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Abstract
Consider the fractional Brownian Motion (fBM) with Hurst index . We construct a probability space supporting both and a fully simulatable process such that with probability one for any user specified error parameter . When , we further enhance our error guarantee to the -H\"older norm for any . This enables us to extend our algorithm to the simulation of fBM driven stochastic differential equations . Under mild regularity conditions on the drift and diffusion coefficients of Y, we construct a probability space supporting both Y and a fully simulatable process such that with probability one. Our algorithms enjoy the tolerance-enforcement feature, i.e., the error bounds can be updated sequentially. Thus, the algorithms can be readily combined with other simulation techniques like multilevel Monte Carlo to estimate expectation of functionals of fBMs efficiently.
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We provide the first generic exact simulation algorithm for multivariate diffusions. Current exact sampling algorithms for diffusions require the existence of a transformation which can be used to reduce the sampling problem to the case of a constant diffusion matrix and a drift which is the gradient of some function. Such transformation, called Lamperti transformation, can be applied in general only in one dimension. So, completely different ideas are required for exact sampling of generic multivariate diffusions. The development of these ideas is the main contribution of this paper. Our strategy combines techniques borrowed from the theory of rough paths, on one hand, and multilevel Monte Carlo on the other.
For a stochastic differential equation driven by a fractional Brownian motion
with Hurst parameter it is known that the existing (naive) Euler
scheme has the rate of convergence , which means no convergence to
zero of the error when H is formally set to (the standard Brownian
motion case). In this paper we introduce a new (modified Euler) approximation
scheme which is closer to the classical Euler scheme for diffusion processes
and it has the rate of convergence , where when , when
and if . In particular, the rate of
convergence becomes when H is formally set to .
Furthermore, we study the asymptotic behavior of the fluctuations of the error.
More precisely, if is the solution of a stochastic
differential equation driven by a fractional Brownian motion and if is its approximation obtained by the new modified Euler scheme,
then we prove that converges stably to the solution of a
linear stochastic differential equation driven by a matrix-valued Brownian
motion, when . In the case , we show
the convergence of and the limiting process is identified
as the solution of a linear stochastic differential equation driven by a
matrix-valued Rosenblatt process. The rate of weak convergence is also deduced
for this scheme. We also apply our approach to the naive Euler scheme. The main
tools are fractional calculus, Malliavin calculus, and the fourth moment
theorem.
Consider a multidimensional diffusion process . Let be a \textit{deterministic}, user
defined, tolerance error parameter. Under standard regularity conditions on the
drift and diffusion coefficients of X, we construct a probability space,
supporting both X and an explicit, piecewise constant, fully simulatable
process such that \[ \sup_{0\leq t\leq1}\left\Vert
X_{\varepsilon}\left(t\right) -X\left(t\right) \right\Vert_{\infty}<\varepsilon
\] with probability one. Moreover, the user can adaptively choose
so that
(also piecewise constant and fully simulatable) can
be constructed conditional on to ensure an error smaller than
with probability one. Our construction requires a
detailed study of continuity estimates of the Ito map using Lyon's theory of
rough paths. We approximate the underlying Brownian motion, jointly with the
L\'{e}vy areas with a deterministic error in the underlying rough
path metric.
We present the first class of perfect sampling (also known as exact
simulation) algorithms for the steady-state distribution of non-Markovian loss
networks. We use a variation of Dominated Coupling From The Past for which we
simulate a stationary infinite server queue backwards in time and analyze the
running time in heavy traffic. In particular, we are able to simulate
stationary renewal marked point processes in unbounded regions. We use the
infinite server queue as an upper bound process to simulate loss systems. The
running time analysis of our perfect sampling algorithm for loss systems is
performed in the Quality-Driven (QD) and the Quality-and-Efficiency-Driven
regimes. In both cases, we show that our algorithm achieves sub-exponential
complexity as both the number of servers and the arrival rate increase.
Moreover, in the QD regime, our algorithm achieves a nearly optimal rate of
convergence.
Suppose that X1, X2, . . . are independent identically distributed Bernoulli random variables with mean p. A Bernoulli factory for a function f takes as input X1, X2, . . . and outputs a random variable that is Bernoulli with mean f(p). A fast algorithm is a function that only depends on the values of X1, . . ., XT
, where T is a stopping time with small mean. When f(p) is a real analytic function the problem reduces to being able to draw from linear functions Cp for a constant C > 1. Also it is necessary that Cp ≤ 1 − ε for known ε > 0. Previous methods for this problem required extensive modification of the algorithm for every value of C and ε. These methods did not have explicit bounds on \\mathbb{E}[T]\ as a function of C and ε. This paper presents the first Bernoulli factory for f(p) = Cp with bounds on \\mathbb{E}[T]\ as a function of the input parameters. In fact, sup
p∈[0,(1−ε)/C]\\mathbb{E}[T]\ ≤ 9.5ε−1C. In addition, this method is very simple to implement. Furthermore, a lower bound on the average running time of any Cp Bernoulli factory is shown. For ε ≤ 1/2, sup
p∈[0,(1−ε)/C]\\mathbb{E}[T]\≥0.004Cε−1, so the new method is optimal up to a constant in the running time.
A stochastic perpetuity takes the form D∞=∑n=0∞
exp(Y1+⋯+Yn)B
n, where Yn:n≥0) and
(Bn:n≥0) are two independent sequences of independent
and identically distributed random variables (RVs). This is an expression for the stationary distribution of the Markov chain defined
recursively by Dn+1=An
Dn+Bn,
n≥0, where
An=eYn; D∞ then satisfies
the stochastic fixed-point equation D∞D̳AD∞+B,
where A and B are independent copies of the
An and Bn (and independent of D∞ on the right-hand
side). In our framework, the quantity Bn, which represents a random
reward at time n, is assumed to be positive, unbounded with EBnp
0, and have a suitably regular continuous positive density. The
quantity Yn is assumed to be light tailed and represents a discount rate from
time n to n-1. The RV D∞ then represents the net present value, in a
stochastic economic environment, of an infinite stream of stochastic rewards.
We provide an exact simulation algorithm for generating samples of D∞. Our
method is a variation of dominated coupling from the past and it
involves constructing a sequence of dominating processes.
This note is devoted to construct a rough path above a multidimensional fractional Brownian motion B with any Hurst parameter H∈(0, 1), by means of its representation as a Volterra Gaussian process. This approach yields some algebraic and computational simplifications with respect to [Stochastic Process. Appl. 120 (2010) 1444–1472], where the construction of a rough path over B was first introduced.
A ubiquitous observation in cell biology is that diffusion of macromolecules and organelles is anomalous, and a description simply based on the conventional diffusion equation with diffusion constants measured in dilute solution fails. This is commonly attributed to macromolecular crowding in the interior of cells and in cellular membranes, summarising their densely packed and heterogeneous structures. The most familiar phenomenon is a power-law increase of the MSD, but there are other manifestations like strongly reduced and time-dependent diffusion coefficients, persistent correlations, non-gaussian distributions of the displacements, heterogeneous diffusion, and immobile particles. After a general introduction to the statistical description of slow, anomalous transport, we summarise some widely used theoretical models: gaussian models like FBM and Langevin equations for visco-elastic media, the CTRW model, and the Lorentz model describing obstructed transport in a heterogeneous environment. Emphasis is put on the spatio-temporal properties of the transport in terms of 2-point correlation functions, dynamic scaling behaviour, and how the models are distinguished by their propagators even for identical MSDs. Then, we review the theory underlying common experimental techniques in the presence of anomalous transport: single-particle tracking, FCS, and FRAP. We report on the large body of recent experimental evidence for anomalous transport in crowded biological media: in cyto- and nucleoplasm as well as in cellular membranes, complemented by in vitro experiments where model systems mimic physiological crowding conditions. Finally, computer simulations play an important role in testing the theoretical models and corroborating the experimental findings. The review is completed by a synthesis of the theoretical and experimental progress identifying open questions for future investigation.
A generalized bridge is the law of a stochastic process that is conditioned
on linear functionals of its path. We consider two types of representations of
such bridges: orthogonal and canonical. In the canonical representation the
filtrations and the linear spaces generated by the bridge process and the
original process coincide. In the orthogonal representation the bridge is
constructed from the entire path of the underlying process. The orthogonal
representation is given for any continuous Gaussian process but the canonical
representation is given only for so-called prediction-invertible Gaussian
processes. Finally, we apply the canonical bridge representation to insider
trading by interpreting the bridge from an initial enlargement of filtration
point of view.
The theory of rough paths allows one to define controlled differential equations driven by a path which is irregular. The most simple case is the one where the driving path has finite p-variations with 1⩽p<2, in which case the integrals are interpreted as Young integrals. The prototypal example is given by stochastic differential equations driven by fractional Brownian motion with Hurst index greater than 1/2. Using simple computations, we give the main results regarding this theory – existence, uniqueness, convergence of the Euler scheme, flow property … – which are spread out among several articles.
This paper develops the first class of algorithms that enables unbiased
estimation of steady-state expectations for multidimensional reflected Brownian
motion. In order to explain our ideas we first consider the case of compound
Poisson input. In this case, we analyze the complexity of our procedure as the
dimension of the network increases and show that under certain assumptions the
algorithm has polynomial expected termination time as the dimension increases.
Our methodology includes procedures that are of interest beyond steady-state
simulation and reflected processes. For instance, we use wavelets to construct
a piece-wise linear function that can be guaranteed to be within
distance (deterministic) in the uniform norm to Brownian motion in any compact
time interval.
In this article, we study the numerical approximation of stochastic
differential equations driven by a multidimensional fractional Brownian motion
(fBm) with Hurst parameter greater than 1/3. We introduce an implementable
scheme for these equations, which is based on a second order Taylor expansion,
where the usual Levy area terms are replaced by products of increments of the
driving fBm. The convergence of our scheme is shown by means of a combination
of rough paths techniques and error bounds for the discretisation of the Levy
area terms.
For 0 < y < 2, let Byd be a d-dimensional y-fractional Brownian sheet with index set [0, 1]d and let (ξ k)k≥1 be an independent sequence of standard normal random variables. We prove the existence of continuous functions uk such that almost surely Byd(t) = Σ k=1∞ξkuk(t), t ε [0, 1]d, and (double-struck E sign suptε[0,1] d|Σk=n∞ξku k(t)|2)1/2 ≈ n-y/2(1+ log n) d(y+1)/2-y/2. This order is shown to be optimal. We obtain small-ball estimates for Byd, extending former results in the case y = 1. Our investigations rest upon basic properties of different kinds of s-numbers of operators.
Generating sample paths of stochastic differential equations SDE using the Monte Carlo method finds wide applications in financial engineering. Discretization is a popular approximate approach to generating those paths: it is easy to implement but prone to simulation bias. This paper presents a new simulation scheme to exactly generate samples for SDEs. The key observation is that the law of a general SDE can be decomposed into a product of the law of standard Brownian motion and the law of a doubly stochastic Poisson process. An acceptance-rejection algorithm is devised based on the combination of this decomposition and a localization technique. The numerical results corroborates that the mean-square error of the proposed method is in the order of Ot-1/2, which is superior to the conventional discretization schemes. Furthermore, the proposed method also can generate exact samples for SDE with boundaries which the discretization schemes usually find difficulty in dealing with.
A method for simulating a stationary Gaussian process on a fine rectangular grid in [0, 1] ⊂ℝ is described. It is assumed that the process is stationary with respect to translations of ℝ, but the method does not require the process to be isotropic. As with some other approaches to this simulation problem, our procedure uses discrete Fourier methods and exploits the efficiency of the fast Fourier transform. However, the introduction of a novel feature leads to a procedure that is exact in principle when it can be applied. It is established that sufficient conditions for it to be possible to apply the procedure are (1) the covariance function is summable on ℝ, and (2) a certain spectral density on the d-dimensional torus, which is determined by the covariance function on ℝ, is strictly positive. The procedure can cope with more than 50,000 grid points in many cases, even on a relatively modest computer. An approximate procedure is also proposed to cover cases where it is not feasible to apply the procedure in its exact form.
We consider the power of tests for distinguishing between fractional Gaussian noise and white noise of a first-order autoregressive process. Our tests are based on the beta-optimal principle (Davies, 1969), local optimality and the rescaled range test.
Consider the fractional Brownian motion process , with parameter . Meyer, Sellan and Taqqu have developed several random wavelet representations for , of the form where are Gaussian random variables and where the functions are not random. Based on the results of Kühn and Linde, we say that the approximation of is optimal if as . We show that the random wavelet representations given in Meyer, Sellan and Taqqu are optimal.
Discrete approximations to solutions of stochastic differential equations are
well-known to converge with "strong" rate 1/2. Such rates have played a
key-role in Giles' multilevel Monte Carlo method [Giles, Oper. Res. 2008] which
gives a substantial reduction of the computational effort necessary for the
evaluation of diffusion functionals. In the present article similar results are
established for large classes of rough differential equations driven by
Gaussian processes (including fractional Brownian motion with as
special case).
This paper focuses on simulating fractional Brownian motion (fBm). Despite the availability of several exact simulation methods, attention has been paid to approximate simulation (i.e., the output is approximately fBm), particularly because of possible time savings. In this paper, we study the class of approximate methods that are based on the spectral properties of fBm's stationary incremental process, usually called fractional Gaussian noise (fGn). The main contribution is a proof of asymptotical exactness (in a sense that is made precise) of these spectral methods. Moreover, we establish the connection between the spectral simulation approach and a widely used method, origi-nally proposed by Paxson, that lacked a formal mathematical justification. The insights enable us to evaluate the Paxson method in more detail. It is also shown that spectral simulation is related to the fastest known exact method.
The aim of this communication is to propose some complementary remarks and interpretation on the wavelet-based synthesis technique for fractional Brownian motion proposed by Sellan in 1995. These comments will lead us to propose a fast and efficient pyramidal filter bank-based Mallat-type algorithm, which permits an easy and efficient imple-mentation of this synthesis technique. c 1996 Academic Press, Inc.
This article introduces and analyzes multilevel Monte Carlo schemes for the evaluation of the expectation , where Y=(Yt)t[set membership, variant][0,1] is a solution of a stochastic differential equation driven by a Lévy process. Upper bounds are provided for the worst case error over the class of all path dependent measurable functions f, which are Lipschitz continuous with respect to the supremum norm. In the case where the Blumenthal-Getoor index of the driving process is smaller than one, one obtains convergence rates of order , when the computational cost n tends to infinity. This rate is optimal up to logarithms in the case where Y is itself a Lévy process. Furthermore, an error estimate for Blumenthal-Getoor indices larger than one is included together with results of numerical experiments.
In this paper we show that the Milstein scheme can be used to improve the convergence of the multilevel Monte Carlo method
for scalar stochastic differential equations. Numerical results for Asian, lookback, barrier and digital options demonstrate
that the computational cost to achieve a root-mean-square error of ε is reduced to O(ε-2). This is achieved through a careful construction of the multilevel estimator which computes the difference in expected payoff
when using different numbers of timesteps.
Let B be a fractional Brownian motion with Hurst index H(0,1). Denote by the positive, real zeros of the Bessel function J–H of the first kind of order –H, and let be the positive zeros of J1–H. In this paper we prove the series representation where X1,X2,... and Y1,Y2,... are independent, Gaussian random variables with mean zero and and the constant cH2 is defined by cH2=–1(1+2H) sin H. We show that with probability 1, both random series converge absolutely and uniformly in t[0,1], and we investigate the rate of convergence.
In this paper we show, by using dyadic approximations, the existence of a geometric rough path associated with a fractional
Brownian motion with Hurst parameter greater than 1/4. Using the integral representation of fractional Brownian motions, we
furthermore obtain a Skohorod integral representation of the geometric rough path we constructed. By the results in [Ly1],
a stochastic integration theory may be established for fractional Brownian motions, and strong solutions and a Wong-Zakai
type limit theorem for stochastic differential equations driven by fractional Brownian motions can be deduced accordingly.
The method can actually be applied to a larger class of Gaussian processes with covariance functions satisfying a simple decay
condition.
A new extension of a fractality concept in financial mathematics has been developed. We have introduced a new fractional Langevin-type stochastic differential equation that differs from the standard Langevin equation: (i) by replacing the first-order derivative with respect to time by the fractional derivative of order μ; and (ii) by replacing “white noise” Gaussian stochastic force by the generalized “shot noise”, each pulse of which has a random amplitude with the α-stable Lévy distribution. As an application of the developed fractional non-Gaussian dynamical approach the expression for the probability distribution function (pdf) of the returns has been established. It is shown that the obtained fractional pdf fits well the central part and the tails of the empirical distribution of S&P 500 returns.
We show that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using Monte Carlo path simulations. In the simplest case of a Lipschitz payoff and a Euler discretisation, the computational cost to achieve an accuracy of O(ε) is reduced from O(ε-3) to O(ε-2(logε)2). The analysis is supported, by numerical results showing significant computational savings.
We present an iterative sampling method which delivers upper and lower
bounding processes for the Brownian path. We develop such processes with
particular emphasis on being able to unbiasedly simulate them on a personal
computer. The dominating processes converge almost surely in the supremum and
norms. In particular, the rate of converge in is of the order
, denoting the computing cost.
The a.s. enfolding of the Brownian path can be exploited in Monte Carlo
applications involving Brownian paths whence our algorithm (termed the
-strong algorithm) can deliver unbiased Monte Carlo estimators
over path expectations, overcoming discretisation errors characterising
standard approaches. We will show analytical results from applications of the
-strong algorithm for estimating expectations arising in option
pricing. We will also illustrate that individual steps of the algorithm can be
of separate interest, giving new simulation methods for interesting Brownian
distributions.
Incluye bibliografía e índice
Fractional Brownian motion (fBm) is a centered self-similar Gaussian process with stationary increments, which depends on a parameter H �¸ (0, 1) called the Hurst index. In this note we will survey some facts about the stochastic calculus with respect to fBm using a pathwise approach and the techniques of the Malliavin calculus. Some applications in turbulence and finance will be discussed.
This paper aims to provide a systematic approach to the treatment of differential equations of the type dyt = Si fi(yt) dxti where the driving signal xt is a rough path. Such equations are very common and occur particularly frequently in probability where the driving signal might be a vector valued Brownian motion, semi-martingale or similar process. However, our approach is deterministic, is totally independent of probability and permits much rougher paths than the Brownian paths usually discussed. The results here are strong enough to treat the main probabilistic examples and significantly widen the class of stochastic processes which can be used to drive stochastic differential equations. (For a simple example see [10], [1]). We hope our results will have an influence on infinite dimensional analysis on path spaces, loop groups, etc. as well as in more applied situations. Variable step size algorithms for the numerical integration of stochastic differential equations [8] have been constructed as a consequence of these results.
An abstract model for aggregated connectionless traffic, based on
the fractional Brownian motion, is presented. Insight into the
parameters is obtained by relating the model to an equivalent burst
model. Results on a corresponding storage process are presented. The
buffer occupancy distribution is approximated by a Weibull distribution.
The model is compared with publicly available samples of real Ethernet
traffic. The degree of the short-term predictability of the traffic
model is studied through an exact formula for the conditional variance
of a future value given the past. The applicability and interpretation
of the self-similar model are discussed extensively, and the notion of
ideal free traffic is introduced
Recent network traffic studies argue that network arrival processes are much more faithfully modeled using statistically self-similar processes instead of traditional Poisson processes [LTWW94, PF95]. One difficulty in dealing with selfsimilar models is how to efficiently synthesize traces (sample paths) corresponding to self-similar traffic. We present a fast Fourier transform method for synthesizing approximate self-similar sample paths for one type of self-similar process, Fractional Gaussian Noise, and assess its performance and validity. We find that the method is as fast or faster than existing methods and appears to generate close approximations to true self-similar sample paths. We also discuss issues in using such synthesized sample paths for simulating network traffic, and how an approximation used by our method can dramatically speed up evaluation of Whittle's estimator for H , the Hurst parameter giving the strength of long-range dependence present in a self-similar time ...
Recent measurement studies have shown that the burstiness of packet traffic is associated with long-range correlations that can be efficiently modeled by terms of fractal or self-similar processes, e.g., fractional Brownian motion (FBM). To gain a better understanding of queuing and networkrelated performance issues based on simulations as well as to determine network element performance and capacity characteristics based on load testing, it is essential to be able to accurately and quickly generate long traces from FBM processes. In this paper, we consider an approximate FBM generation method based on the concept of bisection and interpolation, which is an improvement of a much used but inaccurate method known as the random midpoint displacement (RMD) algorithm. We further extend our new algorithm (referred to as RMDmn ) to be able to generate FBM traces without a priori knowledge of the length of the simulation (i.e., on-the-fly generation), instead of being a pure top-down generation ...
. Geostatistical simulations often require the generation of numerous realizations of a stationary Gaussian process over a regularly meshed sample grid## This paper shows that for many important correlation functions in geostatistics, realizations of the associated process over m +1 equispaced points on a line can be produced at the cost of an initial FFT of length 2m with each new realization requiring an additional FFT of the same length. In particular, the paper first notes that if an (m+1)×(m+1) Toeplitz correlation matrix R can be embedded in a nonnegative definite 2M×2M circulant matrix S, exact realizations of the normal multivariate y #N(0,R) can be generated via FFTs of length 2M . Theoretical results are then presented to demonstrate that for many commonly used correlation structures the minimal embedding in which M = m is nonnegative definite. Extensions to simulations of stationary fields in higher dimensions are also provided and illustrated. Key words. geostatistics, ...
A theory of systems of differential equations of the form dyi = ∑jfij(y)dxi, where the driving path x(t) is nondifferentiable, has recently been developed by Lyons. I develop an alternative approach to this theory, using (modified)
Euler approximations, and investigate its applicability to stochastic differential equations driven by Brownian motion. I
also give some other examples showing that the main results are reasonably sharp.
We describe a new, surprisingly simple algorithm, that simulates exact sample paths of a class of stochastic differential equations. It involves rejection sampling and, when applicable, returns the location of the path at a random collection of time instances. The path can then be completed without further reference to the dynamics of the target process.
In this article, we derive the exact rate of convergence of some approximation schemes associated to scalar stochastic differential equations driven by a fractional Brownian motion with Hurst index H. We consider two cases. If H>1/2, the exact rate of convergence of the Euler scheme is determined. We show that the error of the Euler scheme converges almost surely to a random variable, which in particular depends on the Malliavin derivative of the solution. This result extends those contained in J. Complex. 22(4), 459-474, 2006 and C.R. Acad. Sci. Paris, Ser. I 340(8), 611-614, 2005. When 1/6<H<1/2, the exact rate of convergence of the Crank-Nicholson scheme is determined for a particular equation. Here we show convergence in law of the error to a random variable, which depends on the solution of the equation and an independent Gaussian random variable.
The fitting of time-series models. Revue de l'Institut International de Statistique
- J Durbin
J. Durbin. The fitting of time-series models. Revue de l'Institut International de Statistique,
pages 233-244, 1960.
Self-similar traffic generation: the random midpoint displacement algorithm and its properties
- W Lau
- J L Wang
- A Erramilli
- W Willinger
W. Lau, J.L. Wang A. Erramilli, and W. Willinger. Self-similar traffic generation: the random midpoint displacement algorithm and its properties. In Proceedings IEEE International
Conference on Communications ICC'95, volume 1, pages 466-472. IEEE, 1995.
A new approach to unbiased estimation for SDE's
- C Rhee
- P W Glynn
C. Rhee and P.W. Glynn. A new approach to unbiased estimation for SDE's. In Proceedings
of the Winter Simulation Conference, page 17, 2012.
- Z Liu
- J Blanchet
- A B Dieker
- T Mikosch
Z. Liu, J. Blanchet, A.B. Dieker, and T. Mikosch. On optimal exact simulation of max-stable
and related random fields. arXiv preprint arXiv:1609.06001, 2016.