| 1. |
- Bender, Martin
(författare)
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Global fluctuations in general β Dyson’s Brownian motion
- 2008
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Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149 .- 1879-209X. ; 118:6, s. 1022-1042
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Tidskriftsartikel (refereegranskat)abstract
- We consider a system of diffusing particles on the real line in a quadratic external potential and with a logarithmic interaction potential. The empirical measure process is known to converge weakly to a deterministic measure-valued process as the number of particles tends to infinity. Provided the initial fluctuations are small, the rescaled linear statistics of the empirical measure process converge in distribution to a Gaussian limit for sufficiently smooth test functions. For a large class of analytic test functions, we derive explicit general formulae for the mean and covariance in this central limit theorem by analyzing a partial differential equation characterizing the limiting fluctuations.
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| 2. |
- Ekström, Erik
(författare)
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Properties of American option prices
- 2004
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Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149 .- 1879-209X. ; 114:2, s. 265-278
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Tidskriftsartikel (refereegranskat)abstract
- We investigate some properties of American option prices in the setting of time- and level-dependent volatility. The properties under consideration are convexity in the underlying stock price, monotonicity and continuity in the volatility and time decay. Some properties are direct consequences of the corresponding properties of European option prices that are already known, and some follow by writing solutions of different stochastic differential equations as time changes of the same Brownian motion.
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| 3. |
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| 4. |
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| 5. |
- Hult, Henrik, et al.
(författare)
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Extremal behavior of regularly varying stochastic processes
- 2005
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Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149 .- 1879-209X. ; 115:2, s. 249-274
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Tidskriftsartikel (refereegranskat)abstract
- We study a formulation of regular variation for multivariate stochastic processes on the unit interval with sample paths that are almost surely right-continuous with left limits and we provide necessary and sufficient conditions for such stochastic processes to be regularly varying. A version of the Continuous Mapping Theorem is proved that enables the derivation of the tail behavior of rather general mappings of the regularly varying stochastic process. For a wide class of Markov processes with increments satisfying a condition of weak dependence in the tails we obtain simplified sufficient conditions for regular variation. For such processes we show that the possible regular variation limit measures concentrate on step functions with one step, from which we conclude that the extremal behavior of such processes is due to one big jump or an extreme starting point. By combining this result with the Continuous Mapping Theorem, we are able to give explicit results on the tail behavior of various vectors of functionals acting on such processes. Finally, using the Continuous Mapping Theorem we derive the tail behavior of filtered regularly varying Levy processes.
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| 6. |
- Olsson, Jimmy, et al.
(författare)
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Asymptotic properties of particle filter-based maximum likelihood estimators for state space models
- 2008
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Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 1879-209X .- 0304-4149. ; 118:4, s. 649-680
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Tidskriftsartikel (refereegranskat)abstract
- We study the asymptotic performance of approximate maximum likelihood estimators for state space models obtained via sequential Monte Carlo methods. The state space of the latent Markov chain and the parameter space are assumed to be compact. The approximate estimates are computed by, firstly, running possibly dependent particle filters on a fixed grid in the parameter space, yielding a pointwise approximation of the log-likelihood function. Secondly, extensions of this approximation to the whole parameter space are formed by means of piecewise constant functions or B-spline interpolation, and approximate maximum likelihood estimates are obtained through maximization of the resulting functions. In this setting we formulate criteria for how to increase the number of particles and the resolution of the grid in order to produce estimates that are consistent and asymptotically normal.
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| 7. |
- Rubenthaler, Sylvain, et al.
(författare)
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Fast simulated annealing in Rd with an application to maximum likelihood estimation in state-space models
- 2009
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Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149 .- 1879-209X. ; 119:6, s. 1912-1931
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Tidskriftsartikel (refereegranskat)abstract
- We study simulated annealing algorithms to maximise a function psi on a subset of R(d). In classical simulated annealing, given a current state theta(n) in stage n of the algorithm, the probability to accept a proposed state z at which psi is smaller, is exp(-beta(n+1)(psi(z) - psi (theta(n))) where (beta(n)) is the inverse temperature. With the standard logarithmic increase of (beta(n)) the probability P(psi(theta(n)) <= psi(max) - epsilon), with psi(max) the maximal value of psi, then tends to zero at a logarithmic rate as n increases. We examine variations of this scheme in which (beta(n)) is allowed to grow faster, but also consider other functions than the exponential for determining acceptance probabilities. The main result shows that faster rates of convergence can be obtained, both with the exponential and other acceptance functions. We also show how the algorithm may be applied to functions that cannot be computed exactly but only approximated, and give an example of maximising the log-likelihood function for a state-space model.
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| 8. |
- Lundh, Torbjörn, 1965
(författare)
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Percolation Diffusion
- 2001
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Ingår i: Stochastic Processes and their Applications. - 0304-4149. ; 95:2, s. 235-244
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Tidskriftsartikel (refereegranskat)abstract
- Let a Brownian motion in the unit ball be absorbed if it hits a set generated by a radially symmetric Poisson point process. The point set is fattened by putting a ball with a constant hyperbolic radius on each point. When is the probability non-zero that the Brownian motion hits the boundary of the unit ball? That is, manage to avoid all the Poisson balls and percolate diffusively all the way to the boundary. We will show that if the bounded Poisson intensity at a point z is ν(d(0,z)), where d(· ,·) is the hyperbolic metric, then the Brownian motion percolates diffusively if and only if $\nu \in L^1$.
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| 9. |
- Albin, Patrik, 1960, et al.
(författare)
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On the asymptotic behaviour of L\'evy processes, Part I: Subexponential and exponential processes
- 2009
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Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149. ; 119:1, s. 281-304
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Tidskriftsartikel (refereegranskat)abstract
- We study tail probabilities of suprema of L\'evy processes with subexponential or exponential marginal distributions over compact intervals. Several of the processes for which the asymptotics are studied here for the first time have recently become important to model financial time series. Hence our results should be important, for example, in the assessment of financial risk.
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| 10. |
- Andersson, Sofia, et al.
(författare)
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Linear Optimal Prediction and Innovations Representations of Hidden Markov Models
- 2003
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Ingår i: Stochastic Processes and their Applications. - 1879-209X. ; 108:1, s. 131-149
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Tidskriftsartikel (refereegranskat)abstract
- The topic of this paper is linear optimal prediction of hidden Markov models (HMMs) and innovations representations of HMMs. Our interest in these topics primarily arise from subspace estimation methods, which are intrinsically linked to such representations. For HMMs, derivation of innovations representations is complicated by non-minimality of the corresponding state space representations, and requires the solution of algebraic Riccati equations under non-minimality assumptions.
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| 11. |
- Asmussen, Sören, et al.
(författare)
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Large deviations and fast simulation in the presence of boundaries
- 2002
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Ingår i: Stochastic Processes and their Applications. - 1879-209X. ; 102:1, s. 1-23
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Tidskriftsartikel (refereegranskat)abstract
- Let c(x) = inf {t > 0: Q(t) greater than or equal to x} be the time of first overflow of a queueing process 1001 over level x (the buffer size) and Z = P(T(X) less than or equal to T). Assuming that {Q(t)) is the reflected version of a Levy process {X(t)} or a Markov additive process, we study a variety of algorithms for estimating z by simulation when the event {tau(X) less than or equal to T} is rare, and analyse their performance. In particular, we exhibit an estimator using a filtered Monte Carlo argument which is logarithmically efficient whenever an efficient estimator for the probability of overflow within a busy cycle (i.e., for first passage probabilities for the unrestricted netput process) is available, thereby providing a way out of counterexamples in the literature on the scope of the large deviations approach to rare events simulation. We also add a counterexample of this type and give various theoretical results on asymptotic properties of Z=P(tau(x) less than or equal to T), both in the reflected Levy process setting and more generally for regenerative processes in a regime where T is so small that the exponential approximation for T(x) is not a priori valid.
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| 12. |
- Asmussen, Sören, et al.
(författare)
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Russian and American options under exponential phase-type Lévy models
- 2004
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Ingår i: Stochastic Processes and their Applications. - 1879-209X. ; , s. 79-111
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Tidskriftsartikel (refereegranskat)abstract
- Consider the American put and Russian option (Ann. Appl. Probab. 3(1993)603; Theory Probab. Appl. 39 (1994)103; Ann. Appl. Probab. 3(1993)641) with the stock price modeled as an exponential Lévy process. We find an explicit expression for the price in the dense class of Lévy processes with phase-type jumps in both directions. The solution rests on the reduction to the first passage time problem for (reflected) Lévy processes and on an explicit solution of the latter in the phase-type case via martingale stopping and Wiener-Hopf factorization. The same type of approach is also applied to the more general class of regime switching Lévy processes with phase-type jumps.
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| 13. |
- Axelson-Fisk, Marina, 1972, et al.
(författare)
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Biased random walk in a one-dimensional percolation model
- 2009
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Ingår i: Stochastic processes and their Applications. - : Elsevier BV. - 0304-4149. ; 119:10, s. 3395-3415
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Tidskriftsartikel (refereegranskat)abstract
- We consider random walk with a nonzero bias to the right, on the infinite cluster in the following percolation model: take i.i.d.\ bond percolation with retention parameter $p$ on the so-called infinite ladder, and condition on the event of having a bi-infinite path from $-\infty$ to $\infty$. The random walk is shown to be transient, and to have an asymptotic speed to the right which strictly positive or zero depending on whether the bias is below or above a certain critical value which we compute explicitly.
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| 14. |
- den Hollander, F., et al.
(författare)
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Bad configurations for random walk in random scenery and related subshifts
- 2005
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Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149. ; 115:7, s. 1209-1232
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we consider an arbitrary irreducible random walk on ℤd, d ≥ 1, with i.i.d. increments, together with an arbitrary i.i.d. random scenery. Walk and scenery are assumed to be independent. Random walk in random scenery (RWRS) is the random process where time is indexed by ℤ, and at each unit of time both the step taken by the walk and the scenery value at the site that is visited are registered. Bad configurations for RWRS are the discontinuity points of the conditional probability distribution for the configuration at the origin of time given the configuration at all other times. We show that the set of bad configurations is non-empty. We give a complete description of this set and compute its probability under the random scenery measure. Depending on the type of random walk, this probability may be zero or positive. For simple symmetric random walk we get three different types of behavior depending on whether d = 1, 2, d = 3, 4 or d ≥ 5. Our classification is actually valid for a class of subshifts having a certain determinative property, which we call specifiable, of which RWRS is an example. We also consider bad configurations w.r.t. a finite time interval (replacing the origin) and obtain an almost complete generalization of our results. Remarkably, this extension turns out to be somewhat delicate.
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| 15. |
- Israelsson, S
(författare)
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Asymptotic fluctuations of a particle system with singular interaction
- 2001
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Ingår i: STOCHASTIC PROCESSES AND THEIR APPLICATIONS. - : ELSEVIER SCIENCE BV. - 0304-4149. ; 93:1, s. 25-56
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Tidskriftsartikel (refereegranskat)abstract
- A particle system on the real line with singular interaction consisting of electrostatic repulsion and a linear restoring force is considered. The empirical measure process is known to converge weakly in a space of continuous-measure-valued functions. In
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| 16. |
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| 17. |
- Ricci, Valeria, et al.
(författare)
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On the derivation of a linear Boltzmann equation from a periodic lattice gas
- 2004
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Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149. ; 111:2, s. 281-315
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Tidskriftsartikel (refereegranskat)abstract
- We consider the problem of deriving the linear Boltzmann equation from the Lorentz process with hard spheres obstacles. In a suitable limit (the Boltzmann-Grad limit), it has been proved that the linear Boltzmann equation can be obtained when the position of obstacles are Poisson distributed, while the validation fails, also for the "correct" ratio between obstacle size and lattice parameter, when they are distributed on a purely periodic lattice, because of the existence of very long free trajectories. Here we validate the linear Boltzmann equation, in the limit when the scatterer's radius E vanishes, for a family of Lorentz processes such that the obstacles have a random distribution on a lattice and the probability for an obstacle to be on a given lattice site p=E δ/(1-2δ) and the lattice parameter l=E 1/(1-2δ) , 0
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| 18. |
- Rubenthaler, S, et al.
(författare)
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Improved convergence rate for the simulation of stochastic differential equations driven by subordinated Levy processes
- 2003
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Ingår i: Stochastic Processes and their Applications. - 1879-209X. ; 108:1, s. 1-26
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Tidskriftsartikel (refereegranskat)abstract
- We consider the Euler approximation of stochastic differential equations (SDEs) driven by Levy processes in the case where we cannot simulate the increments of the driving process exactly. In some cases, where the driving process Y is a subordinated stable process, i.e., Y = Z(V) with V a subordinator and Z a stable process, we propose an approximation Y by Z(V-n) where V-n is an approximation of V. We then compute the rate of convergence for the approximation of the solution X of an SDE driven by Y using results about the stability of SDEs. (C) 2003 Elsevier B.V. All rights reserved.
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| 19. |
- Wiktorsson, Magnus, et al.
(författare)
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On the simulation of iterated Itô integrals
- 2001
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Ingår i: Stochastic Processes and their Applications. - 1879-209X. ; 91:1, s. 151-168
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Tidskriftsartikel (refereegranskat)abstract
- We consider algorithms for simulation of iterated Itô integrals with application to simulation of stochastic differential equations. The fact that the iterated Itô integral I_{ij}(t_n,t_n+h)=\int_{t_n}^{t_n+h} \int_{t_n}^{s} dW_{i}(u)dW_{j}(s) conditioned on W_i(t_n+h)-W_i(t_n) and W_j(t_n+h)-W_j(t_n), has an infinitely divisible distribution is utilised for the simultaneous simulation of $I_{ij}(t_n,t_n+h)$,W_{i}(t_n+h)-W_{i}(t_n) and W_j(t_n+h)-W_j(t_n). Different simulation methods for the iterated Itô integrals are investigated. We show mean square convergence rates for approximations of shot-noise type and asymptotic normality of the remainder of the approximations. This together with the fact that the conditional distribution of I_{ij}(t_n,t_n+h), apart from an additive constant, is a Gaussian variance mixture is used to achieve an improved convergence rate. This is done by a coupling method for the remainder of the approximation.
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| 20. |
- Wiktorsson, Magnus
(författare)
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Simulation of stochastic integrals with respect to Levy processes of type G
- 2002
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Ingår i: Stochastic Processes and their Applications. - 1879-209X. ; 101:1, s. 113-125
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Tidskriftsartikel (refereegranskat)abstract
- We study the simulation of stochastic processes defined as stochastic integrals with respect to type G Levy processes for the case where it is not possible to simulate the type G process exactly. The type G Levy process as well as the stochastic integral can on compact intervals be represented as an infinite series. In a practical simulation we must truncate this representation. We examine the approximation of the remaining terms with a simpler process to get an approximation of the stochastic integral. We also show that a stochastic time change representation can be used to obtain an approximation of stochastic integrals with respect to type G Levy processes provided that the integrator and the integrand are independent.
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