Multidimensional Ornstein-Uhlenbeck Processes Research Articles

Continuous‐time stochastic gradient descent for optimizing over the stationary distribution of stochastic differential equations

AbstractWe develop a new continuous‐time stochastic gradient descent method for optimizing over the stationary distribution of stochastic differential equation (SDE) models. The algorithm continuously updates the SDE model's parameters using an estimate for the gradient of the stationary distribution. The gradient estimate is simultaneously updated using forward propagation of the SDE state derivatives, asymptotically converging to the direction of steepest descent. We rigorously prove convergence of the online forward propagation algorithm for linear SDE models (i.e., the multidimensional Ornstein–Uhlenbeck process) and present its numerical results for nonlinear examples. The proof requires analysis of the fluctuations of the parameter evolution around the direction of steepest descent. Bounds on the fluctuations are challenging to obtain due to the online nature of the algorithm (e.g., the stationary distribution will continuously change as the parameters change). We prove bounds for the solutions of a new class of Poisson partial differential equations (PDEs), which are then used to analyze the parameter fluctuations in the algorithm. Our algorithm is applicable to a range of mathematical finance applications involving statistical calibration of SDE models and stochastic optimal control for long time horizons where ergodicity of the data and stochastic process is a suitable modeling framework. Numerical examples explore these potential applications, including learning a neural network control for high‐dimensional optimal control of SDEs and training stochastic point process models of limit order book events.

Impact of Zero-Dispersion Wavelength Fluctuations in a Coupled Dual-Core Fiber Optical Parametric Amplifier

We have studied the impact of zero dispersion wavelength (ZDW) fluctuations in coupled dual-core fiber optical parametric amplifiers (FOPAs) and compared it to the single-core single-pump FOPA. We have designed a Monte Carlo model, that simulates a multidimensional Ornstein-Uhlenbeck process, describing accurately ZDW fluctuations in single and multicore fibers. In order to validate our results we have developed an analytical model, based on Taylor expansion of the average gain and successfully compared both models, with experimental results published in the literature. We have found that a significant reduction of the impact of ZDW fluctuations can be obtained in dual-core FOPAs compared to single-core ones.

Domain of attraction of quasi-stationary distribution for absorbing Markov processes

In this paper, for the general λ0-positive absorbing Markov processes, we obtain a concise result on the domain of attraction of the quasi-stationary distribution. Our conclusion includes some existing results. Finally, we apply our result to multi-dimensional Ornstein–Uhlenbeck process and give a subset of the domain of attraction of its minimal quasi-stationary distribution.

Universal excursion and bridge shapes in ABBM/CIR/Bessel processes

Several years ago, in the context of the physics of hysteresis in magnetic materials, a simple stochastic model has been introduced: the ABBM model. Later, the ABBM model has been advocated as a paradigm for the description of a broad class of diverse phenomena, baptized ‘crackling noise phenomena’. The model reproduces many statistical features of such intermittent signals, as for instance the statistics of burst (or avalanche) durations and sizes, in particular the power law exponents that would characterize the dynamics as critical. In order to go beyond such ‘critical exponents’, the measure of the average shape of the avalanche has also been proposed. Here, the exact calculation of the average (as well as the fluctuations) of the avalanche shape for the ABBM model is presented, showing that its normalised shape does not depend on the external drive. Moreover, the average (and the fluctuations) of the multi-avalanche shape, that is the average shape of a sequence of avalanches of fixed total duration, is also computed. Surprisingly, the two quantities (avalanche and multi-avalanche normalised shapes) are exactly the same. This result is obtained using the exact solution of the ABBM model, which is obtained leveraging the equivalence with the Cox–Ingersoll–Ross process (CIR), rigorously obtained with a so called ‘time change’. A simple presentation of this and other known relevant exact results is provided: notably the correspondence of the ABBM/CIR model with the Rayleigh model and, more importantly, with the generalised Bessel process, which describes the dynamics of the modulus of the multi dimensional Ornstein–Uhlenbeck process (exactly as the Bessel process does for the Brownian process). As a consequence, our main finding, that is the correspondence between the excursion (avalanche) and bridge (multi-avalanche) shape distributions, turns to apply to all the aforementioned stochastic processes. In simple words: if we consider the distance from the origin of such diffusive particles, the (normalised) average shape of its trajectory (and the fluctuations around that) until a return in a time T is the same, whether it has returned before T or not.

Parallel Queues with Discrete-Choice Arrival Pattern: Empirical Evidence and Asymptotic Characterization

We consider a parallel-queue system in which each queue is served by a dedicated service provider. The arrival process is driven by a discrete choice model, that is, customers observe the queue length for each service provider and choose one to join upon arrival. We assume that a customer's utility is difference between the service reward and the waiting cost, both of which are heterogeneous. Empirical analysis of the vehicle queues at the U.S.-Canada border-crossing port of entry supports our model setting. We show that with such a choice model, the arrival rate function satisfies certain properties, which allow us to characterizes the fluid and diffusion limit of the queue-length process. In particular, we show that even without the well-used Lipschitz-continuity assumption, the fluid limit process is unique and is attracted to a unique equilibrium. The diffusion limit process is a reflected multi-dimensional Ornstein-Uhlenbeck process centered at that equilibrium. We prove that the stationary distribution of the diffusion limit is a truncated multivariant Gaussian and interchange of limits holds.

Exact pathwise simulation of multi-dimensional Ornstein–Uhlenbeck processes

The exact pathwise simulation of multidimensional Ornstein–Uhlenbeck processes is considered. We propose two procedures that allow the exact pathwise simulation of this type of processes and, simultaneously, the generation of the underlying Wiener trajectories from the same source of randomness. This is particularly important when both processes are system-components in larger stochastic models, for which the study of pathwise dynamics is required.

Diffusion limits for networks of Markov-modulated infinite-server queues

This paper studies the diffusion limit for a network of infinite-server queues operating under Markov modulation, meaning that the system’s parameters depend on an autonomously evolving Markov chain, called the background process. In previous papers on single-node queues with Markov modulation, two variants were distinguished. In the first variant the arrival rate and the server speed are modulated, whereas in the second variant the arrival rate and the service requirement are modulated. The setup of the present paper, however, is more general: we not only extend single-node systems to a network setting, but also allow both the server speed and the service requirement to depend on the background process. For this model we derive a Functional Central Limit Theorem. In particular, we show that, after accelerating the arrival processes and the background process, a centered and normalized version of the network population vector converges to a multidimensional Ornstein–Uhlenbeck process. The proof of this result relies on weak convergence of stochastic integrals as well as continuous-mapping arguments.

Integrals of Bessel processes and multi-dimensional Ornstein–Uhlenbeck processes: exact asymptotics for -functionals

We prove results on exact asymptotics of the expectations , as for , , , where , , is a Bessel process of order . We also find exact asymptotics of the probabilities , as , where , , is the -dimensional non- stationary Ornstein– Uhlenbeck process with a parameter starting at the origin. We also obtain a number of other results. Numerical values of the asymptotics are given for , .

Integrals of Bessel processes and multi-dimensional Ornstein-Uhlenbeck processes: exact asymptotics for $L^p$-functionals

Для $p>0$, $a \geqslant 0$, $b \geqslant 0$ доказаны результаты о точных асимптотиках при $T \to \infty$ средних $\mathbf{E}_a \exp (-\int_0^T \xi_q^p(t) dt )$, $\mathbf{E}_a [ \exp (-\int_0^T \xi_q^p(t) dt ) | \xi_q(T)=b ]$, где $\xi_q(t)$, $t \geqslant 0$, - бесселевский процесс порядка $q \geqslant-1/2$. Найдены также точные асимптотики при $\varepsilon \to 0$ вероятностей $\mathbf{P} \{ \int_0^1 \sum_{k=1}^n |Y_k(t)|^p dt \leqslant\varepsilon^p \}$, $\mathbf{P} \{ \int_0^1 [ \sum_{k=1}^n Y_k^2(t) ]^{p/2} dt \leqslant\varepsilon^p \}$, где $\mathbf{Y}(t)=(Y_1(t),…, Y_n(t))$, $t \geqslant 0$, - $n$-мерный нестационарный процесс Орнштейна-Уленбека с параметром $\gamma=(\gamma_1, …, \gamma_n)$, выходящий из нуля. Получен также ряд иных результатов. Численные значения для асимптотик приведены в случаях $p=1$, $p=2$. Библиография: 48 наименований.

Conditions of smoothness for the distribution density of a solution of a multidimensional linear stochastic differential equation with lévy noise

We obtain a sufficient condition of smoothness for the distribution density of a multidimensional Ornstein–Uhlenbeck process with Levy noise, i.e., for the solution of a linear stochastic differential equation with Levy noise.

Asymptotic properties of an estimator of the drift coefficients of multidimensional Ornstein-Uhlenbeck processes that are not necessarily stable

In this paper, we investigate the consistency and asymptotic efficiency of an estimator of the drift matrix, $F$, of Ornstein-Uhlenbeck processes that are not necessarily stable. We consider all the cases. (1) The eigenvalues of $F$ are in the right half space (i.e., eigenvalues with positive real parts). In this case the process grows exponentially fast. (2) The eigenvalues of $F$ are on the left half space (i.e., the eigenvalues with negative or zero real parts). The process where all eigenvalues of $F$ have negative real parts is called a stable process and has a unique invariant (i.e., stationary) distribution. In this case the process does not grow. When the eigenvalues of $F$ have zero real parts (i.e., the case of zero eigenvalues and purely imaginary eigenvalues) the process grows polynomially fast. Considering (1) and (2) separately, we first show that an estimator, $\hat{F}$, of $F$ is consistent. We then combine them to present results for the general Ornstein-Uhlenbeck processes. We adopt similar procedure to show the asymptotic efficiency of the estimator.

Fluid models of integrated traffic and multipath routing

In this paper we consider a stochastic model describing the varying number of flows in a network. This model features flows of two types, namely file transfers (with fixed volume) and streaming traffic (with fixed duration), and extends the model of Key, Massoulie, Bain and Kelly [27] by allowing more general bandwidth allocation criteria. We analyse the dynamics of the system under a fluid scaling, and show Lyapunov stability of the fluid limits under a natural stability condition. We provide natural interpretations of the fixed points of these fluid limits. We then compare the fluid dynamics of file transfers under (i) balanced multipath routing and (ii) parallel, uncoordinated routing. We show that for identical traffic demands, parallel uncoordinated routing can be unstable while balanced multipath routing is stable. Finally, we identify multi-dimensional Ornstein-Uhlenbeck processes as second-order approximations to the first-order fluid limit dynamics.

Connection between deriving bridges and radial parts from multidimensional Ornstein-Uhlenbeck processes

First we give a construction of bridges derived from a general Markov process using only its transition densities. We give sufficient conditions for their existence and uniqueness (in law). Then we prove that the law of the radial part of the bridge with endpoints zero derived from a special multidimensional Ornstein--Uhlenbeck process equals the law of the bridge with endpoints zero derived from the radial part of the same Ornstein--Uhlenbeck process. We also construct bridges derived from general multidimensional Ornstein--Uhlenbeck processes.

On multidimensional Ornstein-Uhlenbeck processes driven by a general Lévy process

We prove the following probabilistic properties of a multidimensional Ornstein-Uhlenbeck process driven by a general Levy process, under mild regularity conditions: the strong Feller property; the existence of a smooth transition density; and the exponential β-mixing property. As a class of possible invariant distributions of an Ornstein-Uhlenbeck process, we also discuss centred and non-skewed multidimensional generalized hyperbolic distributions.

Local volatility in the Heston model: a Malliavin calculus approach

We implement the Heston stochastic volatility model by using multidimensional Ornstein‐Uhlenbeck processes and a special Girsanov transformation, and consider the Malliavin calculus of this model. We derive explicit formulas for the Malliavin derivatives of the Heston volatility and the log‐price, and give a formula for the local volatility which is approachable by Monte‐Carlo methods.

Parameter Estimation with Exact Distribution for Multidimensional Ornstein–Uhlenbeck Processes

It is shown that the suitably normalized maximum likelihood estimators of some parameters of multidimensional Ornstein–Uhlenbeck processes with coefficient matrix of a special structure have exactly a normal distribution. This result provides a generalization to an arbitrary dimension of the well-known behavior of the estimator of the period of a complex AR(1) process.

Fluctuations and the Boltzmann equation. I

The evolution of a homogeneous dilute gas is treated as a Markov process in the complete set of $K$ coarsegrained velocity states of all $N$ particles. From the Siegert master equation for the process a Fokker-Planck equation is derived which describes, in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$, the fluctuations in the occupation numbers ${n}_{i}(t)$, whose average behavior is governed by the (appropriately discretized) Boltzmann equation: The continuum limit $K\ensuremath{\rightarrow}\ensuremath{\infty}$ corresponds to fluctuations in the usual molecular distribution function $f(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}};t)$. On similar reasoning, a Fokker-Planck equation is obtained for the fluctuation process near equilibrium, where the average is governed by the linearized Boltzmann equation. The theory of linear irreversible processes, which offers a statistical description of fluctuations on a thermodynamical basis, is applied to the linearized Boltzmann equation---treated as a linear phenomenological equation---following the development given recently by Fox and Uhlenbeck: The resulting stochastic equation is seen to be equivalent to the Fokker-Planck equation obtained from the master equation, yielding a multidimensional Ornstein-Uhlenbeck process which describes the fluctuations in molecular phase space.