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1.	Andersson, Patrik, et al. (författare) Second order probabilistic parametrix method for unbiased simulation of stochastic differential equations 2020 Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149 .- 1879-209X. ; 130:9, s. 5543-5574 Tidskriftsartikel (refereegranskat)abstract In this article, following the paradigm of bias-variance trade-off philosophy, we derive parametrix expansions of order two, based on the Euler-Maruyama scheme with random partitions, for the purpose of constructing an unbiased simulation method for multidimensional stochastic differential equations. These formulas lead to Monte Carlo simulation methods which can be easily parallelized. The second order method proposed here requires further regularity of coefficients in comparison with the first order method but achieves finite moments even when Poisson sampling is used for the partitions, in contrast to Andersson and Kohatsu-Higa (2017). Moreover, using an exponential scaling technique one achieves an unbiased simulation method which resembles a space importance sampling technique which significantly improves the efficiency of the proposed method. A hint of how to derive higher order expansions is also presented.
2.	Bender, Martin (författare) Global fluctuations in general β Dyson’s Brownian motion 2008 Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149 .- 1879-209X. ; 118:6, s. 1022-1042 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.
3.	Björklund, Michael (författare) Ergodic theorems for random clusters 2010 Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149 .- 1879-209X. ; 120:3, s. 296-305 Tidskriftsartikel (refereegranskat)abstract We prove pointwise ergodic theorems for a class of random measures which Occurs in Laplacian growth models, most notably in the anisotropic Hastings-Levitov random cluster models. The proofs are based oil the theory Of quasi-orthogonal functions and uniform Wintner-Wintner theorems.
4.	Bodnariu, Andi, et al. (författare) A controller-stopper-game with hidden controller type 2024 Ingår i: Stochastic Processes and their Applications. - 0304-4149 .- 1879-209X. ; 173 Tidskriftsartikel (refereegranskat)abstract We consider a continuous time stochastic dynamic game between a stopper (the owner of an asset) and a controller (the manager ) who is either effective or non -effective. An effective manager can exert high or low effort which corresponds to a high or a low positive drift for the accumulated income of the owner with random noise in terms of Brownian motion. The manager earns a salary until the owner stops the game. A non -effective manager cannot act but receives a salary. We find a threshold (Nash) equilibrium using stochastic filtering methods in a weak formulation.
5.	Budhiraja, Amarjit, et al. (författare) Quasistationary distributions and ergodic control problems 2022 Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149 .- 1879-209X. ; 145, s. 143-164 Tidskriftsartikel (refereegranskat)abstract We introduce and study the basic properties of two ergodic stochastic control problems associated with the quasistationary distribution (QSD) of a diffusion process X relative to a bounded domain. The two problems are in some sense dual, with one defined in terms of the generator associated with X and the other in terms of its adjoint. Besides proving wellposedness of the associated Hamilton-Jacobi- Bellman equations, we describe how they can be used to characterize important properties of the QSD. Of particular note is that the QSD itself can be identified, up to normalization, in terms of the cost potential of the control problem associated with the adjoint.
6.	Casanova, Adrian Gonzalez, et al. (författare) An individual-based model for the Lenski experiment, and the deceleration of the relative fitness 2016 Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149 .- 1879-209X. ; 126:8, s. 2211-2252 Tidskriftsartikel (refereegranskat)abstract The Lenski experiment investigates the long-term evolution of bacterial populations. In this paper we present an individual-based probabilistic model that captures essential features of the experimental design, and whose mechanism does not include epistasis in the continuous-time (intraday) part of the model, but leads to an epistatic effect in the discrete-time (interday) part. We prove that under some assumptions excluding clonal interference, the rescaled relative fitness process converges in the large population limit to a power law function, similar to the one obtained by Wiser et al. (2013), there attributed to effects of clonal interference and epistasis.
7.	Christensen, Sören, et al. (författare) On time-inconsistent stopping problems and mixed strategy stopping times 2020 Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149 .- 1879-209X. ; 130:5, s. 2886-2917 Tidskriftsartikel (refereegranskat)abstract A game-theoretic framework for time-inconsistent stopping problems where the time-inconsistency is due to the consideration of a non-linear function of an expected reward is developed. A class of mixed strategy stopping times that allows the agents in the game to jointly choose the intensity function of a Cox process is introduced and motivated. A subgame perfect Nash equilibrium is defined. The equilibrium is characterized and other necessary and sufficient equilibrium conditions including a smooth fit result are proved. Existence and uniqueness are investigated. A mean-variance and a variance problem are studied. The state process is a general one-dimensional Ito diffusion.
8.	Crisan, D., et al. (författare) Generalised particle filters with Gaussian mixtures 2015 Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149 .- 1879-209X. ; 125:7, s. 2643-2673 Tidskriftsartikel (refereegranskat)abstract Stochastic filtering is defined as the estimation of a partially observed dynamical system. Approximating the solution of the filtering problem with Gaussian mixtures has been a very popular method since the 1970s. Despite nearly fifty years of development, the existing work is based on the success of the numerical implementation and is not theoretically justified. This paper fills this gap and contains a rigorous analysis of a new Gaussian mixture approximation to the solution of the filtering problem. We deduce the L-2-convergence rate for the approximating system and show some numerical examples to test the new algorithm.
9.	Dareiotis, Konstantinos, et al. (författare) Density symmetries for a class of 2-D diffusions with applications to finance 2019 Ingår i: Stochastic Processes and their Applications. - : ELSEVIER SCIENCE BV. - 0304-4149 .- 1879-209X. ; 129:2, s. 452-472 Tidskriftsartikel (refereegranskat)abstract We study densities of two-dimensional diffusion processes with one non-negative component. For such diffusions, the density may explode at the boundary, thus making a precise specification of the boundary condition in the corresponding forward Kolmogorov equation problematic. We overcome this by extending a classical symmetry result for densities of one-dimensional diffusions to our case, thereby reducing the study of forward equations with exploding boundary data to the study of a related backward equation with non-exploding boundary data. We also discuss applications of this symmetry for option pricing in stochastic volatility models and in stochastic short rate models. (C) 2018 Elsevier B.V. All rights reserved.
10.	Dareiotis, Konstantinos, et al. (författare) Finite difference schemes for linear stochastic integro-differential equations 2016 Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149 .- 1879-209X. ; 126:10, s. 3202-3234 Tidskriftsartikel (refereegranskat)abstract We study the rate of convergence of an explicit and an implicit explicit finite difference scheme for linear stochastic integro-differential equations of parabolic type arising in non-linear filtering of jump diffusion processes. We show that the rate is of order one in space and order one-half in time.
11.	De Angelis, Tiziano, et al. (författare) Playing with ghosts in a Dynkin game 2020 Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149 .- 1879-209X. ; 130:10, s. 6133-6156 Tidskriftsartikel (refereegranskat)abstract We study a class of two-player optimal stopping games (Dynkin games) of preemption type, with uncertainty about the existence of competitors. The set-up is well-suited to model, for example, real options in the context of investors who do not want to publicly reveal their interest in a certain business opportunity. We show that if the underlying process is a R-d-valued, continuous, strong Markov process, and the stopping payoff is a continuous function (with mild integrability properties) there exists a Nash equilibrium in randomised stopping times for the game. Moreover, the equilibrium strategies and the expected payoffs of the two players are computed explicitly in terms of the corresponding one-player game. To the best of our knowledge this is the first paper to address this version of Dynkin games.
12.	Djehiche, Boualem, 1962-, et al. (författare) Optimal portfolio choice with path dependent benchmarked labor income : A mean field model 2022 Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149 .- 1879-209X. ; 145, s. 48-85 Tidskriftsartikel (refereegranskat)abstract We consider the life-cycle optimal portfolio choice problem faced by an agent receiving labor income and allocating her wealth to risky assets and a riskless bond subject to a borrowing constraint. In this paper, to reflect a realistic economic setting, we propose a model where the dynamics of the labor income has two main features. First, labor income adjusts slowly to financial market shocks, a feature already considered in Biffis et al. (2015). Second, the labor income gamma i of an agent i is benchmarked against the labor incomes of a population gamma(n) := (gamma(1),gamma(2), ...,gamma(n)) of n agents with comparable tasks and/or ranks. This last feature has not been considered yet in the literature and is faced taking the limit when n -> +infinity so that the problem falls into the family of optimal control of infinite-dimensional McKean-Vlasov Dynamics, which is a completely new and challenging research field. We study the problem in a simplified case where, adding a suitable new variable, we are able to find explicitly the solution of the associated HJB equation and find the optimal feedback controls. The techniques are a careful and nontrivial extension of the ones introduced in the previous papers of Biffis et al. (2015, 0000). (C) 2021 Elsevier B.V. All rights reserved.
13.	Douc, Randal, et al. (författare) Posterior consistency for partially observed Markov models 2020 Ingår i: Stochastic Processes and their Applications. - : ELSEVIER. - 0304-4149 .- 1879-209X. ; 132:2, s. 733-759 Tidskriftsartikel (refereegranskat)abstract We establish the posterior consistency for parametric, partially observed, fully dominated Markov models. The prior is assumed to assign positive probability to all neighborhoods of the true parameter, for a distance induced by the expected Kullback-Leibler divergence between the parametric family members' Markov transition densities. This assumption is easily checked in general. In addition, we show that the posterior consistency is implied by the consistency of the maximum likelihood estimator. The result is extended to possibly improper priors and non-stationary observations. Finally, we check our assumptions on a linear Gaussian model and a well-known stochastic volatility model.
14.	Dufitinema, Josephine, et al. (författare) Long-range dependent completely correlated mixed fractional Brownian motion 2024 Ingår i: Stochastic Processes and their Applications. - : ELSEVIER. - 0304-4149 .- 1879-209X. ; 170 Tidskriftsartikel (refereegranskat)abstract In this paper we introduce the long-range dependent completely correlated mixed fractional Brownian motion (ccmfBm). This is a process that is driven by a mixture of Brownian motion (Bm) and a long-range dependent completely correlated fractional Brownian motion (fBm, ccfBm) that is constructed from the Brownian motion via the Molchan-Golosov representation. Thus, there is a single Bm driving the mixed process. In the short time-scales the ccmfBm behaves like the Bm (it has Brownian Holder index and quadratic variation). However, in the long time-scales it behaves like the fBm (it has long-range dependence governed by the fBms Hurst index). We provide a transfer principle for the ccmfBm and use it to construct the Cameron-Martin-Girsanov-Hitsuda theorem and prediction formulas. Finally, we illustrate the ccmfBm by simulations.
15.	Ekström, Erik, 1977-, et al. (författare) A detection problem with a monotone observation rate 2024 Ingår i: Stochastic Processes and their Applications. - : Elsevier. - 0304-4149 .- 1879-209X. ; 172 Tidskriftsartikel (refereegranskat)abstract We study a quickest detection problem where the observation rate of the underlying process can be increased at any time for higher precision, but at an observation cost that grows linearly in the observation rate. This leads to a problem of combined control-and-stopping with incomplete information, with a two-dimensional sufficient statistic comprised of the current observation rate together with the conditional probability that disorder has already happened. The problem is shown to have a semi-explicit solution, where for some parameter values it is too costly to exert control at all, whereas for other parameter values the optimal strategy is to increase the observation rate in such a way that the sufficient statistic reflects at a certain boundary until the optimal stopping time. In both cases we fully characterise the optimal strategy with the help of appropriate smooth fit conditions.
16.	Ekström, Erik, 1977-, et al. (författare) Bayesian sequential least-squares estimation for the drift of a Wiener process 2022 Ingår i: Stochastic Processes and their Applications. - : Elsevier. - 0304-4149 .- 1879-209X. ; 145, s. 335-352 Tidskriftsartikel (refereegranskat)abstract Given a Wiener process with unknown and unobservable drift, we try to estimate this drift as effectively but also as quickly as possible, in the presence of a quadratic penalty for the estimation error and of a fixed, positive cost per unit of observation time. In a Bayesian framework, where the unobservable drift is assumed to have a known "prior " distribution, this question reduces to choosing judiciously a stopping time for an appropriate diffusion process in natural scale. We establish structural properties of the solution for the corresponding problem of optimal stopping. In particular, we show that, regardless of the prior distribution, the continuation region is monotonically shrinking in time. Moreover, we provide conditions on the prior distribution that guarantee a one-sided stopping region. Lastly, some concrete prior distributions are studied to illustrate the theoretical results.
17.	Ekström, Erik, 1977-, et al. (författare) Optimal stopping of a Brownian bridge with an unknown pinning point 2020 Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149 .- 1879-209X. ; 130:2, s. 806-823 Tidskriftsartikel (refereegranskat)abstract The problem of stopping a Brownian bridge with an unknown pinning point to maximise the expected value at the stopping time is studied. A few general properties, such as continuity and various bounds of the value function, are established. However, structural properties of the optimal stopping region are shown to crucially depend on the prior, and we provide a general condition for a one-sided stopping region. Moreover, a detailed analysis is conducted in the cases of the two-point and the mixed Gaussian priors, revealing a rich structure present in the problem.
18.	Ekström, Erik (författare) Properties of American option prices 2004 Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149 .- 1879-209X. ; 114:2, s. 265-278 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.
19.	Fan, Ming (författare) Some inequalities for p-variation of martingales 2004 Ingår i: Stochastic Processes and their Applications. - : Elsevier. - 0304-4149 .- 1879-209X. ; :109, s. 189-201 Tidskriftsartikel (refereegranskat)
20.	Fleermann, Michael, et al. (författare) Large sample covariance matrices of Gaussian observations with uniform correlation decay 2023 Ingår i: Stochastic Processes and their Applications. - 0304-4149 .- 1879-209X. ; 162, s. 456-480 Tidskriftsartikel (refereegranskat)abstract We derive the Marchenko–Pastur (MP) law for sample covariance matrices of the form , where X is a p × n data matrix and p/n → y ∈ (0,∞) as n, p → ∞. We assume the data in X stems from a correlated joint normal distribution. In particular, the correlation acts both across rows and across columns of X, and we do not assume a specific correlation structure, such as separable dependencies. Instead, we assume that correlations converge uniformly to zero at a speed of an/n, where an may grow mildly to infinity. We employ the method of moments tightly: We identify the exact condition on the growth of an which will guarantee that the moments of the empirical spectral distributions (ESDs) converge to the MP moments. If the condition is not met, we can construct an ensemble for which all but finitely many moments of the ESDs diverge. We also investigate the operator norm of Vn under a uniform correlation bound of C/nδ, where C, δ > 0 are fixed, and observe a phase transition at δ = 1. In particular, convergence of the operator norm to the maximum of the support of the MP distribution can only be guaranteed if δ > 1. The analysis leads to an example for which the MP law holds almost surely, but the operator norm remains stochastic in the limit, and we provide its exact limiting distribution.
21.	Gusakova, Anna, et al. (författare) The volume of random simplices from elliptical distributions in high dimension 2023 Ingår i: Stochastic Processes and their Applications. - 0304-4149 .- 1879-209X. ; 164, s. 357-382 Tidskriftsartikel (refereegranskat)abstract Random simplices and more general random convex bodies of dimension p in with p ≤ n are considered, which are generated by random vectors having an elliptical distribution. In the high-dimensional regime, that is, if p → ∞ and n → ∞ in such a way that p/n → γ ∈ (0, 1), a central and a stable limit theorem for the logarithmic volume of random simplices and random convex bodies is shown. The result follows from a related central limit theorem for the log-determinant of p × n random matrices whose rows are copies of a random vector with an elliptical distribution, which is established as well.
22.	Hult, Henrik, 1975- (författare) Approximating some Volterra type stochastic integrals with applications to parameter estimation 2003 Ingår i: Stochastic Processes and their Applications. - 0304-4149 .- 1879-209X. ; 105:1, s. 1-32 Tidskriftsartikel (refereegranskat)
23.	Hult, Henrik, et al. (författare) Extremal behavior of regularly varying stochastic processes 2005 Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149 .- 1879-209X. ; 115:2, s. 249-274 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.
24.	Hult, Henrik, 1975-, et al. (författare) Large deviations for weighted empirical measures arising in importance sampling 2016 Ingår i: Stochastic Processes and their Applications. - : Elsevier. - 0304-4149 .- 1879-209X. ; 126:1 Tidskriftsartikel (refereegranskat)abstract Importance sampling is a popular method for efficient computation of various properties of a distribution such as probabilities, expectations, quantiles etc. The output of an importance sampling algorithm can be represented as a weighted empirical measure, where the weights are given by the likelihood ratio between the original distribution and the sampling distribution. In this paper the efficiency of an importance sampling algorithm is studied by means of large deviations for the weighted empirical measure. The main result, which is stated as a Laplace principle for the weighted empirical measure arising in importance sampling, can be viewed as a weighted version of Sanov's theorem. The main theorem is applied to quantify the performance of an importance sampling algorithm over a collection of subsets of a given target set as well as quantile estimates. The proof of the main theorem relies on the weak convergence approach to large deviations developed by Dupuis and Ellis.
25.	Ibragimov, Ildar, et al. (författare) Some extensions of linear approximation and prediction problems for stationary processes 2019 Ingår i: Stochastic Processes and their Applications. - : ELSEVIER SCIENCE BV. - 0304-4149 .- 1879-209X. ; 129:8, s. 2758-2782 Tidskriftsartikel (refereegranskat)abstract Let (B(t))(t is an element of Theta) with Theta = Z or Theta = R be a wide sense stationary process with discrete or continuous time. The classical linear prediction problem consists of finding an element in span{B(s), s amp;lt;= t} providing the best possible mean square approximation to the variable B(tau) with tau amp;gt; t. In this article we investigate this and some other similar problems where, in addition to prediction quality, optimization takes into account other features of the objects we search for. One of the most motivating examples of this kind is an approximation of a stationary process B by a stationary differentiable process X taking into account the kinetic energy that X spends in its approximation efforts. (C) 2018 Elsevier B.V. All rights reserved.
26.	Janson, Svante, 1955-, et al. (författare) Sesqui-type branching processes 2018 Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149 .- 1879-209X. ; 128:11, s. 3628-3655 Tidskriftsartikel (refereegranskat)abstract We consider branching processes consisting of particles (individuals) of two types (type L and type S) in which only particles of type L have offspring, proving estimates for the survival probability and the (tail of) the distribution of the total number of particles. Such processes are in some sense closer to single than to multi-type branching processes. Nonetheless, the second, barren, type complicates the analysis significantly. The results proved here (about point and survival probabilities) are a key ingredient in the analysis of bounded-size Achlioptas processes in a recent paper by the last two authors.
27.	Kaj, Ingemar (författare) Time reversal and stationarity of infinite-dimensional Markov birth-and-death processes. 1988 Ingår i: Stochastic Processes and their Applications. - 0304-4149 .- 1879-209X. ; 29, s. 1-50 Tidskriftsartikel (refereegranskat)
28.	Kavallaris, Nikos I., et al. (författare) Finite-time blow-up of a non-local stochastic parabolic problem 2020 Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149 .- 1879-209X. ; 130:9, s. 5605-5635 Tidskriftsartikel (refereegranskat)abstract The main aim of the current work is the study of the conditions under which (finite-time) blow-up of a non-local stochastic parabolic problem occurs. We first establish the existence and uniqueness of the local-in-time weak solution for such problem. The first part of the manuscript deals with the investigation of the conditions which guarantee the occurrence of noise-induced blow-up. In the second part we first prove the C1-spatial regularity of the solution. Then, based on this regularity result, and using a strong positivity result we derive, for first in the literature of SPDEs, a Hopf’s type boundary value point lemma. The preceding results together with Kaplan’s eigenfunction method are then employed to provide a (non-local) drift term induced blow-up result. In the last part of the paper, we present a method which provides an upper bound of the probability of (non-local) drift term induced blow-up.
29.	Lifshits, Mikhail, et al. (författare) Energy of taut strings accompanying Wiener process 2015 Ingår i: Stochastic Processes and their Applications. - : Elsevier. - 0304-4149 .- 1879-209X. ; 125:2, s. 401-427 Tidskriftsartikel (refereegranskat)abstract Let W be a Wiener process. For r greater than 0 and T greater than 0 let I-W (T, r)(2) denote the minimal value of the energy integral(T)(0) h(t)(2)dt taken among all absolutely continuous functions h(.) defined on [0, T], starting at zero and satisfying W(t) - r less than= h(t) less than= W(t) + r, 0 less than= t less than= T. The function minimizing energy is a taut string, a classical object well known in Variational Calculus, in Mathematical Statistics, and in a broad range of applications. We show that there exists a constant C E (0, infinity) such that for any q greater than 0 r/T-1/2 I-W (T, r) -greater than(Lq) C, as r/T-1/2 -greater than 0, and for any fixed r greater than 0, r/(TIW)-I-1/2 (T, r)-greater than(a.s.) C, as T -greater than infinity. Although precise value of C remains unknown, we give various theoretical bounds for it, as well as rather precise results of computer simulation. While the taut string clearly depends on entire trajectory of W, we also consider an adaptive version of the problem by giving a construction (called Markovian pursuit) of a random function h(t) based only on the values W(s), s less than= t, and having minimal asymptotic energy. The solution, i.e. an optimal pursuit strategy, turns out to be related with a classical minimization problem for Fisher information on the bounded interval.
30.	Lundström, Niklas, 1980-, et al. (författare) Stochastic and partial differential equations on non-smooth time-dependent domains 2019 Ingår i: Stochastic Processes and their Applications. - : Elsevier. - 0304-4149 .- 1879-209X. ; 129:4, s. 1097-1131 Tidskriftsartikel (refereegranskat)abstract In this article, we consider non-smooth time-dependent domains whose boundary is W1,p in time and single-valued, smoothly varying directions of reflection at the boundary. In this setting, we first prove existence and uniqueness of strong solutions to stochastic differential equations with oblique reflection. Secondly, we prove, using the theory of viscosity solutions, a comparison principle for fully nonlinear second-order parabolic partial differential equations with oblique derivative boundary conditions. As a consequence, we obtain uniqueness, and, by barrier construction and Perron’s method, we also conclude existence of viscosity solutions. Our results generalize two articles by Dupuis and Ishii to time-dependent domains.
31.	Mikulevicius, Remigijus, et al. (författare) On the Rate of Convergence of Weak Euler Approximation for Nondegenerate SDEs Driven by Levy Processes 2011 Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149 .- 1879-209X. ; 121:8, s. 1720-1748 Tidskriftsartikel (refereegranskat)
32.	Milinanni, Federica, 1996-, et al. (författare) A large deviation principle for the empirical measures of Metropolis–Hastings chains 2024 Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149 .- 1879-209X. ; 170 Tidskriftsartikel (refereegranskat)abstract To sample from a given target distribution, Markov chain Monte Carlo (MCMC) sampling relies on constructing an ergodic Markov chain with the target distribution as its invariant measure. For any MCMC method, an important question is how to evaluate its efficiency. One approach is to consider the associated empirical measure and how fast it converges to the stationary distribution of the underlying Markov process. Recently, this question has been considered from the perspective of large deviation theory, for different types of MCMC methods, including, e.g., non-reversible Metropolis–Hastings on a finite state space, non-reversible Langevin samplers, the zig-zag sampler, and parallel tempering. This approach, based on large deviations, has proven successful in analysing existing methods and designing new, efficient ones. However, for the Metropolis–Hastings algorithm on more general state spaces, the workhorse of MCMC sampling, the same techniques have not been available for analysing performance, as the underlying Markov chain dynamics violate the conditions used to prove existing large deviation results for empirical measures of a Markov chain. This also extends to methods built on the same idea as Metropolis–Hastings, such as the Metropolis-Adjusted Langevin Method or ABC-MCMC. In this paper, we take the first steps towards such a large-deviations based analysis of Metropolis–Hastings-like methods, by proving a large deviation principle for the empirical measures of Metropolis–Hastings chains. In addition, we also characterize the rate function and its properties in terms of the acceptance- and rejection-part of the Metropolis–Hastings dynamics.
33.	Noble, John, 1966- (författare) Time homogeneous diffusions with a given marginal at a deterministic time 2013 Ingår i: Stochastic Processes and their Applications. - : Elsevier. - 0304-4149 .- 1879-209X. ; 123:3, s. 675-718 Tidskriftsartikel (refereegranskat)abstract In this article, it is proved that for any probability law μ over R with finite first moment and a given deterministic time t>0, there exists a gap diffusion with law μ at the prescribed time t.The method starts by constructing a discrete time process X on a finite state space, where Xτ has law μ, for a geometric time τ, independent of the diffusion. This argument is developed, using a fixed point theorem, to give conditions for the existence of a process with prescribed law when stopped at an independent time with negative binomial distribution. Reducing the time mesh gives a continuous time diffusion with prescribed law for τ with Gamma distribution. Keeping E[τ]=t fixed, the parameters of the Gamma distribution are altered, giving the prescribed law for the deterministic time. An approximating sequence establishes the result for arbitrary probability measure over R.
34.	Olsson, Jimmy, et al. (författare) Asymptotic properties of particle filter-based maximum likelihood estimators for state space models 2008 Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 1879-209X .- 0304-4149. ; 118:4, s. 649-680 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.
35.	Perninge, Magnus, 1980- (författare) Optimal stopping of BSDEs with constrained jumps and related zero-sum games 2024 Ingår i: Stochastic Processes and their Applications. - : Elsevier. - 0304-4149 .- 1879-209X. ; 173 Tidskriftsartikel (refereegranskat)abstract In this paper, we introduce a non-linear Snell envelope which at each time represents the maximal value that can be achieved by stopping a BSDE with constrained jumps. We establish the existence of the Snell envelope by employing a penalization technique and the primary challenge we encounter is demonstrating the regularity of the limit for the scheme. Additionally, we relate the Snell envelope to a finite horizon, zero-sum stochastic differential game, where one player controls a path-dependent stochastic system by invoking impulses, while the opponent is given the opportunity to stop the game prematurely. Importantly, by developing new techniques within the realm of control randomization, we demonstrate that the value of the game exists and is precisely characterized by our non-linear Snell envelope.
36.	Privault, Nicolas, et al. (författare) Large deviations for Bernstein bridges 2016 Ingår i: Stochastic Processes and their Applications. - : Elsevier. - 0304-4149 .- 1879-209X. ; 126:5, s. 1285-1305 Tidskriftsartikel (refereegranskat)abstract Bernstein processes over a finite time interval are simultaneously forward and backward Markov processes with arbitrarily fixed initial and terminal probability distributions. In this paper, a large deviation principle is proved for a family of Bernstein processes (depending on a small parameter ħ which is called the Planck constant) arising naturally in Euclidean quantum physics. The method consists in nontrivial Girsanov transformations of integral forms, suitable equivalence forms for large deviations and the (local and global) estimates on the parabolic kernel of the Schrödinger operator.
37.	Rubenthaler, Sylvain, et al. (författare) Fast simulated annealing in Rd with an application to maximum likelihood estimation in state-space models 2009 Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149 .- 1879-209X. ; 119:6, s. 1912-1931 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.
38.	van den Berg, J., et al. (författare) Sharpness versus robustness of the percolation transition in 2d contact processes 2015 Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149 .- 1879-209X. ; 125:2, s. 513-537 Tidskriftsartikel (refereegranskat)abstract We study versions of the contact process with three states, and with infections occurring at a rate depending on the overall infection density. Motivated by a model described in Kefi et al. (2007) for vegetation patterns in arid landscapes, we focus on percolation under invariant measures of such processes. We prove that the percolation transition is sharp (for one of our models this requires a reasonable assumption). This is shown to contradict a form of 'robust critical behaviour' with power law cluster size distribution for a range of parameter values, as suggested in Kefi et al. (2007).
39.	Lundh, Torbjörn, 1965 (författare) Percolation Diffusion 2001 Ingår i: Stochastic Processes and their Applications. - 0304-4149. ; 95:2, s. 235-244 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 $\nu \in L^1$.
40.	Agapiou, Sergios, et al. (författare) Posterior contraction rates for the Bayesian approach to linear ill-posed inverse problems 2013 Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149. ; 123:10, s. 3828-3860 Tidskriftsartikel (refereegranskat)abstract We consider a Bayesian nonparametric approach to a family of linear inverse problems in a separable Hilbert space setting with Gaussian noise. We assume Gaussian priors, which are conjugate to the model, and present a method of identifying the posterior using its precision operator. Working with the unbounded precision operator enables us to use partial differential equations (PDE) methodology to obtain rates of contraction of the posterior distribution to a Dirac measure centered on the true solution. Our methods assume a relatively weak relation between the prior covariance, noise covariance and forward operator, allowing for a wide range of applications.
41.	Albin, Patrik, 1960, et al. (författare) On the asymptotic behaviour of L\'evy processes, Part I: Subexponential and exponential processes 2009 Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149. ; 119:1, s. 281-304 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.
42.	Andersson, Sofia, et al. (författare) Linear Optimal Prediction and Innovations Representations of Hidden Markov Models 2003 Ingår i: Stochastic Processes and their Applications. - 1879-209X. ; 108:1, s. 131-149 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.
43.	Asmussen, Sören, et al. (författare) Large deviations and fast simulation in the presence of boundaries 2002 Ingår i: Stochastic Processes and their Applications. - 1879-209X. ; 102:1, s. 1-23 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.
44.	Asmussen, Sören, et al. (författare) Russian and American options under exponential phase-type Lévy models 2004 Ingår i: Stochastic Processes and their Applications. - 1879-209X. ; , s. 79-111 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.
45.	Axelson-Fisk, Marina, 1972, et al. (författare) Biased random walk in a one-dimensional percolation model 2009 Ingår i: Stochastic processes and their Applications. - : Elsevier BV. - 0304-4149. ; 119:10, s. 3395-3415 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.
46.	Barth, A., et al. (författare) Lp and almost sure convergence of a Milstein scheme for stochastic partial differential equations 2013 Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149. ; 123:5, s. 1563-1587 Tidskriftsartikel (refereegranskat)abstract In this paper, Lp convergence and almost sure convergence of the Milstein approximation of a partial differential equation of advection-diffusion type driven by a multiplicative continuous martingale is proven. The (semidiscrete) approximation in space is a projection onto a finite dimensional function space. The considered space approximation has to have an order of convergence fitting to the order of convergence of the Milstein approximation and the regularity of the solution. The approximation of the driving noise process is realized by the truncation of the Karhunen-Loève expansion of the driving noise according to the overall order of convergence. Convergence results in Lp and almost sure convergence bounds for the semidiscrete approximation as well as for the fully discrete approximation are provided. © 2013 Elsevier B.V. All rights reserved.
47.	Christensen, Sören, 1982, et al. (författare) Convergence of switching diffusions 2015 Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149. ; 125:9, s. 3623-3635 Tidskriftsartikel (refereegranskat)abstract © 2015 Elsevier B.V. This paper studies the asymptotic behavior of processes with switching. More precisely, the stability under fast switching for diffusion processes and discrete state space Markovian processes is considered. The proofs are based on semimartingale techniques, so that no Markovian assumption for the modulating process is needed.
48.	Christensen, Sören, 1982 (författare) On the solution of general impulse control problems using superharmonic functions 2014 Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149. ; 124:1, s. 709-729 Tidskriftsartikel (refereegranskat)abstract In this paper, a characterization of the solution of impulse control problems in terms of superharmonic functions is given. In a general Markovian framework, the value function of the impulse control problem is shown to be the minimal function in a convex set of superharmonic functions. This characterization also leads to optimal impulse control strategies and can be seen as the corresponding characterization to the description of the value function for optimal stopping problems as a smallest superharmonic majorant of the reward function. The results are illustrated with examples from different fields, including multiple stopping and optimal switching problems. © 2013 Elsevier B.V. All rights reserved.
49.	Christensen, Sören, 1982, et al. (författare) Optimal stopping of strong Markov processes 2013 Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149. ; 123:3, s. 1138-1159 Tidskriftsartikel (refereegranskat)abstract We characterize the value function and the optimal stopping time for a large class of optimal stopping problems where the underlying process to be stopped is a fairly general Markov process. The main result is inspired by recent findings for Lévy processes obtained essentially via the Wiener-Hopf factorization. The main ingredient in our approach is the representation of the β-excessive functions as expected suprema. A variety of examples is given. © 2012 Elsevier B.V. All rights reserved.
50.	Cleanthous, G., et al. (författare) Regularity, continuity and approximation of isotropic Gaussian random fields on compact two-point homogeneous spaces 2020 Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149. ; 130:8, s. 4873-4891 Tidskriftsartikel (refereegranskat)abstract Gaussian random fields defined over compact two-point homogeneous spaces are considered and Sobolev regularity and Hölder continuity are explored through spectral representations. It is shown how spectral properties of the covariance function associated to a given Gaussian random field are crucial to determine such regularities and geometric properties. Furthermore, fast approximations of random fields on compact two-point homogeneous spaces are derived by truncation of the series expansion, and a suitable bound for the error involved in such an approximation is provided. © 2020 Elsevier B.V.

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