Valuation of mortgage-backed securities using Brownian bridges to reduce effective dimension

Let $X:=(X_1, \ldots, X_p)$ be random objects (the inputs), defined on some probability space $(\Omega,{\mathcal{F}}, \mathbb P)$ and valued in some measurable space $E=E_1\times\ldots \times E_p$. Further, let $Y:=Y = f(X_1, \ldots, X_p)$ be the output. Here, $f$ is a measurable function from $E$ to some Hilbert space $\mathbb{H}$ ($\mathbb{H}$ could be either of finite or infinite dimension). In this work, we give a natural generalization of the Sobol indices (that are classically defined when $Y\in\R$ ), when the output belongs to $\mathbb{H}$. These indices have very nice properties. First, they are invariant. under isometry and scaling. Further they can be, as in dimension $1$, easily estimated by using the so-called Pick and Freeze method. We investigate the asymptotic behaviour of such estimation scheme.

/pdf/sensitivity-analysis-for-multidimensional-and-functional-455xqwutia.pdf

Sensitivity analysis for multidimensional and functional outputs

In this paper, we first study a new sensitivity index that is based on higher moments and generalizes the so-called Sobol one. Further, following an idea of Borgonovo ([3]), we define and study a new sensitivity index based on the Cramer von Mises distance. This new index appears to be more general than the Sobol one as it takes into account, not only the variance, but the whole distribution of the random variable. Furthermore, we study the statistical properties of a Monte Carlo estimate of this new index.

/pdf/sensitivity-analysis-based-on-cramer-von-mises-distance-4ika2zwiw2.pdf

Sensitivity analysis based on Cramér von Mises distance

Let $X:=(X_1, \ldots, X_p)$ be random objects (the inputs), defined on some probability space $(\Omega,{\mathcal{F}}, \mathbb P)$ and valued in some measurable space $E=E_1\times\ldots \times E_p$. Further, let $Y:=Y = f(X_1, \ldots, X_p)$ be the output. Here, $f$ is a measurable function from $E$ to some Hilbert space $\mathbb{H}$ ($\mathbb{H}$ could be either of finite or infinite dimension). In this work, we give a natural generalization of the Sobol indices (that are classically defined when $Y\in\mathbb R$ ), when the output belongs to $\mathbb{H}$. These indices have very nice properties. First, they are invariant. under isometry and scaling. Further they can be, as in dimension $1$, easily estimated by using the so-called Pick and Freeze method. We investigate the asymptotic behaviour of such estimation scheme.

Introduction to Sensitivity Analysis

We present some new results on incorporating quasi-Monte Carlo rules into Markov chain Monte Carlo. First, we present some new constructions of points, fully equidistributed LFSRs, which are small enough that the entire point set can be used in a Monte Carlo calculation. Second, we introduce some antithetic and round trip sampling constructions and show that they preserve the completely uniformly distributed property necessary for QMC in MCMC. Finally, we also give some new empirical results. We see large improvements in sampling some GARCH and stochastic volatility models.

/pdf/new-inputs-and-methods-for-markov-chain-quasi-monte-carlo-3mrvxiotf9.pdf

New Inputs and Methods for Markov Chain Quasi-Monte Carlo

Sobol' indices measure the dependence of a high dimensional function on groups of variables defined on the unit cube $[0,1]^d$. They are based on the ANOVA decomposition of functions, which is an $L^2$ decomposition. In this paper we discuss generalizations of Sobol' indices which yield $L^p$ measures of the dependence of $f$ on subsets of variables. Our interest is in values $p>2$ because then variable importance becomes more about reaching the extremes of $f$. We introduce two methods. One based on higher order moments of the ANOVA terms and another based on higher order norms of a spectral decomposition of $f$, including Fourier and Haar variants. Both of our generalizations have representations as integrals over $[0,1]^{kd}$ for $k\ge 1$, allowing direct Monte Carlo or quasi-Monte Carlo estimation. We find that they are sensitive to different aspects of $f$, and thus quantify different notions of variable importance.

Higher order Sobol' indices

Sobol’ indices measure the dependence of a high dimensional function on groups of variables dened on the unit cube [0 ; 1] d . They are based on the ANOVA decomposition of functions, which is an L 2 decomposition. In this paper we discuss generalizations of Sobol’ indices which yield L p measures of the dependence of f on subsets of variables. Our interest is in values p > 2 because then variable importance becomes more about reaching the extremes of f. We introduce two methods. One based on higher order moments of the ANOVA terms and another based on higher order norms of a spectral decomposition of f, including Fourier and Haar variants. Both of our generalizations have representations as integrals over [0; 1] kd for k > 1, allowing direct Monte Carlo or quasi-Monte Carlo estimation. We

/pdf/higher-order-sobol-indices-13uoroxgls.pdf

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New Inputs and Methods for Markov Chain Quasi-Monte Carlo

Higher order Sobol' indices

Higher order Sobol' indices