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Register a new userExact Simulation of the Extrema of Stable Processes
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Jorge Ignacio Gonz\\'alez C\\'azares
Publication date
2018
fields
Mathematical Statistics
and research's language is
English
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We exhibit an exact simulation algorithm for the supremum of a stable process over a finite time interval using dominated coupling from the past (DCFTP). We establish a novel perpetuity equation for the supremum (via the representation of the concave majorants of Levy processes) and apply it to construct a Markov chain in the DCFTP algorithm. We prove that the number of steps taken backwards in time before the coalescence is detected is finite. We analyse numerically the performance of the algorithm (the code, written in Julia 1.0, is available on GitHub).
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The simulation of exit times for diffusion processes is a challenging task since it concerns many applications in different fields like mathematical finance, neuroscience, reliability... The usual procedure is to use discretiza-tion schemes which unfortunately introduce some error in the target distribution. Our aim is to present a new algorithm which simulates exactly the exit time for one-dimensional diffusions. This acceptance-rejection algorithm requires to simulate exactly the exit time of the Brownian motion on one side and the Brownian position at a given time, constrained not to have exit before, on the other side. Crucial tools in this study are the Girsanov transformation, the convergent series method for the simulation of random variables and the classical rejection sampling. The efficiency of the method is described through theoretical results and numerical examples.
Using a generalization of the skew-product representation of planar Brownian motion and the analogue of Spitzers celebrated asymptotic Theorem for stable processes due to Bertoin and Werner, for which we provide a new easy proof, we obtain some limit Theorems for the exit time from a cone of stable processes of index $alphain(0,2)$. We also study the case $trightarrow0$ and we prove some Laws of the Iterated Logarithm (LIL) for the (well-defined) winding process associated to our planar stable process.
We consider a new family of $R^d$-valued L{e}vy processes that we call Lamperti stable. One of the advantages of this class is that the law of many related functionals can be computed explicitely (see for instance cite{cc}, cite{ckp}, cite{kp} and cite{pp}). This family of processes shares many properties with the tempered stable and the layered stable processes, defined in Rosinski cite{ro} and Houdre and Kawai cite{hok} respectively, for instance their short and long time behaviour. Additionally, in the real valued case we find a series representation which is used for sample paths simulation. In this work we find general properties of this class and we also provide many examples, some of which appear in recent literature.
Eigenproblems frequently arise in theory and applications of stochastic processes, but only a few have explicit solutions. Those which do, are usually solved by reduction to the generalized Sturm--Liouville theory for differential operators. This includes the Brownian motion and a whole class of processes, which derive from it by means of linear transformations. The more general eigenproblem for the {em fractional} Brownian motion (f.B.m.) is not solvable in closed form, but the exact asymptotics of its eigenvalues and eigenfunctions can be obtained, using a method based on analytic properties of the Laplace transform. In this paper we consider two processes closely related to the f.B.m.: the fractional Ornstein--Uhlenbeck process and the integrated fractional Brownian motion. While both derive from the f.B.m. by simple linear transformations, the corresponding eigenproblems turn out to be much more complex and their asymptotic structure exhibits new effects.
Since diffusion processes arise in so many different fields, efficient tech-nics for the simulation of sample paths, like discretization schemes, represent crucial tools in applied probability. Such methods permit to obtain approximations of the first-passage times as a by-product. For efficiency reasons, it is particularly challenging to simulate directly this hitting time by avoiding to construct the whole paths. In the Brownian case, the distribution of the first-passage time is explicitly known and can be easily used for simulation purposes. The authors introduce a new rejection sampling algorithm which permits to perform an exact simulation of the first-passage time for general one-dimensional diffusion processes. The efficiency of the method, which is essentially based on Girsanovs transformation , is described through theoretical results and numerical examples.
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