In this paper we consider the filtering of partially observed multidimensional diffusion processes that are observed regularly at discrete times. This is a challenging problem which requires the use of advanced numerical schemes based upon time-discretization of the diffusion process and then the application of particle filters. Perhaps the state-of-the-art method for moderate-dimensional problems is the multilevel particle filter of Jasra et al. (SIAM J. Numer. Anal. 55 (2017), 3068–3096). This is a method that combines multilevel Monte Carlo and particle filters. The approach in that article is based intrinsically upon an Euler discretization method. We develop a new particle filter based upon the antithetic truncated Milstein scheme of Giles and Szpruch (Ann. Appl. Prob. 24 (2014), 1585–1620). We show empirically for a class of diffusion problems that, for $\epsilon>0$ given, the cost to produce a mean squared error (MSE) of $\mathcal{O}(\epsilon^2)$ in the estimation of the filter is $\mathcal{O}(\epsilon^{-2}\log(\epsilon)^2)$. In the case of multidimensional diffusions with non-constant diffusion coefficient, the method of Jasra et al. (2017) requires a cost of $\mathcal{O}(\epsilon^{-2.5})$ to achieve the same MSE.

We propose a discrete-time discrete-space Markov chain approximation with a Brownian bridge correction for computing curvilinear boundary crossing probabilities of a general diffusion process on a finite time interval. For broad classes of curvilinear boundaries and diffusion processes, we prove the convergence of the constructed approximations in the form of products of the respective substochastic matrices to the boundary crossing probabilities for the process as the time grid used to construct the Markov chains is getting finer. Numerical results indicate that the convergence rate for the proposed approximation with the Brownian bridge correction is $O(n^{-2})$ in the case of $C^2$ boundaries and a uniform time grid with n steps.

The survival energy model (SEM) is a recently introduced novel approach to mortality prediction, which offers a cohort-wise distribution function of the time of death as the first hitting time of a “survival energy” diffusion process to zero. In this study, we propose a novel SEM that can serve as a suitable candidate in the family of prediction models. We also proposed a method to improve the prediction in an earlier work. We further examine the practical advantages of SEM over existing mortality models.

Oscillatory systems of interacting Hawkes processes with Erlang memory kernels were introduced by Ditlevsen and Löcherbach (Stoch. Process. Appl., 2017). They are piecewise deterministic Markov processes (PDMP) and can be approximated by a stochastic diffusion. In this paper, first, a strong error bound between the PDMP and the diffusion is proved. Second, moment bounds for the resulting diffusion are derived. Third, approximation schemes for the diffusion, based on the numerical splitting approach, are proposed. These schemes are proved to converge with mean-square order 1 and to preserve the properties of the diffusion, in particular the hypoellipticity, the ergodicity, and the moment bounds. Finally, the PDMP and the diffusion are compared through numerical experiments, where the PDMP is simulated with an adapted thinning procedure.

We investigate the impact of Knightian uncertainty on the optimal timing policy of an ambiguity-averse decision-maker in the case where the underlying factor dynamics follow a multidimensional Brownian motion and the exercise payoff depends on either a linear combination of the factors or the radial part of the driving factor dynamics. We present a general characterization of the value of the optimal timing policy and the worst-case measure in terms of a family of explicitly identified excessive functions generating an appropriate class of supermartingales. In line with previous findings based on linear diffusions, we find that ambiguity accelerates timing in comparison with the unambiguous setting. Somewhat surprisingly, we find that ambiguity may lead to stationarity in models which typically do not possess stationary behavior. In this way, our results indicate that ambiguity may act as a stabilizing mechanism.

We propose a new approach to mortality prediction under survival energy hypothesis (SEH). We assume that a human is born with initial energy, which changes stochastically in time and the human dies when the energy vanishes. Then, the time of death is represented by the first hitting time of the survival energy (SE) process to zero. This study assumes that SE follows a time-inhomogeneous diffusion process and defines the mortality function, which is the first hitting time distribution function of the SE process. Although SEH is a fictitious construct, we illustrate that this assumption has the potential to yield a good parametric family of cumulative probability of death, and the parametric family yields surprisingly good predictions for future mortality rates.

We consider the Halfin–Whitt diffusion process Xd(t), which is used, for example, as an approximation to the m-server M/M/m queue. We use recently obtained integral representations for the transient density p(x,t) of this diffusion process, and obtain various asymptotic results for the density. The asymptotic limit assumes that a drift parameter β in the model is large, and the state variable x and the initial condition x0 (with Xd(0) = x0 > 0) are also large. We obtain some alternate representations for the density, which involve sums and/or contour integrals, and expand these using a combination of the saddle point method, Laplace method and singularity analysis. The results give some insight into how steady state is achieved, and how if x0 > 0 the probability mass migrates from Xd(t) > 0 to the range Xd(t) < 0, which is where it concentrates as t → ∞, in the limit we consider. We also discuss an alternate approach to the asymptotics, based on geometrical optics and singular perturbation techniques.

Let M be acomplete Riemannian manifold, M ∈ ℕ andp ≥ 1. Weprove that almost everywhere on x = (x1,...,xN) ∈ MNfor Lebesgue measure in MN, the measure \hbox{$\di \mu(x)=\f1N\sum_{k=1}^N\d_{x_k}$}$\mathit{\mu }\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{\mathit{N}}\sum _{\mathit{k}\mathrm{=}\mathrm{1}}^{\mathit{N}}{\mathit{\delta }}_{{\mathit{x}}_{\mathit{k}}}$ has a unique p–mean ep(x).As a consequence, if X = (X1,...,XN)is a MN-valued randomvariable with absolutely continuous law, then almost surely μ(X(ω)) has aunique p–mean. In particular if (Xn)n ≥ 1is an independent sample of an absolutely continuous law in M, then the processep,n(ω) = ep(X1(ω),...,Xn(ω))is well-defined. Assume M is compact and consider a probability measureν inM. Usingpartial simulated annealing, we define a continuous semimartingale which converges inprobability to the set of minimizers of the integral of distance at power p with respect toν. When theset is a singleton, it converges to the p–mean.

In this article, our aim is to estimate the successive derivatives of the stationarydensity f of a strictly stationary and β-mixing process (Xt)t≥0. This process is observed at discrete timest = 0,Δ,...,nΔ. Thesampling interval Δ can be fixed or small. We use a penalizedleast-square approach to compute adaptive estimators. If the derivativef(j) belongs to the Besov space \hbox{$\rond{B}_{2,\infty}^{\alpha}$}$B_{\mathrm{2}\mathit{,}\mathrm{\infty }}^{\mathit{\alpha }}$, then our estimator converges at rate (nΔ)−α/(2α+2j+1). Then we consider a diffusion with known diffusioncoefficient. We use the particular form of the stationary density to compute an adaptiveestimator of its first derivative f′. When the sampling intervalΔ tends to 0, and when the diffusion coefficient is known, theconvergence rate of our estimator is (nΔ)−α/(2α+1). When the diffusion coefficient is known, we alsoconstruct a quotient estimator of the drift for low-frequency data.

Using an empirical approach to capital market returns analogous to that used for mortality rates by Halley more than three centuries ago to establish life assurance on a sound and scientific footing, a theory of option pricing is built up in terms of the same three key components as for life assurance premiums, namely the expected cost of claims, an allowance for expenses, and a contingency margin as a reserve against the risk of insolvency. The dimensionality of the process describing security returns to any future point in time is increased from two to three by the addition of systematic variability around ‘central values’ to the standard descriptors of expected return and variance of return. It is shown that this approach, which involves only common sense principles and elementary mathematics, has important theoretical, practical and regulatory advantages over the Black-Scholes and related methodologies of modern finance theory.

We consider a Wright-Fisher diffusion (x(t)) whose current state cannot be observed directly. Instead, at times t 1 < t 2 < ..., the observations y(ti) are such that, given the process (x(t)), the random variables (y(ti)) are independent and the conditional distribution of y(ti) only depends on x(ti). When this conditional distribution has a specific form, we prove that the model ((x(ti),y(ti)), i ≥1) is a computable filter in the sense that all distributions involved in filtering, prediction and smoothing are exactly computable. These distributions are expressed as finite mixtures of parametric distributions. Thus, the number of statistics to compute at each iteration is finite, but this number may vary along iterations.

The dynamics of sensors operated devices such as Automated Mobile Robots and more generally automated target seeking devices is studied in presence of noise. We introduce a simple and analytically tractable class of dynamics which permits to classify qualitatively and somehow quantitatively also the approach to the targets when fluctuations corrupt the ideal trajectories. Our model constitute a first evaluation of the feasibility of an efficient approach when the parameters of the model (statistics of the noise, lengths of the path and progressing steps and heading velocity) are known.

In a deregulated insurance market, insurance carriers have an incentive to be innovative in their pricing decisions by segmenting their portfolios and designing new bonus-malus systems (BMS). This paper examines the evolution of market shares and claim frequencies in a two-company market, when one insurer breaks off the existing stability by introducing a super-discount class in its BMS. Several assumptions concerning policyholders and insurers behavior are tested. Diffusion theory is used to model the spread of the information concerning the new BMS among prospective customers. A wide variety of market outcomes results: one company may take over the market or the two may survive with equal or unequal market shares, each specializing in a specific niche of the market. Before engaging in an aggressive competitive behavior, insurers should consequently be reasonably confident in their assumptions concerning the reactions of their policyholders to the new BMS.

In this paper a new multifractal stochastic process called Limit of theIntegrated Superposition of Diffusion processes with Linear differencialGenerator (LISDLG) is presented which realistically characterizes the networktraffic multifractality. Several properties of the LISDLG model are presentedincluding long range dependence, cumulants, logarithm of the characteristicfunction, dilative stability, spectrum and bispectrum. The model captureshigher-order statistics by the cumulants. The relevance and validation of theproposed model are demonstrated by real data of Internet traffic.


This paper introduces a fractional heat equation, where the diffusion operator is the composition of the Bessel and Riesz potentials. Sharp bounds are obtained for the variance of the spatial and temporal increments of the solution. These bounds establish the degree of singularity of the sample paths of the solution. In the case of unbounded spatial domain, a solution is formulated in terms of the Fourier transform of its spatially and temporally homogeneous Green function. The spectral density of the resulting solution is then obtained explicitly. The result implies that the solution of the fractional heat equation may possess spatial long-range dependence asymptotically.

We consider a diffusing particle in one dimension that is subject to a time-dependent drift or potential field. A reflecting barrier constrains the particle's position to the half-line X ≥ 0. Such models arise naturally in the study of queues with time-dependent arrival rates, as well as in advection-diffusion problems of mathematical physics. We solve for the probability distribution of the particle as a function of space and time. Then we do a detailed study of the asymptotic properties of the solution, for various ranges of space and time. We also relate our asymptotic results to those obtained by probabilistic approaches, such as central limit theorems and large deviations. We consider drifts that are either piecewise constant or linear functions of time.

Let (Xt ) be a diffusion on the interval (l,r) and Δn a sequence of positive numbers tending to zero. We define J i as the integral between iΔn and (i + 1)Δn of X s .We give an approximation of the law of (J0,...,Jn-1)by means of a Euler scheme expansion for the process (Ji ). In some special cases, an approximation by anexplicit Gaussian ARMA(1,1) process is obtained.When Δn = n-1 we deduce from this expansion estimatorsof the diffusion coefficient of X based on (Ji ). These estimatorsare shown to be asymptotically mixed normal as n tends to infinity.

This paper provides a detailed stochastic analysis of leucocyte cell movement based on the dynamics of a rigid body. The cell's behavior is studied in two relevant anisotropic environments displaying adhesion mediated movement (haptotaxis) and stimulus mediated movement (chemotaxis). This behavior is modeled by diffusion processes on three successively longer time scales, termed locomotion, translocation, and migration.

The series expansion for the solution of the integral equation for the first-passage-time probability density function, obtained by resorting to the fixed point theorem, is used to achieve approximate evaluations for which error bounds are indicated. A different use of the fixed point theorem is then made to determine lower and upper bounds for asymptotic approximations, and to examine their range of validity.