Multidimensional Diffusion Processes Research Articles

Estimation for Discretely Observed Small Diffusions Based on Approximate Martingale Estimating Functions

Abstract. We consider an asymptotically efficient estimator of the drift parameter for a multi‐dimensional diffusion process with small dispersion parameter ɛ. In the situation where the sample path is observed at equidistant times k/n, k = 0, 1, …, n, we study asymptotic properties of an M‐estimator derived from an approximate martingale estimating function as ɛ tends to 0 and n tends to ∞ simultaneously.

Recurrence and transience of multi-dimensional diffusion processes in reflected Brownian environments

We discuss limiting behaviors of multi-dimensional diffusion processes in new types of random environments. The environments we treat are: (1) d independent non-negative-reflected Brownian environments and (2) d independent non-positive-reflected Brownian environments. Their behaviors are quite different from those of d-dimensional Brownian motion.

Exact approximation rate of killed hypoelliptic diffusions using the discrete Euler scheme

We are interested in approximating a multidimensional hypoelliptic diffusion process ( X t ) t⩾0 killed when it leaves a smooth domain D. When a discrete Euler scheme with time step h is used, we prove under a noncharacteristic boundary condition that the weak error is upper bounded by C 1 h , generalizing the result obtained by Gobet in (Stoch. Proc. Appl. 87 (2000) 167) for the uniformly elliptic case. We also obtain a lower bound with the same rate h , thus proving that the order of convergence is exactly 1/2. This provides a theoretical explanation of the well-known bias that we can numerically observe in that kind of procedure.

Small-diffusion asymptotics for discretely sampled stochastic differential equations

1051-1069: The minimum-contrast estimation of drift and diffusion coefficient parameters for a multi-dimensional diffusion process with a small dispersion parameter $\varepsilon$ based on a Gaussian approximation to transition density is presented when the sample path is observed at equidistant times $k/n$, $k=0,1,\rm \dots,n$. We study asymptotic results for the minimum-contrast estimator as $\varepsilon$ goes to $0$ and $n$ goes to $\infty$ simultaneously.

Subordinated Lèvy–Feldheim motion as a model of anomalous self-similar diffusion

Multidimensional self-similar diffusion processes are treated as subordinated symmetrical Lèvy–Feldheim motion. Multivariate LF distributions are obtained by numerical calculation.

Local Asymptotic Mixed Normality Property for Elliptic Diffusion: A Malliavin Calculus Approach

We address the problem of the validity of the local asymptotic mixed normality (LAMN) property when the model is a multidimensional diffusion process X whose coefficients depend on a scalar parameter θ: the sample (Xk/n)0≤ k≤ n corresponds to an observation of X at equidistant times in the interval [0,1]. We prove that the LAMN property holds true for the likelihood under an ellipticity condition and some suitable smoothness assumptions on the coefficients of the stochastic differential equation. Our method is based on Malliavin calculus techniques: in particular, we derive for the log-likelihood ratio a tractable representation involving conditional expectations.

Mathematical formalism for isothermal linear irreversibility

Mathematical formalism for isothermal linear irreversibility

A stochastic modeling of early HIV-1 population dynamics

In a recent paper, Tuckwell and Le Corfec [J. Theor. Biol. 195 (1998) 450–463] applied the multi-dimensional diffusion process to model early human immunodeficiency virus type-1 (HIV-1) population dynamics. The purpose of this paper is to assess certain features and consequences of their model in the context of Tan and Wu's stochastic approach [Math. Biosci. 147 (1998) 173–205].

First passage time to detection in stochastic population dynamical models for HIV-1

The detection of HIV-1 levels in human hosts is cast as a first exit time problem for a multidimensional diffusion process. We consider a four-component model for early HIV-1 dynamics including uninfected CD4+ T-cells, latently infected cells, actively infected cells, and HIV-1 virions. An analytical framework is presented for the distribution of the time at which a given virion level is attained. A one-dimensional diffusion approximation for a branching process leads to an estimate for the distribution of the virion density and an expression for the mean detection time for any given detection threshold.

Passage time moments for multidimensional diffusions

Let τ r denote the hitting time of B(0:r) for a multidimensional diffusion process. We give verifiable criteria for finiteness/infiniteness of As an application we exhibit classes of diffusion processes which are recurrent but is infinite for all p > 0, |x| > r > 0; this includes the two-dimensional Brownian motion and the reflecting Brownian motion in a wedge with a certain parameter α = 0.

Passage time moments for multidimensional diffusions

Let τr denote the hitting time of B(0:r) for a multidimensional diffusion process. We give verifiable criteria for finiteness/infiniteness of As an application we exhibit classes of diffusion processes which are recurrent but is infinite for all p > 0, |x| > r > 0; this includes the two-dimensional Brownian motion and the reflecting Brownian motion in a wedge with a certain parameter α = 0.

Applications of Malliavin calculus to Monte-Carlo methods in finance. II

This paper presents an original probabilistic method for the numerical computations of Greeks (i.e. price sensitivities) in finance. Our approach is based on the {\it integration-by-parts} formula, which lies at the core of the theory of variational stochastic calculus, as developed in the Malliavin calculus. The Greeks formulae, both with respect to initial conditions and for smooth perturbations of the local volatility, are provided for general discontinuous path-dependent payoff functionals of multidimensional diffusion processes. We illustrate the results by applying the formula to exotic European options in the framework of the Black and Scholes model. Our method is compared to the Monte Carlo finite difference approach and turns out to be very efficient in the case of discontinuous payoff functionals.

‘Path’ and ‘time’ estimates in the Monte Carlo method

We propose new weight modifications of a 'path* estimate for the calculation of linear func- tionals (Φ,/ι) of radiation intensity Φ. In the framework of the 'collision* model of a transport process we mathematicall y justify the estimate, including variance finiteness, in the case of the function /ι with alternating signs. We construct and justify the analogous 'time' estimate for the calculation of linear functionals of the con- centration of particles moving along trajectories of multidimensional diffusion processes (the concentration of trajectories for brevity). In this paper we consider a problem of estimating the linear functionals (Φ, h) of radia- tion intensity Φ by new weight modifications of a 'path' estimate, which implies that the mean path of a particle in a domain is equal to the integral of radiation intensity in this domain. We use the 'collision' model of a transport process, i.e. the Markov homoge- neous chain terminating with probability one. Its states are the points in the phase space of coordinate velocities at which instantaneous velocity changes occur along a random particle trajectory. In the framework of this model we mathematicall y justify 'path es- tiamates' in the case of the function h with alternating signs. Due to this justification we can study the problem of variance finiteness for the new weight modifications of the estimate as well, i.e. when an auxiliary 'nonphysicaF Markov chain is modelled and the estimate is multiplied by a special weight multiplier. We construct and justify the analogous 'time estimate' for the calculation of linear functionals of the concentration of particles moving along trajectories of multidimen- sional diffusion processes (the concentration of trajectories for brevity). This estimate is the trajectory time integral of a given weight function. We study the problem of the variance finiteness of this estimate and a possibility of its approximate numerical implementation by the Euler scheme. We establish a relation between the time es- timate and the probabilistic representations of solutions of boundary value problems. Under certain conditions this allows us to justify the variance finiteness of the estimate if there is particle multiplication. It is shown that we can approximately calculate the first eigenvalue of a diffusion operator by the easily realizable parametric differentiation of a special weight estimate. 1. STANDARD WEIGHT ESTIMATES OF THE MONTE CARLO METHOD AND THEIR PARAMETRIC DIFFERENTIATION

A Stochastic Model for Early HIV-1 Population Dynamics

A simple stochastic mathematical model is developed and investigated for early human immunodeficiency virus type-1 (HIV-1) population dynamics. The model, which is a multi-dimensional diffusion process, includes activated uninfected CD4+T cells, latently and actively infected CD4+T cells and free virions occurring in plasma. Stochastic effects are assumed to arise in the process of infection of CD4+T cells and transitions may occur from uninfected to latently or actively infected cells by chance mechanisms. Using the best currently available parameter values, the intrinsic variability in response to a given initial infection is examined by solving the stochastic system numerically. We estimate the statistical distributions of the time of occurrence and the magnitude of the early peak in viral concentration. The maximum of the viral load has a value in the experimental range and its time of occurrence has a 95% confidence interval from 19.4 to 25.1 days. The stochastic nature of the growth of viral density is extremely pronounced in the first few days after initial infection. Threshold effects are noted at virion levels of about 3–5×10−5mm−3. In addition to modeling the intrinsic variability in HIV-1 growth, we have explored the effects of perturbations in the parameter values in order to assess the additional stochastic effects of between-patient variability. We found that changes in the initial number of virions or dose size, the rate at which latently infected CD4+T cells are converted to the actively infected form and the fraction of latent cells has only minor effects on the size, speed and variability of the response. In contrast, decreased speed and magnitude but greater variability in response were obtained when the death rate of uninfected CD4+T cells, the death rate of actively infected cells and the clearance rate of the virus were increased or when the appearance rate of uninfected CD4+T cells, the number of virions produced by infected cells, the infection rate of CD4+T cells and the initial number of uninfected activated CD4+T cells were decreased. We also determined the distribution of the time to reach a given virion density. From this distribution the probability of detection of the virus as a function of time can be estimated. The numerical results obtained are in the range of experimental values and are discussed in relation to recently proposed detection and testing procedures.

Noisy spiking neurons and networks: useful approximations for firing probabilities and global behavior

Electrophysiological properties of spiking neurons receiving complex stimuli perturbed by noise are investigated. A semi-analytical estimate of firing probabilities and subthreshold behavior of the stochastic system can be made in terms of the solution of a purely deterministic system. The method comes from an approximation for the distribution function and moments of the underlying non linear multidimensional diffusion process. This so called moment method works for general conductance-based systems and an application is presented for the Hodgkin–Huxley neuronal model. Statistical properties obtained from the moment method are compared with direct numerical integration of the stochastic system. The firing probability due to external noise is derived as a closed formula. Results are given for different forms of the deterministic component of the stimulus. A generalization to neural networks of conductance-based systems with internal currents perturbed by noise can be obtained using the same approach. In the case of fully connected networks, a mean field population equation is derived which may be compared to Kuramoto’s master equation for weakly coupled neural oscillators.

Heavy Traffic Convergence of a Controlled, Multiclass Queueing System

This paper provides a rigorous proof of the connection between the optimal sequencing problem for a two-station, two-customer-class queueing network and the problem of control of a multidimensional diffusion process, obtained as a heavy traffic limit of the queueing problem. In particular, the diffusion problem, which is one of ``singular control'' of a Brownian motion, is used to develop policies which are shown to be asymptotically nearly optimal as the traffic intensity approaches one in the queueing network. The results are proved by a viscosity solution analysis of the related Hamilton--Jacobi--Bellman equations.

Maximum Likelihood Estimation of Non-Linear Continuous-Time Term-Structure Models

This paper presents an exact maximum likelihood method which is designed for estimating multi-dimensional time-homogeneous diffusion processes, using discretely sampled observations. The essence of this method is to evaluate the transition density by means of simulating the diffusion processes. The advantages of this approach are that we do not have to derive the distribution for the diffusion process described by the stochastic differential equation, and that we can use all the standard results from the likelihood theory. We apply this exact maximum likelihood method to consider a class of dynamic one factor continuous-time interest rate models all nested within the CKLS-model (Chan, Karolyi, Longstaff and Sanders 1992), which will enable us to compare the different models. The general superiority of this method to existing methods (e.g. QML, GMM) is documented through Monte Carlo studies. We also show that the method can be applied with great benefit to multi-factor continuous-time term-structure models, and we look at a non-linear special case of the Brennan & Schwartz model (1982). Finally, empirical results from the U.S. market will be presented. NOTE: This is a revised version of the paper Estimation of a Dynamic One Factor Continuous-Time Term-Structure Model.

A diffusion approximation for a network of reservoirs with power law release rule

A diffusion approximation for a network of continuous time reservoirs with power law release rules is examined. Under a mild assumption on the inflow processes, we show that for physically reasonable values of the power law constants, the system of processes converges to a multi-dimensional Gaussian diffusion process. We also illustrate how the limiting Gaussian process may be used to compute approximations to the original system of reservoirs. In addition, we study the quality of our approximations by comparing them to results obtained by simulations of the original watershed model. The simulations offer support for the use of the approximation developed here.

Almost self-optimizing strategies for the adaptive control of diffusion processes

The ergodic control of a multidimensional diffusion process described by a stochastic differential equation that has some unknown parameters appearing in the drift is investigated. The invariant measure of the diffusion process is shown to be a continuous function of the unknown parameters. For the optimal ergodic cost for the known system, an almost optimal adaptive control is constructed for the unknown system.

The Novikov and entropy conditions of multidimensional diffusion processes with singular drift

We consider multidimensional stochastic differential equations of the form dX t =b(t,X t )dt+dW t ,0≤t≤R,R<∞,(1) with arbitrary initial (probability) distribution μ on ℝ d , d≥1. The first aim of this paper is to give handy-to- verify analytic (i.e. nonstochastic) conditions for the existence of a weak solution of (1), where the drift b will be allowed to have singularities. These investigations are illustrated by various examples. We first concentrate on (a uniform form of) the Novikov condition E μ exp1 2∫ 0 R |b(s,X s )| 2 ds<∞(2) and then investigate further sufficient conditions for the applicability of the Girsanov- Maruyama theorem which are not covered by (2). The outcoming results improve some of those of H. J. Engelbert and W. Schmidt [Math. Nachr. 119, 97-115] (for time-independent drifts b(x)) and N. I. Portenko [Theory Probab. Appl. 20, 27-37 (1975); translation from Teor. Veroyatn. Primen. 20, 29-39] (for time-dependent drifts b(t,x)). One of the examples involves a drift which is singular on a dense set in ℝ d but nevertheless satisfies (2). The second aim of this paper is to discuss some general properties and applications of (2). For instance, we investigate whether the factor 1/2 in the Novikov condition (2) “can be replaced” by 1/2±e (e>0). Furthermore, we give several equivalence characterizations of (2) [being connected to the well-known R. Z. Khas'minskij lemma, ibid. 4, 309-318 (1960) resp. ibid. 4, 332-341]. Finally, it is shown that under the Novikov condition (2), the diffusion process with drift b has finite relative entropy with respect to Wiener measure (and thus finite“energy”).