Showing papers in "Journal of Applied Probability in 2019"
TL;DR: Wasserstein Profile Inference is introduced, a novel inference methodology which extends the use of methods inspired by Empirical Likelihood to the setting of optimal transport costs (of which Wasserstein distances are a particular case).
Abstract: We show that several machine learning estimators, including square-root least absolute shrinkage and selection and regularized logistic regression, can be represented as solutions to distributionally robust optimization problems. The associated uncertainty regions are based on suitably defined Wasserstein distances. Hence, our representations allow us to view regularization as a result of introducing an artificial adversary that perturbs the empirical distribution to account for out-of-sample effects in loss estimation. In addition, we introduce RWPI (robust Wasserstein profile inference), a novel inference methodology which extends the use of methods inspired by empirical likelihood to the setting of optimal transport costs (of which Wasserstein distances are a particular case). We use RWPI to show how to optimally select the size of uncertainty regions, and as a consequence we are able to choose regularization parameters for these machine learning estimators without the use of cross validation. Numerical experiments are also given to validate our theoretical findings.
110 citations
TL;DR: The construction principle of phase-type (PH) distributions is extended to allow for inhomogeneous transition rates and it is shown that this naturally leads to direct probabilistic descriptions of certain transformations of PH distributions.
Abstract: We extend the construction principle of phase-type (PH) distributions to allow for inhomogeneous transition rates and show that this naturally leads to direct probabilistic descriptions of certain transformations of PH distributions. In particular, the resulting matrix distributions enable the carrying over of fitting properties of PH distributions to distributions with heavy tails, providing a general modelling framework for heavy-tail phenomena. We also illustrate the versatility and parsimony of the proposed approach in modelling a real-world heavy-tailed fire insurance dataset.
33 citations
TL;DR: Depending on the situation, increasing the number of control variates may or may not be computationally more efficient than increasing the Monte Carlo sample size.
Abstract: It is well known that Monte Carlo integration with variance reduction by means of control variates can be implemented by the ordinary least squares estimator for the intercept in a multiple linear regression model. A central limit theorem is established for the integration error if the number of control variates tends to infinity. The integration error is scaled by the standard deviation of the error term in the regression model. If the linear span of the control variates is dense in a function space that contains the integrand, the integration error tends to zero at a rate which is faster than the square root of the number of Monte Carlo replicates. Depending on the situation, increasing the number of control variates may or may not be computationally more efficient than increasing the Monte Carlo sample size.
29 citations
TL;DR: For spectrally negative Lévy processes, several fluctuation results involving a general draw-down time are proved, which is a downward exit time from a dynamic level that depends on the running maximum of the process.
Abstract: For spectrally negative Levy processes, we prove several fluctuation results involving a general draw-down time, which is a downward exit time from a dynamic level that depends on the running maximum of the process. In particular, we find expressions of the Laplace transforms for the two-sided exit problems involving the draw-down time. We also find the Laplace transforms for the hitting time and creeping time over the running-maximum related draw-down level, respectively, and obtain an expression for a draw-down associated potential measure. The results are expressed in terms of scale functions for the spectrally negative Levy processes.
25 citations
TL;DR: In this paper, the existence and uniqueness of a stationary distribution and absolute regularity for nonlinear GARCH and INGARCH models of order (p, q) were proved, besides a geometric drift condition, only a semi-contractive condition which allows to include models which would be ruled out by a fully contractive condition.
Abstract: We prove existence and uniqueness of a stationary distribution and absolute regularity for nonlinear GARCH and INGARCH models of order (p, q). In contrast to previous work we impose, besides a geometric drift condition, only a semi-contractive condition which allows us to include models which would be ruled out by a fully contractive condition. This results in a subgeometric rather than the more usual geometric decay rate of the mixing coefficients. The proofs are heavily based on a coupling of two versions of the processes.
22 citations
TL;DR: Both finite-dimensional and functional limit theorems for the fractional nonhomogeneous Poisson process and the fractionsal compound Poissonprocess are given.
Abstract: The fractional nonhomogeneous Poisson process was introduced by a time change of the nonhomogeneous Poisson process with the inverse α-stable subordinator. We propose a similar definition for the (nonhomogeneous) fractional compound Poisson process. We give both finite-dimensional and functional limit theorems for the fractional nonhomogeneous Poisson process and the fractional compound Poisson process. The results are derived by using martingale methods, regular variation properties and Anscombe’s theorem. Eventually, some of the limit results are verified in a Monte Carlo simulation.
19 citations
TL;DR: It is shown how some results on sample path large deviations for such diffusions characterise the small-time and tail estimates of the implied volatility for rough volatility models, recently proposed in mathematical finance.
Abstract: We study the asymptotic behaviour of a class of small-noise diffusions driven by fractional Brownian motion, with random starting points. Different scalings allow for different asymptotic properties of the process (small-time and tail behaviours in particular). In order to do so, we extend some results on sample path large deviations for such diffusions. As an application, we show how these results characterise the small-time and tail estimates of the implied volatility for rough volatility models, recently proposed in mathematical finance.
18 citations
TL;DR: A lumping method based on a notion of local symmetry, which compares only local neighbourhoods of vertices ensures approximate lumpability, and it is proved the approximation error decreases monotonically.
Abstract: We study a Markovian agent-based model (MABM) in this paper. Each agent is endowed with a local state that changes over time as the agent interacts with its neighbours. The neighbourhood structure is given by a graph. Recently, Simon, Taylor, and Kiss [40] used the automorphisms of the underlying graph to generate a lumpable partition of the joint state space, ensuring Markovianness of the lumped process for binary dynamics. However, many large random graphs tend to become asymmetric, rendering the automorphism-based lumping approach ineffective as a tool of model reduction. In order to mitigate this problem, we propose a lumping method based on a notion of local symmetry, which compares only local neighbourhoods of vertices. Since local symmetry only ensures approximate lumpability, we quantify the approximation error by means of the Kullback–Leibler divergence rate between the original Markov chain and a lifted Markov chain. We prove the approximation error decreases monotonically. The connections to fibrations of graphs are also discussed.
13 citations
TL;DR: It is shown that the average nearest-neighbour degree a ( k ) indeed decays with k in three simple random graph models with power-law degrees: the erased configuration model, the rank-1 inhomogeneous random graph, and the hyperbolic random graph.
Abstract: We study the average nearest-neighbour degree a(k) of vertices with degree k. In many real-world networks with power-law degree distribution, a(k) falls off with k, a property ascribed to the constraint that any two vertices are connected by at most one edge. We show that a(k) indeed decays with k in three simple random graph models with power-law degrees: the erased configuration model, the rank-1 inhomogeneous random graph, and the hyperbolic random graph. We find that in the large-network limit for all three null models, a(k) starts to decay beyond the degree exponent.
12 citations
TL;DR: A model of Branching Brownian Motion in which the usual spatially-homogeneous and catalytic branching at a single point are simultaneously present is considered, establishing the almost sure growth rates of population in certain time-dependent regions and as a consequence the first-order asymptotic behaviour of the rightmost particle.
Abstract: We consider a model of branching Brownian motion in which the usual spatially homogeneous branching and catalytic branching at a single point are simultaneously present. We establish the almost sure growth rates of population in certain time-dependent regions and as a consequence the first-order asymptotic behaviour of the rightmost particle.
12 citations
TL;DR: In this paper, an interacting particle system representing aggregation at the level of individuals was investigated and it was shown that the empirical density of the individual converges to the solution of the PDE-ODE system.
Abstract: Inspired by a PDE–ODE system of aggregation developed in the biomathematical literature, we investigate an interacting particle system representing aggregation at the level of individuals. We prove that the empirical density of the individual converges to the solution of the PDE–ODE system.
TL;DR: A novel method for exactly simulating the generalised Vervaat perpetuity is proposed using a distributional decomposition technique and is much faster than existing algorithms illustrated in Chi (2012), Cloud and Huber (2017), Devroye and Fawzi (2010), and Fill and Huer (2010).
Abstract: We consider a generalised Vervaat perpetuity of the form X = Y1W1 +Y2W1W2 + · · ·, where and (Yi)i≥0 is an independent and identically distributed sequence of random variables independent from (Wi)i≥0. Based on a distributional decomposition technique, we propose a novel method for exactly simulating the generalised Vervaat perpetuity. The general framework relies on the exact simulation of the truncated gamma process, which we develop using a marked renewal representation for its paths. Furthermore, a special case arises when Yi = 1, and X has the generalised Dickman distribution, for which we present an exact simulation algorithm using the marked renewal approach. In particular, this new algorithm is much faster than existing algorithms illustrated in Chi (2012), Cloud and Huber (2017), Devroye and Fawzi (2010), and Fill and Huber (2010), as well as being applicable to the general payments case. Examples and numerical analysis are provided to demonstrate the accuracy and effectiveness of our method.
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TL;DR: Monte Carlo Fusion as mentioned in this paper is the first general approach which avoids any form of approximation error in obtaining the unified inference, and it is based on the Monte Carlo theory and methodology proposed in this paper.
Abstract: This paper proposes a new theory and methodology to tackle the problem of unifying distributed analyses and inferences on shared parameters from multiple sources, into a single coherent inference. This surprisingly challenging problem arises in many settings (for instance, expert elicitation, multi-view learning, distributed ‘big data’ problems etc.), but to-date the framework and methodology proposed in this paper (Monte Carlo Fusion) is the first general approach which avoids any form of approximation error in obtaining the unified inference. In this paper we focus on the key theoretical underpinnings of this new methodology, and simple (direct) Monte Carlo interpretations of the theory. There is considerable scope to tailor the theory introduced in this paper to particular application settings (such as the big data setting), construct efficient parallelised schemes, understand the approximation and computational efficiencies of other such unification paradigms, and explore new theoretical and methodological directions.
TL;DR: In this article, a Poisson limit theorem for the number of large probability nearest-neighbour balls is derived for nPn − ln n as n → ∞.
Abstract: Let X1, …, Xn be independent random points drawn from an absolutely continuous probability measure with density f in ℝd. Under mild conditions on f, wederive a Poisson limit theorem for the number of large probability nearest-neighbour balls. Denoting by Pn the maximum probability measure of nearest-neighbour balls, this limit theorem implies a Gumbel extreme value distribution for nPn − ln n as n → ∞. Moreover, we derive a tight upper bound on the upper tail of the distribution of nPn − ln n, which does not depend on f.
TL;DR: A discrete-type approximation scheme to solve continuous-time optimal stopping problems based on fully non-Markovian continuous processes adapted to the Brownian motion filtration that allows for concrete Monte Carlo schemes for non- Markovian optimal stopping time problems.
Abstract: We present a discrete-type approximation scheme to solve continuous-time optimal stopping problems based on fully non-Markovian continuous processes adapted to the Brownian motion filtration. The approximations satisfy suitable variational inequalities which allow us to construct -optimal stopping times and optimal values in full generality. Explicit rates of convergence are presented for optimal values based on reward functionals of path-dependent stochastic differential equations driven by fractional Brownian motion. In particular, the methodology allows us to design concrete Monte Carlo schemes for non-Markovian optimal stopping time problems as demonstrated in the companion paper by Bezerra et al.
TL;DR: A wide generalization of known results related to the telegraph process on a straight line and their generalizations on an arbitrary state space are proposed.
Abstract: We propose a wide generalization of known results related to the telegraph process. Functionals of the simple telegraph process on a straight line and their generalizations on an arbitrary state space are studied.
TL;DR: For a suitable set I and sequence (sn), the Hausdorff dimension of the set t is established, for t in the boundary of a Galton–Watson tree by the generation-n cylinder.
Abstract: Abstract We consider, for t in the boundary of a Galton–Watson tree $(\\partial \\textsf{T})$, the covering number $(\\textsf{N}_n(t))$ by the generation-n cylinder. For a suitable set I and sequence (sn), we almost surely establish the Hausdorff dimension of the set $\\{ t \\in \\partial {\\textsf{T}}:{{\\textsf{N}}_n}(t) - nb \\ {\\sim} \\ {s_n}\\} $ for b ∈ I.
TL;DR: An idealised model for overland flow generated by rain falling on a hillslope shows how the coalescence of runoff streams promotes the total generation of runoff and identifies the critical point at which the phase change occurs.
Abstract: We introduce an idealised model for overland flow generated by rain falling on a hill-slope. Our prime motivation is to show how the coalescence of runoff streams promotes the total generation of runoff. We show that, for our model, as the rate of rainfall increases in relation to the soil infiltration rate, there is a distinct phase-change. For low rainfall (the subcritical case) only the bottom of the hill-slope contributes to the total overland runoff, while for high rainfall (the supercritical case) the whole slope contributes and the total runoff increases dramatically. We identify the critical point at which the phase-change occurs, and show how it depends on the degree of coalescence. When there is no stream coalescence the critical point occurs when the rainfall rate equals the average infiltration rate, but when we allow coalescence the critical point occurs when the rainfall rate is less than the average infiltration rate, and increasing the amount of coalescence increases the total expected runoff.
TL;DR: In this article, the authors find explicit estimates for the exponential rate of long-term convergence for the ruin probability in a level-dependent Levy-driven risk model, as time goes to infinity.
Abstract: We find explicit estimates for the exponential rate of long-term convergence for the ruin probability in a level-dependent Levy-driven risk model, as time goes to infinity. Siegmund duality allows us to reduce the problem to long-term convergence of a reflected jump-diffusion to its stationary distribution, which is handled via Lyapunov functions.
TL;DR: The findings thus shed extra light on related results concerning first passage times downwards of continuous-state branching processes (resp. upwards) of spectrally negative positive self-similar Markov processes.
Abstract: For a spectrally negative Levy process X, killed according to a rate that is a function ω of its position, we complement the recent findings of [12] by analysing (in greater generality) the exit probability of the one-sided upwards passage problem. When ω is strictly positive, this problem is related to the determination of the Laplace transform of the first passage time upwards for X that has been time-changed by the inverse of the additive functional . In particular, our findings thus shed extra light on related results concerning first passage times downwards (resp. upwards) of continuous-state branching processes (resp. spectrally negative positive self-similar Markov processes).
TL;DR: In this paper, moment identities for the stochastic integrals of multiparameter processes in a random-connection model based on a point process admitting a Papangelou intensity were derived.
Abstract: We derive moment identities for the stochastic integrals of multiparameter processes in a random-connection model based on a point process admitting a Papangelou intensity. The identities are written using sums over partitions, and they reduce to sums over non-flat partition diagrams if the multiparameter processes vanish on diagonals. As an application, we obtain general identities for the moments of k-hop counts in the random-connection model, which simplify the derivations available in the literature.
TL;DR: The Cramér type moderate deviation is studied by applying the conjugate method to the partial sums of linear random fields with short or long memory and to nonparametric regression with random field errors.
Abstract: We study the Cramer type moderate deviation for partial sums of random fields by applying the conjugate method. The results are applicable to the partial sums of linear random fields with short or long memory and to nonparametric regression with random field errors.
TL;DR: In this paper, the persistence probability of a random polynomial arising from evolutionary game theory has been derived for a multi-player two-strategy random evolutionary game with no internal equilibria.
Abstract: We obtain an asymptotic formula for the persistence probability in the positive real line of a random polynomial arising from evolutionary game theory. It corresponds to the probability that a multi-player two-strategy random evolutionary game has no internal equilibria. The key ingredient is to approximate the sequence of random polynomials indexed by their degrees by an appropriate centered stationary Gaussian process.
TL;DR: This work provides different sets of sufficient conditions for the corresponding stochastic comparisons and considers various scenarios, namely, two different coherent systems operate under the same random environment, whereas the other under a deterministic one.
Abstract: We study the impact of a random environment on lifetimes of coherent systems with dependent components. There are two combined sources of this dependence. One results from the dependence of the components of the coherent system operating in a deterministic environment and the other is due to dependence of components of the system sharing the same random environment. We provide different sets of sufficient conditions for the corresponding stochastic comparisons and consider various scenarios, namely, (i) two different (as a specific case, identical) coherent systems operate in the same random environment; (ii) two coherent systems operate in two different random environments; (iii) one of the coherent systems operates in a random environment and the other in a deterministic environment. Some examples are given to illustrate the proposed reasoning.
TL;DR: Failure rate properties of this family of survival functions are studied and closures under monotone failure rates of the mixture’s components are established.
Abstract: This paper presents a flexible family which we call the -mixture of survival functions. This family includes the survival mixture, failure rate mixture, models that are stochastically closer to each of these conventional mixtures, and many other models. The -mixture is endowed by the stochastic order and uniquely possesses a mathematical property known in economics as the constant elasticity of substitution, which provides an interpretation for . We study failure rate properties of this family and establish closures under monotone failure rates of the mixture’s components. Examples include potential applications for comparing systems.
TL;DR: For 0 < a < 1, the Sibuya distribution is concentrated on the set ℕ+ of positive integers and is defined by the generating function as discussed by the authors, and the main point is to find the values of b = 1/a such that the power series expansion of u(1 − (1 − u)b)−1 has nonnegative coefficients.
Abstract: For 0 < a < 1, the Sibuya distribution sa is concentrated on the set ℕ+ of positive integers and is defined by the generating function . The main point is to find the values of b = 1/a such that the power series expansion of u(1 − (1 − u)b)−1 has nonnegative coefficients.
TL;DR: Asset allocation with a derivative security is studied in a hidden, Markovian regime-switching, economy using filtering theory and the martingale approach to derive a solution for a general utility function which includes an exponential utility, a power utility, and a logarithmic utility.
Abstract: Asset allocation with a derivative security is studied in a hidden, Markovian regime-switching, economy using filtering theory and the martingale approach. A generalized delta-hedged ratio and a generalized elasticity of an option are introduced to accommodate the presence of the information state process and the derivative security. Malliavin calculus is applied to derive a solution for a general utility function which includes an exponential utility, a power utility, and a logarithmic utility. A compact solution is obtained for a logarithmic utility. Some economic implications of the solutions are discussed.
TL;DR: Under a mixing hypothesis, a rate of convergence result is obtained for E[LCn]/n, the length of the longest common subsequences of X1, Xn and Y1,..., Yn.
Abstract: Let (X, Y) = (Xn, Yn)n≥1 be the output process generated by a hidden chain Z = (Zn)n≥1, where Z is a finite-state, aperiodic, time homogeneous, and irreducible Markov chain. Let LCn be the length of the longest common subsequences of X1,..., Xn and Y1,..., Yn. Under a mixing hypothesis, a rate of convergence result is obtained for E[LCn]/n.
TL;DR: In this paper, the authors introduce the notion of sure profits via flash strategies, consisting of a high-frequency limit of buy-and-hold trading strategies, and prove that there are no sure profits in flash strategies if and only if asset prices do not exhibit predictable jumps.
Abstract: We introduce and study the notion of sure profits via flash strategies, consisting of a high-frequency limit of buy-and-hold trading strategies. In a fully general setting, without imposing any semimartingale restriction, we prove that there are no sure profits via flash strategies if and only if asset prices do not exhibit predictable jumps. This result relies on the general theory of processes and provides the most general formulation of the well-known fact that, in an arbitrage-free financial market, asset prices (including dividends) should not exhibit jumps of a predictable direction or magnitude at predictable times. We furthermore show that any price process is always right-continuous in the absence of sure profits. Our results are robust under small transaction costs and imply that, under minimal assumptions, price changes occurring at scheduled dates should only be due to unanticipated information releases.
TL;DR: In this article, a weakly dependent stationary random field with maxima for finite and for infinite sets of size, where and, are determined by a translation-invariant total order on.
Abstract: Let be a weakly dependent stationary random field with maxima for finite and for . In a general setting we prove that for some increasing sequence of sets of size , where and . The sets are determined by a translation-invariant total order on . For a class of fields satisfying a local mixing condition, including m-dependent ones, the main theorem holds with a constant finite A replacing . The above results lead to new formulas for the extremal index for random fields. The new method for calculating limiting probabilities for maxima is compared with some known results and applied to the moving maximum field.