of a response variable for a subject, given previously observed values. In this article, we formulate Bayesian credibility premium under a change of probability measure within the copula framework. Such reformulation is demonstrated using the multivariate generalized beta of the second kind (GB2) distribution. Within this family of GB2 copulas, we are able to derive explicit form of Bayesian credibility premium. Numerical illustrations show the application of these estimators in determining experience-rated insurance premium. We consider generalized Pareto as a special case.

Representation of exchange option prices under stochastic volatility jump-diffusion dynamics

In this article, we provide representations of European and American exchange option prices under stochastic volatility jump-diffusion (SVJD) dynamics following models by Merton [Option pricing when underlying stock returns are discontinuous. J. Financ. Econ., 1976, 3(1-2), 125–144], Heston [A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud., 1993, 6(2), 327–343], and Bates [Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche Mark options. Rev. Financ. Stud., 1996, 9(1), 69–107]. A Radon–Nikodým derivative process is also introduced to facilitate the shift from the objective market measure to other equivalent probability measures, including the equivalent martingale measure. Under the equivalent martingale measure, we derive the integro-partial differential equation that characterizes the exchange option prices. We also derive representations of the European exchange option price using the change-of-numéraire technique proposed by Geman et al. [Changes of numéraire, changes of probability measure and option pricing. J. Appl. Probab., 1995, 32(2), 443–458] and the Fourier inversion formula derived by Caldana and Fusai [A general closed-form spread option pricing formula. J. Bank. Finance, 2013, 37, 4893–4906], and show that these two representations are comparable. Lastly, we show that the American exchange option price can be decomposed into the price of the European exchange option and an early exercise premium.

Penalization of Galton–Watson processes

We apply the penalization technique introduced by Roynette, Vallois, Yor for Brownian motion to Galton–Watson processes with a penalizing function of the form P(x)sx where P is a polynomial of degree p and s∈[0,1]. We prove that the limiting martingales obtained by this method are most of the time classical ones, except in the super-critical case for s=1 (or s→1) where we obtain new martingales. If we make a change of probability measure with this martingale, we obtain a multi-type Galton–Watson tree with p distinguished infinite spines.

An efficient new PDEM-COM based approach for time-variant reliability assessment of structures with monotonically deteriorating materials

Life-cycle time-variant reliability of engineering structures with material deterioration is of paramount importance but still a great challenge. Generally, the computational costs due to embedded structural analyses tracing the life-cycle performance process with monotonically deteriorating material, are prohibitively large. In the present paper, a new method by synthesizing the probability density evolution method (PDEM) and the change of probability measure (COM) is proposed for this purpose. Besides, a functional perspective is provided. The theoretical basis is established by involving the Frobenius-Perron operator, which characterizes the change of random variables, and the Radon-Nikodym operator, which characterizes the change of probability measure of the original source random variables. The material deterioration characterized by time-variant random variables can then be equivalently captured by the time-variant change of probability measure of the basic random variables. Thereby the deterministic analyses at different time instances can be shared in the process of capturing the time-variant probability density of the performance function, and thus the efficiency can be improved considerably. Numerical algorithms are elaborated. Three examples in terms of time-variant reliability evaluation of structures with monotonically deteriorating material and parametric stochastic loading are illustrated, showing the effectiveness of the proposed method. Problems to be further studied are also outlined.

Bayesian Shrinkage Estimators with GB2 Copulas

Bayesian shrinkage estimation is a familiar concept that arises from minimizing the Bayes risk under the squared error loss. For observations over a period of time, Bayesian shrinkage estimators may be used to predict the value of a response variable for a subject, given previously observed values. In this article, we formulate such estimators under a change of probability measure within the copula framework. Such reformulation is demonstrated using the multivariate generalized beta of the second kind (GB2) distribution. Within this family of GB2 copulas, we are able to derive explicit form of Bayesian shrinkage estimators. Numerical illustrations show the application of these estimators in determining experience-rated insurance premium. We consider generalized Pareto as a special case.

Downside risk-neutral probabilities

We show that there exists a probability measure under which the CAPM formula for expected returns holds for general utility functions and probability distributions. This probability measure, the “downside risk-neutral” measure, is adjusted to incorporate the effects of downside risk and higher degree risks. It thus belongs to the same family as the risk-neutral measure, which is also a risk-adjusted measure. Using risk preference theory, we interpret this change in probability measure in terms of risk substitution.

A compatible probabilistic framework for quantification of simultaneous aleatory and epistemic uncertainty of basic parameters of structures by synthesizing the change of measure and change of random variables

Uncertainty has been attached increasing importance in performance evaluation and reliability assessment of engineering structures. However, the logical framework for the quantification of simultaneous aleatory uncertainty and epistemic uncertainty of basic parameters of structures in a compatible probabilistic sense is still not readily available as yet, and the computational efforts are also usually prohibitively large. In the present paper, a compatible probabilistic framework is proposed for this purpose. Limited to the epistemic uncertainty that characterizing the uncertainty in aleatory uncertainty, i.e., the uncertainty in the shape or parameters of probability density function of the source random variables, it is found that the quantification and propagation of aleatory uncertainty is a problem of change of random variables, and the principle of preservation of probability holds. For dynamical systems the probability density evolution method (PDEM) can be adopted for this purpose. Whereas, the quantification of epistemic uncertainty is essentially a problem of change of probability measure, and thus the Radon-Nikodym theorem holds. Therefore, synthesizing the change of measure (COM) and the change of random variables (CRV) will provide a logically clear compatible framework for the quantification of simultaneous aleatory and epistemic uncertainties. The numerical algorithm by changing the assigned probabilities of representative points in the PDEM is then proposed. A nonlinear equation, the Riccati equation, is investigated to illustrate the proposed method. The result is verified by the exact analytical solution. Moreover, a 3-span 10-storey reinforced concrete (RC) frame structure modelled by the finite element method is studied. This exemplifies the quantification of simultaneous aleatory and epistemic uncertainties of basic parameters of real-world civil engineering structures. The examples demonstrate the effectiveness of the proposed method. Problems to be further studied are also outlined.

Efficient Monte Carlo resampling for probability measure changes from Bayesian updating

The objective of Bayesian inference is often to infer, from data, a probability measure for a random variable that can be used as input for Monte Carlo simulation. When datasets for Bayesian inference are small, a principle challenge is that, as additional data are collected, the probability measure inferred from Bayesian inference may change significantly. In such cases, expensive Monte Carlo simulations may have already been performed using the original distribution and it is infeasible to start again and perform a new Monte Carlo analysis using the updated density due to the large added computational cost. This work explores four strategies for updating Monte Carlo simulations for such a change in probability measure. The efficiency of each strategy is compared and the ultimate aim is to achieve the change in distribution with a minimal number of added computational simulations. The results show that, when the change in measure is small, importance sampling reweighting can be very effective. Otherwise, a proposed mixed augmenting-filtering algorithm can robustly and efficiently accommodate a measure change in Monte Carlo simulation. The strategy is then applied for uncertainty quantification in the buckling strength of a simple plate given ongoing data collection to estimate uncertainty in the yield stress.