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Peter Spreij

Publications and source records attributed to Peter Spreij.

At least 37 records · Page 2Linked to original sources
arXiv2019

Nonparametric Bayesian inference for Gamma-type Lévy subordinators

Given discrete time observations over a growing time interval, we consider a nonparametric Bayesian approach to estimation of the Lévy density of a Lévy process belonging to a flexible class of infinite activity subordinators. Posterior inference is performed via MCMC, and we circumvent the problem of the intractable likelihood via the data augmentation device, that in our case relies on bridge process sampling via Gamma process bridges. Our approach also requires the use of a new infinite-dimensional form of a reversible jump MCMC algorithm. We show that our method leads to good practical results in challenging simulation examples. On the theoretical side, we establish that our nonparametric Bayesian procedure is consistent: in the low frequency data setting, with equispaced in time observations and intervals between successive observations remaining fixed, the posterior asymptotically, as the sample size $n\rightarrow\infty$, concentrates around the Lévy density under which the data have been generated. Finally, we test our method on a classical insurance dataset.

math.ST↗
arXiv2018

Approximation of nonnegative systems by moving averages of fixed order

We pose the approximation problem for scalar nonnegative input-output systems via impulse response convolutions of finite order, i.e. finite order moving averages, based on repeated observations of input/output signal pairs. The problem is converted into a nonnegative matrix factorization with special structure for which we use Csiszár's I-divergence as the criterion of optimality. Conditions are given, on the input/output data, that guarantee the existence and uniqueness of the minimum. We propose an algorithm of the alternating minimization type for I-divergence minimization, and present its asymptotic behavior. For the case of noisy observations we give the large sample properties of the statistical version of the minimization problem for different observation regimes. Numerical experiments confirm the asymptotic results and exhibit fast convergence of the proposed algorithm.

math.OC↗
arXiv2018

Accounting Noise and the Pricing of CoCos

Contingent Convertible bonds (CoCos) are debt instruments that convert into equity or are written down in times of distress. Existing pricing models assume conversion triggers based on market prices and on the assumption that markets can always observe all relevant firm information. But all Cocos issued so far have triggers based on accounting ratios and/or regulatory intervention. We incorporate that markets receive information through noisy accounting reports issued at discrete time instants, which allows us to distinguish between market and accounting values, and between automatic triggers and regulator-mandated conversions. Our second contribution is to incorporate that coupon payments are contingent too: their payment is conditional on the Maximum Distributable Amount not being exceeded. We examine the impact of CoCo design parameters, asset volatility and accounting noise on the price of a CoCo; and investigate the interaction between CoCo design features, the capital structure of the issuing bank and their implications for risk taking and investment incentives. Finally, we use our model to explain the crash in CoCo prices after Deutsche Bank's profit warning in February 2016.

q-fin.PR↗
arXiv2016

A non-parametric Bayesian approach to decompounding from high frequency data

Given a sample from a discretely observed compound Poisson process, we consider non-parametric estimation of the density $f_0$ of its jump sizes, as well as of its intensity $λ_0.$ We take a Bayesian approach to the problem and specify the prior on $f_0$ as the Dirichlet location mixture of normal densities. An independent prior for $λ_0$ is assumed to be compactly supported and to possess a positive density with respect to the Lebesgue measure. We show that under suitable assumptions the posterior contracts around the pair $(λ_0,f_0)$ at essentially (up to a logarithmic factor) the $\sqrt{nΔ}$-rate, where $n$ is the number of observations and $Δ$ is the mesh size at which the process is sampled. The emphasis is on high frequency data, $Δ\to 0$, but the obtained results are also valid for fixed $Δ$. In either case we assume that $nΔ\rightarrow\infty$. Our main result implies existence of Bayesian point estimates converging (in the frequentist sense, in probability) to $(λ_0,f_0)$ at the same rate. We also discuss a practical implementation of our approach. The computational problem is dealt with by inclusion of auxiliary variables and we develop a Markov Chain Monte Carlo algorithm that samples from the joint distribution of the unknown parameters in the mixture density and the introduced auxiliary variables. Numerical examples illustrate the feasibility of this approach.

math.ST↗
arXiv2016

Explicit computations for some Markov modulated counting processes

In this paper we present elementary computations for some Markov modulated counting processes, also called counting processes with regime switching. Regime switching has become an increasingly popular concept in many branches of science. In finance, for instance, one could identify the background process with the `state of the economy', to which asset prices react, or as an identification of the varying default rate of an obligor. The key feature of the counting processes in this paper is that their intensity processes are functions of a finite state Markov chain. This kind of processes can be used to model default events of some companies. Many quantities of interest in this paper, like conditional characteristic functions, can all be derived from conditional probabilities, which can, in principle, be analytically computed. We will also study limit results for models with rapid switching, which occur when inflating the intensity matrix of the Markov chain by a factor tending to infinity. The paper is largely expository in nature, with a didactic flavor.

math.PR↗
arXiv2015

Nonparametric Bayesian inference for multidimensional compound Poisson processes

Given a sample from a discretely observed multidimensional compound Poisson process, we study the problem of nonparametric estimation of its jump size density $r_0$ and intensity $λ_0$. We take a nonparametric Bayesian approach to the problem and determine posterior contraction rates in this context, which, under some assumptions, we argue to be optimal posterior contraction rates. In particular, our results imply the existence of Bayesian point estimates that converge to the true parameter pair $(r_0,λ_0)$ at these rates. To the best of our knowledge, construction of nonparametric density estimators for inference in the class of discretely observed multidimensional Lévy processes, and the study of their rates of convergence is a new contribution to the literature.

math.ST↗
arXiv2015

Large deviations for Markov-modulated diffusion processes with rapid switching

In this paper, we study small noise asymptotics of Markov-modulated diffusion processes in the regime that the modulating Markov chain is rapidly switching. We prove the joint sample-path large deviations principle for the Markov-modulated diffusion process and the occupation measure of the Markov chain (which evidently also yields the large deviations principle for each of them separately by applying the contraction principle). The structure of the proof is such that we first prove exponential tightness, and then establish a local large deviations principle (where the latter part is split into proving the corresponding upper bound and lower bound).

math.PR↗
arXiv2014

Markov-modulated Ornstein-Uhlenbeck processes

In this paper we consider an Ornstein-Uhlenbeck (OU) process $(M(t))_{t\geqslant 0}$ whose parameters are determined by an external Markov process $(X(t))_{t\geqslant 0}$ on a finite state space $\{1,\ldots,d\}$; this process is usually referred to as Markov-modulated Ornstein-Uhlenbeck (MMOU). We use stochastic integration theory to determine explicit expressions for the mean and variance of $M(t)$. Then we establish a system of partial differential equations (PDEs) for the Laplace transform of $M(t)$ and the state $X(t)$ of the background process, jointly for time epochs $t=t_1,\ldots,t_K.$ Then we use this PDE to set up a recursion that yields all moments of $M(t)$ and its stationary counterpart; we also find an expression for the covariance between $M(t)$ and $M(t+u)$. We then establish a functional central limit theorem for $M(t)$ for the situation that certain parameters of the underlying OU processes are scaled, in combination with the modulating Markov process being accelerated; interestingly, specific scalings lead to drastically different limiting processes. We conclude the paper by considering the situation of a single Markov process modulating multiple OU processes.

math.PR↗
arXiv2013

Approximation of Nonnegative Systems by Finite Impulse Response Convolutions

We pose the deterministic, nonparametric, approximation problem for scalar nonnegative input/output systems via finite impulse response convolutions, based on repeated observations of input/output signal pairs. The problem is converted into a nonnegative matrix factorization with special structure for which we use Csiszár's I-divergence as the criterion of optimality. Conditions are given, on the input/output data, that guarantee the existence and uniqueness of the minimum. We propose a standard algorithm of the alternating minimization type for I-divergence minimization, and study its asymptotic behavior. We also provide a statistical version of the minimization problem and give its large sample properties.

math.OC↗
arXiv2013

Limit theorems for reflected Ornstein-Uhlenbeck processes

This paper studies one-dimensional Ornstein-Uhlenbeck processes, with the distinguishing feature that they are reflected on a single boundary (put at level 0) or two boundaries (put at levels 0 and d>0). In the literature they are referred to as reflected OU (ROU) and doubly-reflected OU (DROU) respectively. For both cases, we explicitly determine the decay rates of the (transient) probability to reach a given extreme level. The methodology relies on sample-path large deviations, so that we also identify the associated most likely paths. For DROU, we also consider the `idleness process' $L_t$ and the `loss process' $U_t$, which are the minimal nondecreasing processes which make the OU process remain $\geqslant 0$ and $\leqslant d$, respectively. We derive central limit theorems for $U_t$ and $L_t$, using techniques from stochastic integration and the martingale central limit theorem.

math.PR↗
arXiv2013

A block Hankel generalized confluent Vandermonde matrix

Vandermonde matrices are well known. They have a number of interesting properties and play a role in (Lagrange) interpolation problems, partial fraction expansions, and finding solutions to linear ordinary differential equations, to mention just a few applications. Usually, one takes these matrices square, $q\times q$ say, in which case the $i$-th column is given by $u(z_i)$, where we write $u(z)=(1,z,...,z^{q-1})^\top$. If all the $z_i$ ($i=1,...,q$) are different, the Vandermonde matrix is non-singular, otherwise not. The latter case obviously takes place when all $z_i$ are the same, $z$ say, in which case one could speak of a confluent Vandermonde matrix. Non-singularity is obtained if one considers the matrix $V(z)$ whose $i$-th column ($i=1,...,q$) is given by the $(i-1)$-th derivative $u^{(i-1)}(z)^\top$. We will consider generalizations of the confluent Vandermonde matrix $V(z)$ by considering matrices obtained by using as building blocks the matrices $M(z)=u(z)w(z)$, with $u(z)$ as above and $w(z)=(1,z,...,z^{r-1})$, together with its derivatives $M^{(k)}(z)$. Specifically, we will look at matrices whose $ij$-th block is given by $M^{(i+j)}(z)$, where the indices $i,j$ by convention have initial value zero. These in general non-square matrices exhibit a block-Hankel structure. We will answer a number of elementary questions for this matrix. What is the rank? What is the null-space? Can the latter be parametrized in a simple way? Does it depend on $z$? What are left or right inverses? It turns out that answers can be obtained by factorizing the matrix into a product of other matrix polynomials having a simple structure. The answers depend on the size of the matrix $M(z)$ and the number of derivatives $M^{(k)}(z)$ that is involved. The results are obtained by mostly elementary methods, no specific knowledge of the theory of matrix polynomials is needed.

math.RA↗
arXiv2013

Non-parametric Bayesian drift estimation for stochastic differential equations

We consider non-parametric Bayesian estimation of the drift coefficient of a one-dimensional stochastic differential equation from discrete-time observations on the solution of this equation. Under suitable regularity conditions that are weaker than those previosly suggested in the literature, we establish posterior consistency in this context. Furthermore, we show that posterior consistency extends to the multidimensional setting as well, which, to the best of our knowledge, is a new result in this setting.

math.ST↗
arXiv2012

Parametric inference for stochastic differential equations: a smooth and match approach

We study the problem of parameter estimation for a univariate discretely observed ergodic diffusion process given as a solution to a stochastic differential equation. The estimation procedure we propose consists of two steps. In the first step, which is referred to as a smoothing step, we smooth the data and construct a nonparametric estimator of the invariant density of the process. In the second step, which is referred to as a matching step, we exploit a characterisation of the invariant density as a solution of a certain ordinary differential equation, replace the invariant density in this equation by its nonparametric estimator from the smoothing step in order to arrive at an intuitively appealing criterion function, and next define our estimator of the parameter of interest as a minimiser of this criterion function. Our main results show that under suitable conditions our estimator is $\sqrt{n}$-consistent, and even asymptotically normal. We also discuss a way of improving its asymptotic performance through a one-step Newton-Raphson type procedure and present results of a small scale simulation study.

math.ST↗
arXiv2011

Deconvolution for an atomic distribution: rates of convergence

Let $X_1,..., X_n$ be i.i.d.\ copies of a random variable $X=Y+Z,$ where $ X_i=Y_i+Z_i,$ and $Y_i$ and $Z_i$ are independent and have the same distribution as $Y$ and $Z,$ respectively. Assume that the random variables $Y_i$'s are unobservable and that $Y=AV,$ where $A$ and $V$ are independent, $A$ has a Bernoulli distribution with probability of success equal to $1-p$ and $V$ has a distribution function $F$ with density $f.$ Let the random variable $Z$ have a known distribution with density $k.$ Based on a sample $X_1,...,X_n,$ we consider the problem of nonparametric estimation of the density $f$ and the probability $p.$ Our estimators of $f$ and $p$ are constructed via Fourier inversion and kernel smoothing. We derive their convergence rates over suitable functional classes. By establishing in a number of cases the lower bounds for estimation of $f$ and $p$ we show that our estimators are rate-optimal in these cases.

math.ST↗