Journal articles: 'Pathwise approach'

1

Kühn, C., A. E. Kyprianou, and K. van Schaik. "Pricing Israeli options: a pathwise approach." Stochastics 79, no. 1-2 (February 2007): 117–37. http://dx.doi.org/10.1080/17442500600976442.

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2

Willinger, Walter. "A pathwise approach to stochastic integration." Stochastic Processes and their Applications 26 (1987): 236. http://dx.doi.org/10.1016/0304-4149(87)90177-3.

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Cattiaux, Patrick. "A Pathwise Approach of Some Classical Inequalities." Potential Analysis 20, no. 4 (June 2004): 361–94. http://dx.doi.org/10.1023/b:pota.0000009847.84908.6f.

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4

Abdullin, Marat Airatovich, Niyaz Salavatovich Ismagilov, and Farit Sagitovich Nasyrov. "One dimensional stochastic differential equations: pathwise approach." Ufimskii Matematicheskii Zhurnal 5, no. 4 (2013): 3–15. http://dx.doi.org/10.13108/2013-5-4-3.

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Korytowski, Adam, and Maciej Szymkat. "Necessary Optimality Conditions for a Class of Control Problems with State Constraint." Games 12, no. 1 (January 18, 2021): 9. http://dx.doi.org/10.3390/g12010009.

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An elementary approach to a class of optimal control problems with pathwise state constraint is proposed. Based on spike variations of control, it yields simple proofs and constructive necessary conditions, including some new characterizations of optimal control. Two examples are discussed.

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Jin, Xing, Dan Luo, and Xudong Zeng. "Dynamic Asset Allocation with Uncertain Jump Risks: A Pathwise Optimization Approach." Mathematics of Operations Research 43, no. 2 (May 2018): 347–76. http://dx.doi.org/10.1287/moor.2017.0854.

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7

BOUHADOU, S., and Y. OUKNINE. "STOCHASTIC EQUATIONS OF PROCESSES WITH JUMPS." Stochastics and Dynamics 14, no. 01 (December 29, 2013): 1350006. http://dx.doi.org/10.1142/s0219493713500068.

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We consider one-dimensional stochastic differential equations driven by white noises and Poisson random measure. We introduce new techniques based on local time prove new results on pathwise uniqueness and comparison theorems. Our approach is very easy to handle and do not need any approximation approach. Similar equations without jumps were studied in the same context by [8, 12] and other authors.

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8

Catuogno, Pedro, and Christian Olivera. "Renormalized-generalized solutions for the KPZ equation." Infinite Dimensional Analysis, Quantum Probability and Related Topics 17, no. 04 (November 25, 2014): 1450027. http://dx.doi.org/10.1142/s0219025714500271.

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This work introduces a new notion of solution for the KPZ equation, in particular, our approach encompasses the Cole–Hopf solution. We set in the context of the distribution theory the proposed results by Bertini and Giacomin from the mid '90s. This new approach provides a pathwise notion of solution as well as a structured approximation theory. The developments are based on regularization arguments from the theory of distributions.

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Bianchi, A., A. Gaudillière, and P. Milanesi. "On Soft Capacities, Quasi-stationary Distributions and the Pathwise Approach to Metastability." Journal of Statistical Physics 181, no. 3 (August 8, 2020): 1052–86. http://dx.doi.org/10.1007/s10955-020-02618-9.

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10

Westphal, U., and T. Schwartz. "Farthest points and monotone operators." Bulletin of the Australian Mathematical Society 58, no. 1 (August 1998): 75–92. http://dx.doi.org/10.1017/s0004972700032019.

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We apply the theory of monotone operators to study farthest points in closed bounded subsets of real Banach spaces. This new approach reveals the intimate connection between the farthest point mapping and the subdifferential of the farthest distance function. Moreover, we prove that a typical exception set in the Baire category sense is pathwise connected. Stronger results are obtained in Hilbert spaces.

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11

LIM, ENG LIAN, JOHN McCALLUM, and KWOK HUNG CHAN. "PRODUCTION-GRAPH: A GRAPH THEORETICAL MODEL FOR CHECKING KNOWLEDGE BASE ANOMALIES." International Journal on Artificial Intelligence Tools 01, no. 04 (December 1992): 563–95. http://dx.doi.org/10.1142/s0218213092000065.

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Knowledge acquisition is tedious and error-prone. Consequently, a knowledge base may be inconsistent, and contains unreachable rules, redundant rules, and rules which may lead to deadends and infinite loops. There are three approaches for checking these anomalies: interactive, non-interactive pairwise and non-interactive pathwise. In this article, we will present a graph theoretical model called Production-graph for checking knowledge base anomalies along the non-interactive pathwise approach. Production-graph uses graph theoretical constructions to represent facts and rules, as well as relevant properties of the knowledge base that leads to anomalies. Distinctive features of Production-graph include: (i) Using Production-graph, we are able to check on groups of problem instances rather than on individual problem instances. This eliminates the problem of having infinitely many problem instances. (ii) Empirical knowledge is used to limit the problem instances to practically realizable problems. (iii) Effects of chaining both rules and facts are considered.

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Deng, Mengting, Guo Jiang, and Ting Ke. "Numerical Solution of Nonlinear Stochastic Itô–Volterra Integral Equations Driven by Fractional Brownian Motion Using Block Pulse Functions." Discrete Dynamics in Nature and Society 2021 (October 30, 2021): 1–11. http://dx.doi.org/10.1155/2021/4934658.

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This paper presents a valid numerical method to solve nonlinear stochastic Itô–Volterra integral equations (SIVIEs) driven by fractional Brownian motion (FBM) with Hurst parameter H ∈ 1 / 2 , 1 . On the basis of FBM and block pulse functions (BPFs), a new stochastic operational matrix is proposed. The nonlinear stochastic integral equation is converted into a nonlinear algebraic equation by this method. Furthermore, error analysis is given by the pathwise approach. Finally, two numerical examples exhibit the validity and accuracy of the approach.

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13

Braunsteins, Peter, Geoffrey Decrouez, and Sophie Hautphenne. "A pathwise approach to the extinction of branching processes with countably many types." Stochastic Processes and their Applications 129, no. 3 (March 2019): 713–39. http://dx.doi.org/10.1016/j.spa.2018.03.013.

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14

Alnafisah, Yousef. "A New Approach to Compare the Strong Convergence of the Milstein Scheme with the Approximate Coupling Method." Fractal and Fractional 6, no. 6 (June 17, 2022): 339. http://dx.doi.org/10.3390/fractalfract6060339.

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Milstein and approximate coupling approaches are compared for the pathwise numerical solutions to stochastic differential equations (SDE) driven by Brownian motion. These methods attain an order one convergence under the nondegeneracy assumption of the diffusion term for the approximate coupling method. We use MATLAB to simulate these methods by applying them to a particular two-dimensional SDE. Then, we analyze the performance of both methods and the amount of time required to obtain the result. This comparison is essential in several areas, such as stochastic analysis, financial mathematics, and some biological applications.

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15

Wang, Peiguang, and Yan Xu. "Averaging Method for Neutral Stochastic Delay Differential Equations Driven by Fractional Brownian Motion." Journal of Function Spaces 2020 (May 29, 2020): 1–7. http://dx.doi.org/10.1155/2020/5212690.

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In this paper, we investigate the stochastic averaging method for neutral stochastic delay differential equations driven by fractional Brownian motion with Hurst parameter H∈1/2,1. By using the linear operator theory and the pathwise approach, we show that the solutions of neutral stochastic delay differential equations converge to the solutions of the corresponding averaged stochastic delay differential equations. At last, an example is provided to illustrate the applications of the proposed results.

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16

Posilicano, Andrea, and Stefania Ugolini. "Scattering into cones and flux across surfaces in quantum mechanics: A pathwise probabilistic approach." Journal of Mathematical Physics 43, no. 11 (November 2002): 5386–99. http://dx.doi.org/10.1063/1.1504884.

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17

León, Jorge A., Josep L. Solé, and Josep Vives. "A pathwise approach to backward and forward stochastic differential equations on the poisson space*." Stochastic Analysis and Applications 19, no. 5 (October 15, 2001): 0. http://dx.doi.org/10.1081/sap-120000223.

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18

Caruana, Michael, Peter K. Friz, and Harald Oberhauser. "A (rough) pathwise approach to a class of non-linear stochastic partial differential equations." Annales de l'Institut Henri Poincare (C) Non Linear Analysis 28, no. 1 (January 2011): 27–46. http://dx.doi.org/10.1016/j.anihpc.2010.11.002.

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19

Bocar, Ba Demba, Diop Bou, and Thioune Moussa. "AN APPROACH TO PATHWISE STOCHASTIC INTEGRATION IN FRACTIONAL BESOV-TYPE SPACES BY KRYLOV INEQUALITY." Universal Journal of Mathematics and Mathematical Sciences 18 (January 6, 2023): 67–83. http://dx.doi.org/10.17654/2277141723005.

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20

Sheppard, Patrick W., Muruhan Rathinam, and Mustafa Khammash. "A pathwise derivative approach to the computation of parameter sensitivities in discrete stochastic chemical systems." Journal of Chemical Physics 136, no. 3 (January 21, 2012): 034115. http://dx.doi.org/10.1063/1.3677230.

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21

Landriault, David, Bin Li, and Hongzhong Zhang. "A unified approach for drawdown (drawup) of time-homogeneous Markov processes." Journal of Applied Probability 54, no. 2 (June 2017): 603–26. http://dx.doi.org/10.1017/jpr.2017.20.

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AbstractDrawdown (respectively, drawup) of a stochastic process, also referred as the reflected process at its supremum (respectively, infimum), has wide applications in many areas including financial risk management, actuarial mathematics, and statistics. In this paper, for general time-homogeneous Markov processes, we study the joint law of the first passage time of the drawdown (respectively, drawup) process, its overshoot, and the maximum of the underlying process at this first passage time. By using short-time pathwise analysis, under some mild regularity conditions, the joint law of the three drawdown quantities is shown to be the unique solution to an integral equation which is expressed in terms of fundamental two-sided exit quantities of the underlying process. Explicit forms for this joint law are found when the Markov process has only one-sided jumps or is a Lévy process (possibly with two-sided jumps). The proposed methodology provides a unified approach to study various drawdown quantities for the general class of time-homogeneous Markov processes.

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22

Crisan, D., P. Dobson, and M. Ottobre. "Uniform in time estimates for the weak error of the Euler method for SDEs and a pathwise approach to derivative estimates for diffusion semigroups." Transactions of the American Mathematical Society 374, no. 5 (February 26, 2021): 3289–330. http://dx.doi.org/10.1090/tran/8301.

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We present a criterion for uniform in time convergence of the weak error of the Euler scheme for Stochastic Differential equations (SDEs). The criterion requires (i) exponential decay in time of the space-derivatives of the semigroup associated with the SDE and (ii) bounds on (some) moments of the Euler approximation. We show by means of examples (and counterexamples) how both (i) and (ii) are needed to obtain the desired result. If the weak error converges to zero uniformly in time, then convergence of ergodic averages follows as well. We also show that Lyapunov-type conditions are neither sufficient nor necessary in order for the weak error of the Euler approximation to converge uniformly in time and clarify relations between the validity of Lyapunov conditions, (i) and (ii). Conditions for (ii) to hold are studied in the literature. Here we produce sufficient conditions for (i) to hold. The study of derivative estimates has attracted a lot of attention, however not many results are known in order to guarantee exponentially fast decay of the derivatives. Exponential decay of derivatives typically follows from coercive-type conditions involving the vector fields appearing in the equation and their commutators; here we focus on the case in which such coercive-type conditions are non-uniform in space. To the best of our knowledge, this situation is unexplored in the literature, at least on a systematic level. To obtain results under such space-inhomogeneous conditions we initiate a pathwise approach to the study of derivative estimates for diffusion semigroups and combine this pathwise method with the use of Large Deviation Principles.

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23

BERGLUND, NILS, and BARBARA GENTZ. "METASTABILITY IN SIMPLE CLIMATE MODELS: PATHWISE ANALYSIS OF SLOWLY DRIVEN LANGEVIN EQUATIONS." Stochastics and Dynamics 02, no. 03 (September 2002): 327–56. http://dx.doi.org/10.1142/s0219493702000455.

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We consider simple stochastic climate models, described by slowly time-dependent Langevin equations. We show that when the noise intensity is not too large, these systems can spend substantial amounts of time in metastable equilibrium, instead of adiabatically following the stationary distribution of the frozen system. This behavior can be characterized by describing the location of typical paths, and bounding the probability of atypical paths. We illustrate this approach by giving a quantitative description of phenomena associated with bistability, for three famous examples of simple climate models: Stochastic resonance in an energy balance model describing the Ice Ages; hysteresis in a box model for the Atlantic thermohaline circulation; and bifurcation delay in the case of the Lorenz model for Rayleigh–Bénard convection.

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24

Ceci, Claudia, and Katia Colaneri. "Nonlinear Filtering for Jump Diffusion Observations." Advances in Applied Probability 44, no. 03 (September 2012): 678–701. http://dx.doi.org/10.1017/s0001867800005838.

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We deal with the filtering problem of a general jump diffusion process,X, when the observation process,Y, is a correlated jump diffusion process having common jump times withX. In this setting, at any timetthe σ-algebraprovides all the available information aboutXt, and the central goal is to characterize the filter, πt, which is the conditional distribution ofXtgiven observations. To this end, we prove that πtsolves the Kushner-Stratonovich equation and, by applying the filtered martingale problem approach (see Kurtz and Ocone (1988)), that it is the unique weak solution to this equation. Under an additional hypothesis, we also provide a pathwise uniqueness result.

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25

Ceci, Claudia, and Katia Colaneri. "Nonlinear Filtering for Jump Diffusion Observations." Advances in Applied Probability 44, no. 3 (September 2012): 678–701. http://dx.doi.org/10.1239/aap/1346955260.

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We deal with the filtering problem of a general jump diffusion process, X, when the observation process, Y, is a correlated jump diffusion process having common jump times with X. In this setting, at any time t the σ-algebra provides all the available information about Xt, and the central goal is to characterize the filter, πt, which is the conditional distribution of Xt given observations . To this end, we prove that πt solves the Kushner-Stratonovich equation and, by applying the filtered martingale problem approach (see Kurtz and Ocone (1988)), that it is the unique weak solution to this equation. Under an additional hypothesis, we also provide a pathwise uniqueness result.

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26

BEVERIDGE, CHRISTOPHER, and MARK JOSHI. "THE EFFICIENT COMPUTATION OF PRICES AND GREEKS FOR CALLABLE RANGE ACCRUALS USING THE DISPLACED-DIFFUSION LMM." International Journal of Theoretical and Applied Finance 17, no. 01 (February 2014): 1450001. http://dx.doi.org/10.1142/s0219024914500010.

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We study the simulation of range accrual coupons when valuing callable range accruals in the displaced-diffusion LIBOR market model (DDLMM). We introduce a number of new improvements that lead to significant efficiency improvements, and explain how to apply the adjoint-improved pathwise method to calculate deltas and vegas under the new improvements, which was not previously possible for callable range accruals. One new improvement is based on using a Brownian-bridge-type approach for simulating the range accrual coupons. We consider a variety of examples, including when the reference rate is a LIBOR rate, when it is a spread between swap rates, and when the multiplier for the range accrual coupon is stochastic.

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27

Boumezoued, Alexandre. "Population viewpoint on Hawkes processes." Advances in Applied Probability 48, no. 2 (June 2016): 463–80. http://dx.doi.org/10.1017/apr.2016.10.

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AbstractIn this paper we focus on a class of linear Hawkes processes with general immigrants. These are counting processes with shot-noise intensity, including self-excited and externally excited patterns. For such processes, we introduce the concept of the age pyramid which evolves according to immigration and births. The virtue of this approach that combines an intensity process definition and a branching representation is that the population age pyramid keeps track of all past events. This is used to compute new distribution properties for a class of Hawkes processes with general immigrants which generalize the popular exponential fertility function. The pathwise construction of the Hawkes process and its underlying population is also given.

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Angiuli, Andrea, Christy V. Graves, Houzhi Li, Jean-François Chassagneux, François Delarue, and René Carmona. "Cemracs 2017: numerical probabilistic approach to MFG." ESAIM: Proceedings and Surveys 65 (2019): 84–113. http://dx.doi.org/10.1051/proc/201965084.

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This project investigates numerical methods for solving fully coupled forward-backward stochastic differential equations (FBSDEs) of McKean-Vlasov type. Having numerical solvers for such mean field FBSDEs is of interest because of the potential application of these equations to optimization problems over a large population, say for instance mean field games (MFG) and optimal mean field control problems. Theory for this kind of problems has met with great success since the early works on mean field games by Lasry and Lions, see [29], and by Huang, Caines, and Malhamé, see [26]. Generally speaking, the purpose is to understand the continuum limit of optimizers or of equilibria (say in Nash sense) as the number of underlying players tends to infinity. When approached from the probabilistic viewpoint, solutions to these control problems (or games) can be described by coupled mean field FBSDEs, meaning that the coefficients depend upon the own marginal laws of the solution. In this note, we detail two methods for solving such FBSDEs which we implement and apply to five benchmark problems. The first method uses a tree structure to represent the pathwise laws of the solution, whereas the second method uses a grid discretization to represent the time marginal laws of the solutions. Both are based on a Picard scheme; importantly, we combine each of them with a generic continuation method that permits to extend the time horizon (or equivalently the coupling strength between the two equations) for which the Picard iteration converges.

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Nika, Zsolt, and Tamás Szabados. "Strong approximation of Black-Scholes theory based on simple random walks." Studia Scientiarum Mathematicarum Hungarica 53, no. 1 (March 2016): 93–129. http://dx.doi.org/10.1556/012.2016.53.1.1331.

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A basic model in financial mathematics was introduced by Black, Scholes and Merton in 1973. A classical discrete approximation in distribution is the binomial model given by Cox, Ross and Rubinstein in 1979. In this work we give a strong (almost sure, pathwise) discrete approximation of the BSM model using a suitable nested sequence of simple, symmetric random walks. The approximation extends to the stock price process, the value process, the replicating portfolio, and the greeks. An important tool in the approximation is a discrete version of the Feynman-Kac formula as well. Our aim is to show that from an elementary discrete approach, by taking simple limits, one may get the continuous versions. We think that such an approach can be advantageous for both research and applications. Moreover, it is hoped that this approach has pedagogical merits as well: gives insight and seems suitable for teaching students whose mathematical background may not contain e.g. measure theory or stochastic analysis.

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Vaddireddy, Harsha, and Omer San. "Equation Discovery Using Fast Function Extraction: a Deterministic Symbolic Regression Approach." Fluids 4, no. 2 (June 15, 2019): 111. http://dx.doi.org/10.3390/fluids4020111.

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Advances in machine learning (ML) coupled with increased computational power have enabled identification of patterns in data extracted from complex systems. ML algorithms are actively being sought in recovering physical models or mathematical equations from data. This is a highly valuable technique where models cannot be built using physical reasoning alone. In this paper, we investigate the application of fast function extraction (FFX), a fast, scalable, deterministic symbolic regression algorithm to recover partial differential equations (PDEs). FFX identifies active bases among a huge set of candidate basis functions and their corresponding coefficients from recorded snapshot data. This approach uses a sparsity-promoting technique from compressive sensing and sparse optimization called pathwise regularized learning to perform feature selection and parameter estimation. Furthermore, it recovers several models of varying complexity (number of basis terms). FFX finally filters out many identified models using non-dominated sorting and forms a Pareto front consisting of optimal models with respect to minimizing complexity and test accuracy. Numerical experiments are carried out to recover several ubiquitous PDEs such as wave and heat equations among linear PDEs and Burgers, Korteweg–de Vries (KdV), and Kawahara equations among higher-order nonlinear PDEs. Additional simulations are conducted on the same PDEs under noisy conditions to test the robustness of the proposed approach.

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31

Milstein, Grigori N., and John Schoenmakers. "Uniform approximation of the Cox-Ingersoll-Ross process." Advances in Applied Probability 47, no. 4 (December 2015): 1132–56. http://dx.doi.org/10.1239/aap/1449859803.

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The Doss-Sussmann (DS) approach is used for uniform simulation of the Cox-Ingersoll-Ross (CIR) process. The DS formalism allows us to express trajectories of the CIR process through solutions of some ordinary differential equation (ODE) depending on realizations of a Wiener process involved. By simulating the first-passage times of the increments of the Wiener process to the boundary of an interval and solving the ODE, we uniformly approximate the trajectories of the CIR process. In this respect special attention is payed to simulation of trajectories near 0. From a conceptual point of view the proposed method gives a better quality of approximation (from a pathwise point of view) than standard, even exact, simulation of the stochastic differential equation at some deterministic time grid.

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Milstein, Grigori N., and John Schoenmakers. "Uniform approximation of the Cox-Ingersoll-Ross process." Advances in Applied Probability 47, no. 04 (December 2015): 1132–56. http://dx.doi.org/10.1017/s0001867800049041.

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The Doss-Sussmann (DS) approach is used for uniform simulation of the Cox-Ingersoll-Ross (CIR) process. The DS formalism allows us to express trajectories of the CIR process through solutions of some ordinary differential equation (ODE) depending on realizations of a Wiener process involved. By simulating the first-passage times of the increments of the Wiener process to the boundary of an interval and solving the ODE, we uniformly approximate the trajectories of the CIR process. In this respect special attention is payed to simulation of trajectories near 0. From a conceptual point of view the proposed method gives a better quality of approximation (from a pathwise point of view) than standard, even exact, simulation of the stochastic differential equation at some deterministic time grid.

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33

Catuogno, Pedro José, Sebastián Esteban Ferrando, and Alfredo Lázaro González. "Efficient Hedging of Options with Probabilistic Haar Wavelets." ISRN Probability and Statistics 2012 (September 18, 2012): 1–37. http://dx.doi.org/10.5402/2012/946415.

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The paper brings forward the issue of efficient representations of financial claims; in particular it addresses the problem of large transaction costs in hedging replications. Inspired by the localized properties of wavelets basis, Haar systems associated with space-time discretizations of continuous stochastic processes are proposed as a means to address the issue of efficient pathwise approximation. Theoretical developments are presented that justify the use of these approximations to construct self-financing portfolios by means of binary options. Upper bounds on the volume of transactions required to implement these portfolios are then established to illustrate the quality of the proposed approximations. The approach is applicable to general financial claims of European type, including path-dependent ones, for continuous underlying processes. Several numerical results and comparisons with delta hedging are also presented.

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Perry, D., W. Stadje, and S. Zacks. "A Duality Approach to Queues with Service Restrictions and Storage Systems with State-Dependent Rates." Journal of Applied Probability 50, no. 3 (September 2013): 612–31. http://dx.doi.org/10.1239/jap/1378401226.

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Based on pathwise duality constructions, several new results on truncated queues and storage systems of the G/M/1 type are derived by transforming the workload (content) processes into certain ‘dual’ M/G/1-type processes. We consider queueing systems in which (a) any service requirement that would increase the total workload beyond the capacity is truncated so as to keep the associated sojourn time below a certain constant, or (b) new arrivals do not enter the system if they have to wait more than one time unit in line. For these systems, we derive the steady-state distributions of the workload and the numbers of customers present in the systems as well as the distributions of the lengths of busy and idle periods. Moreover, we use the duality approach to study finite capacity storage systems with general state-dependent outflow rates. Here our duality leads to a Markovian finite storage system with state-dependent jump sizes whose content level process can be analyzed using level crossing techniques. We also derive a connection between the steady-state densities of the non-Markovian continuous-time content level process of the G/M/1 finite storage system with state-dependent outflow rule and the corresponding embedded sequence of peak points (local maxima).

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Perry, D., W. Stadje, and S. Zacks. "A Duality Approach to Queues with Service Restrictions and Storage Systems with State-Dependent Rates." Journal of Applied Probability 50, no. 03 (September 2013): 612–31. http://dx.doi.org/10.1017/s0021900200009748.

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Based on pathwise duality constructions, several new results on truncated queues and storage systems of the G/M/1 type are derived by transforming the workload (content) processes into certain ‘dual’ M/G/1-type processes. We consider queueing systems in which (a) any service requirement that would increase the total workload beyond the capacity is truncated so as to keep the associated sojourn time below a certain constant, or (b) new arrivals do not enter the system if they have to wait more than one time unit in line. For these systems, we derive the steady-state distributions of the workload and the numbers of customers present in the systems as well as the distributions of the lengths of busy and idle periods. Moreover, we use the duality approach to study finite capacity storage systems with general state-dependent outflow rates. Here our duality leads to a Markovian finite storage system with state-dependent jump sizes whose content level process can be analyzed using level crossing techniques. We also derive a connection between the steady-state densities of the non-Markovian continuous-time content level process of the G/M/1 finite storage system with state-dependent outflow rule and the corresponding embedded sequence of peak points (local maxima).

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El-Taha, Muhammad, and Shaler Stidham. "Sample-path stability conditions for multiserver input-output processes." Journal of Applied Mathematics and Stochastic Analysis 7, no. 3 (January 1, 1994): 437–56. http://dx.doi.org/10.1155/s1048953394000353.

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We extend our studies of sample-path stability to multiserver input-output processes with conditional output rates that may depend on the state of the system and other auxiliary processes. Our results include processes with countable as well as uncountable state spaces. We establish rate stability conditions for busy period durations as well as the input during busy periods. In addition, stability conditions for multiserver queues with possibly heterogeneous servers are given for the workload, attained service, and queue length processes. The stability conditions can be checked from parameters of primary processes, and thus can be verified a priori. Under the rate stability conditions, we provide stable versions of Little's formula for single server as well as multiserver queues. Our approach leads to extensions of previously known results. Since our results are valid pathwise, non-stationary as well as stationary processes are covered.

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ROUSSET, MATHIAS, and GIOVANNI SAMAEY. "INDIVIDUAL-BASED MODELS FOR BACTERIAL CHEMOTAXIS IN THE DIFFUSION ASYMPTOTICS." Mathematical Models and Methods in Applied Sciences 23, no. 11 (July 23, 2013): 2005–37. http://dx.doi.org/10.1142/s0218202513500243.

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We discuss velocity-jump models for chemotaxis of bacteria with an internal state that allows the velocity-jump rate to depend on the memory of the chemoattractant concentration along their path of motion. Using probabilistic techniques, we provide a pathwise result that shows that the considered process converges to an advection-diffusion process in the (long-time) diffusion limit. We also (re-)prove using the same approach that the same limiting equation arises for a related, simpler process with direct sensing of the chemoattractant gradient. Additionally, we propose a time discretization technique that retains these diffusion limits exactly, i.e. without error that depends on the time discretization. In the companion paper,21these results are used to construct a coupling technique that allows numerical simulation of the process with internal state with asymptotic variance reduction, in the sense that the variance vanishes in the diffusion limit.

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JOSHI, MARK, and OH KANG KWON. "LEAST SQUARES MONTE CARLO CREDIT VALUE ADJUSTMENT WITH SMALL AND UNIDIRECTIONAL BIAS." International Journal of Theoretical and Applied Finance 19, no. 08 (December 2016): 1650048. http://dx.doi.org/10.1142/s0219024916500485.

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Credit value adjustment (CVA) and related charges have emerged as important risk factors following the Global Financial Crisis. These charges depend on uncertain future values of underlying products, and are usually computed by Monte Carlo simulation. For products that cannot be valued analytically at each simulation step, the standard market practice is to use the regression functions from least squares Monte Carlo method to approximate their values. However, these functions do not necessarily provide accurate approximations to product values over all simulated paths and can result in biases that are difficult to control. Motivated by a novel characterization of the CVA as the value of an option with an early exercise opportunity at a stochastic time, we provide an approximation for CVA and other credit charges that rely only on the sign of the regression functions. The values are determined, instead, by pathwise deflated cash flows. A comparison of CVA for Bermudan swaptions and cancellable swaps shows that the proposed approximation results in much smaller errors than the standard approach of using the regression function values.

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Cathcart, Mark J., Hsiao Yen Lok, Alexander J. McNeil, and Steven Morrison. "CALCULATING VARIABLE ANNUITY LIABILITY “GREEKS” USING MONTE CARLO SIMULATION." ASTIN Bulletin 45, no. 2 (January 5, 2015): 239–66. http://dx.doi.org/10.1017/asb.2014.31.

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AbstractThe implementation of hedging strategies for variable annuity products requires the calculation of market risk sensitivities (or “Greeks”). The complex, path-dependent nature of these products means that these sensitivities are typically estimated by Monte Carlo methods. Standard market practice is to use a “bump and revalue” method in which sensitivities are approximated by finite differences. As well as requiring multiple valuations of the product, this approach is often unreliable for higher-order Greeks, such as gamma, and alternative pathwise (PW) and likelihood-ratio estimators should be preferred. This paper considers a stylized guaranteed minimum withdrawal benefit product in which the reference equity index follows a Heston stochastic volatility model in a stochastic interest rate environment. The complete set of first-order sensitivities with respect to index value, volatility and interest rate and the most important second-order sensitivities are calculated using PW, likelihood-ratio and mixed methods. It is observed that the PW method delivers the best estimates of first-order sensitivities while mixed estimation methods deliver considerably more accurate estimates of second-order sensitivities; moreover there are significant computational gains involved in using PW and mixed estimators rather than simple BnR estimators when many Greeks have to be calculated.

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Komyakov, B. K., B. G. Guliev, A. V. Zagazezhev, and R. V. Aliev. "SURGICAL TREATMENT OF PATIENTS WITH OBSTRUCTION OF PYELOURETERAL SEGMENT." Grekov's Bulletin of Surgery 174, no. 3 (June 28, 2015): 24–28. http://dx.doi.org/10.24884/0042-4625-2015-174-3-24-28.

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The article presents the operation results of 380 patients (170 (44,7%) men and 210 (55,3%) women) with obstruction of pyeloureteral segment at the period from 1996 to 2014. The middle age was 43,2±7,6 years old. Primary strictures took place in 302 (79,5%) patients and recurrent strictures - in 78 (20,5%). Open and laparoscopic plastic operations were performed in 248 (65,2%), endoscopic surgery - in 112 (29,5%), organo-removal surgery - in 20 (5,3%) cases out of 380 patients with obstruction of pyeloureteral segment. The hines Andersen open pyeloplasty was carried out in 142 (37,4%) patients with primary hydronephrosis out of 152. In addition, an antevasal pyeloureteral anastomosis was formed in 65 (17,1%) cases. Neivert operation was used in 8 (2,1%) cases because of extensive stricture of pyeloureteral segment. Kalp-De Vird pathwise pyeloplasty was fulfilled in 2 (0,5%) patients. Laparoscopic pyeloplasty was performed on 96 (16,7%) patients. These surgeries were carried out by transperitoneal approach using lateral position and 3 or 4 trocars. Endoscopic treatment of pyeloureteral segment was completed in 112 (29,5%) patients. Percutaneous endopyelotomy was used in 42 (11,0%) and retrograde - in 64 (16,8%) cases. Endoplasty of pyeloureteral segment was applied in 6 (1,6%). The efficacy of open pyeloplasty consisted of 93,7% and laparoscopic pyeloplasty - 94,6%. Good results after endopyelotomy were noted in recurrent strictures of pyelouretal segment.

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Fekete, D., J. Fontbona, and A. E. Kyprianou. "Skeletal stochastic differential equations for continuous-state branching processes." Journal of Applied Probability 56, no. 4 (December 2019): 1122–50. http://dx.doi.org/10.1017/jpr.2019.67.

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AbstractIt is well understood that a supercritical continuous-state branching process (CSBP) is equal in law to a discrete continuous-time Galton–Watson process (the skeleton of prolific individuals) whose edges are dressed in a Poissonian way with immigration which initiates subcritical CSBPs (non-prolific mass). Equally well understood in the setting of CSBPs and superprocesses is the notion of a spine or immortal particle dressed in a Poissonian way with immigration which initiates copies of the original CSBP, which emerges when conditioning the process to survive eternally. In this article we revisit these notions for CSBPs and put them in a common framework using the well-established language of (coupled) stochastic differential equations (SDEs). In this way we are able to deal simultaneously with all types of CSBPs (supercritical, critical, and subcritical) as well as understanding how the skeletal representation becomes, in the sense of weak convergence, a spinal decomposition when conditioning on survival. We have two principal motivations. The first is to prepare the way to expand the SDE approach to the spatial setting of superprocesses, where recent results have increasingly sought the use of skeletal decompositions to transfer results from the branching particle setting to the setting of measure valued processes. The second is to provide a pathwise decomposition of CSBPs in the spirit of genealogical coding of CSBPs via Lévy excursions, albeit precisely where the aforesaid coding fails to work because the underlying CSBP is supercritical.

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van der Laan, Mark J., and Alexander R. Luedtke. "Targeted Learning of the Mean Outcome under an Optimal Dynamic Treatment Rule." Journal of Causal Inference 3, no. 1 (March 1, 2015): 61–95. http://dx.doi.org/10.1515/jci-2013-0022.

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AbstractWe consider estimation of and inference for the mean outcome under the optimal dynamic two time-point treatment rule defined as the rule that maximizes the mean outcome under the dynamic treatment, where the candidate rules are restricted to depend only on a user-supplied subset of the baseline and intermediate covariates. This estimation problem is addressed in a statistical model for the data distribution that is nonparametric beyond possible knowledge about the treatment and censoring mechanism. This contrasts from the current literature that relies on parametric assumptions. We establish that the mean of the counterfactual outcome under the optimal dynamic treatment is a pathwise differentiable parameter under conditions, and develop a targeted minimum loss-based estimator (TMLE) of this target parameter. We establish asymptotic linearity and statistical inference for this estimator under specified conditions. In a sequentially randomized trial the statistical inference relies upon a second-order difference between the estimator of the optimal dynamic treatment and the optimal dynamic treatment to be asymptotically negligible, which may be a problematic condition when the rule is based on multivariate time-dependent covariates. To avoid this condition, we also develop TMLEs and statistical inference for data adaptive target parameters that are defined in terms of the mean outcome under the estimate of the optimal dynamic treatment. In particular, we develop a novel cross-validated TMLE approach that provides asymptotic inference under minimal conditions, avoiding the need for any empirical process conditions. We offer simulation results to support our theoretical findings.

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Pu, Shusen, and Peter J. Thomas. "Fast and Accurate Langevin Simulations of Stochastic Hodgkin-Huxley Dynamics." Neural Computation 32, no. 10 (October 2020): 1775–835. http://dx.doi.org/10.1162/neco_a_01312.

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Fox and Lu introduced a Langevin framework for discrete-time stochastic models of randomly gated ion channels such as the Hodgkin-Huxley (HH) system. They derived a Fokker-Planck equation with state-dependent diffusion tensor [Formula: see text] and suggested a Langevin formulation with noise coefficient matrix [Formula: see text] such that SS[Formula: see text]. Subsequently, several authors introduced a variety of Langevin equations for the HH system. In this article, we present a natural 14-dimensional dynamics for the HH system in which each directed edge in the ion channel state transition graph acts as an independent noise source, leading to a 14 [Formula: see text] 28 noise coefficient matrix [Formula: see text]. We show that (1) the corresponding 14D system of ordinary differential equations is consistent with the classical 4D representation of the HH system; (2) the 14D representation leads to a noise coefficient matrix [Formula: see text] that can be obtained cheaply on each time step, without requiring a matrix decomposition; (3) sample trajectories of the 14D representation are pathwise equivalent to trajectories of Fox and Lu's system, as well as trajectories of several existing Langevin models; (4) our 14D representation (and those equivalent to it) gives the most accurate interspike interval distribution, not only with respect to moments but under both the [Formula: see text] and [Formula: see text] metric-space norms; and (5) the 14D representation gives an approximation to exact Markov chain simulations that are as fast and as efficient as all equivalent models. Our approach goes beyond existing models, in that it supports a stochastic shielding decomposition that dramatically simplifies [Formula: see text] with minimal loss of accuracy under both voltage- and current-clamp conditions.

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Kong, Benjamin Y., Hao-Wen Sim, Anna K. Nowak, Sonia Yip, Elizabeth H. Barnes, Bryan W. Day, Michael E. Buckland, et al. "LUMOS - Low and Intermediate Grade Glioma Umbrella Study of Molecular Guided TherapieS at relapse: Protocol for a pilot study." BMJ Open 11, no. 12 (December 2021): e054075. http://dx.doi.org/10.1136/bmjopen-2021-054075.

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IntroductionGrades 2 and 3 gliomas (G2/3 gliomas), when combined, are the second largest group of malignant brain tumours in adults. The outcomes for G2/3 gliomas at progression approach the dismal outcomes for glioblastoma (GBM), yet there is a paucity of trials for Australian patients with relapsed G2/3 gliomas compared with patients with GBM. LUMOS will be a pilot umbrella study for patients with relapsed G2/3 gliomas that aims to match patients to targeted therapies based on molecular screening with contemporaneous tumour tissue. Participants in whom no actionable or no druggable mutation is found, or in whom the matching drug is not available, will form a comparator arm and receive standard of care chemotherapy. The objective of the LUMOS trial is to assess the feasibility of this approach in a multicentre study across five sites in Australia, with a view to establishing a national molecular screening platform for patient treatment guided by the mutational analysis of contemporaneous tissue biopsiesMethods and analysisThis study will be a multicentre pilot study enrolling patients with recurrent grade 2/3 gliomas that have previously been treated with radiotherapy and chemotherapy at diagnosis or at first relapse. Contemporaneous tumour tissue at the time of first relapse, defined as tissue obtained within 6 months of relapse and without subsequent intervening therapy, will be obtained from patients. Molecular screening will be performed by targeted next-generation sequencing at the reference laboratory (PathWest, Perth, Australia). RNA and DNA will be extracted from representative formalin-fixed paraffin embedded tissue scrolls or microdissected from sections on glass slides tissue sections following a review of the histology by pathologists. Extracted nucleic acid will be quantified by Qubit Fluorometric Quantitation (Thermo Fisher Scientific). Library preparation and targeted capture will be performed using the TruSight Tumor 170 (TST170) kit and samples sequenced on NextSeq 550 (Illumina) using NextSeq V.2.5 hi output reagents, according to the manufacturer’s instructions. Data analysis will be performed using the Illumina BaseSpace TST170 app v1.02 and a custom tertiary pipeline, implemented within the Clinical Genomics Workspace software platform from PierianDx (also refer to section 3.2). Primary outcomes for the study will be the number of patients enrolled and the number of patients who complete molecular screening. Secondary outcomes will include the proportion of screened patients enrolled; proportion of patients who complete molecular screening; the turn-around time of molecular screening; and the value of a brain tumour specific multi-disciplinary tumour board, called the molecular tumour advisory panel as measured by the proportion of patients in whom the treatment recommendation was refined compared with the recommendations from the automated bioinformatics platform of the reference laboratory testing.Ethics and disseminationThe study was approved by the lead Human Research Ethics Committee of the Sydney Local Health District: Protocol No. X19-0383. The study will be conducted in accordance with the principles of the Declaration of Helsinki 2013, guidelines for Good Clinical Practice and the National Health and Medical Research Council National Statement on Ethical Conduct in Human Research (2007, updated 2018 and as amended periodically). Results will be disseminated using a range of media channels including newsletters, social media, scientific conferences and peer-reviewed publications.Trial registration numberACTRN12620000087954; Pre-results.

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Duc, Luu Hoang. "Exponential stability of stochastic systems: A pathwise approach." Stochastics and Dynamics, April 18, 2022. http://dx.doi.org/10.1142/s0219493722400123.

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We provide a pathwise approach using semigroup technique to study the asymptotic stability for stochastic differential equations which admit a unique equilibrium. The driving noises in consideration are [Formula: see text] — Hölder continuous with [Formula: see text], so that the perturbed systems can be solved using rough path theory, where the rough integrals are interpreted in the Gubinelli sense for controlled rough paths. Our approach suggests an alternative method for stochastic systems with standard Brownian noises, by not using Itô formula but a relaxed isometry property of Itô stochastic integrals.

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Gubinelli, Massimiliano, Peter Imkeller, and Nicolas Perkowski. "A Fourier analytic approach to pathwise stochastic integration." Electronic Journal of Probability 21 (2016). http://dx.doi.org/10.1214/16-ejp3868.

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Fernandez, Roberto, Francesco Manzo, Francesca Nardi, and Elisabetta Scoppola. "Asymptotically exponential hitting times and metastability: a pathwise approach without reversibility." Electronic Journal of Probability 20 (2015). http://dx.doi.org/10.1214/ejp.v20-3656.

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Barth, Andrea, and Andreas Stein. "Numerical analysis for time-dependent advection-diffusion problems with random discontinuous coefficients." ESAIM: Mathematical Modelling and Numerical Analysis, June 10, 2022. http://dx.doi.org/10.1051/m2an/2022054.

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As an extension to the well-established stationary elliptic partial differential equation (PDE) with random continuous coefficients we study a time-dependent advection-diffusion problem, where the coefficients may have random spatial discontinuities. In a subsurface flow model, the randomness in a parabolic equation may account for insufficient measurements or uncertain material procurement, while the discontinuities could represent transitions in heterogeneous media. Specifically, a scenario with coupled advection and diffusion coefficients that are modeled as sums of continuous random fields and discontinuous jump components are considered. The respective coefficient functions allow a very flexible modeling, however, they also complicate the analysis and numerical approximation of the corresponding random parabolic PDE. We show that the model problem is indeed well-posed under mild assumptions and show measurability of the pathwise solution. For the numerical approximation we employ a sample-adapted, pathwise discretization scheme based on a finite element approach. This semi-discrete method accounts for the discontinuities in each sample, but leads to stochastic, finite-dimensional approximation spaces. We ensure measurability of the semi-discrete solution, which in turn enables us to derive moments bounds on the mean-squared approximation error. By coupling this semi-discrete approach with suitable coefficient approximation and a stable time stepping, we obtain a fully discrete algorithm to solve the random parabolic PDE. We provide an overall error bound for this scheme and illustrate our results with several numerical experiments.

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Duc, Luu Hoang, and Phan Thanh Hong. "Asymptotic Dynamics of Young Differential Equations." Journal of Dynamics and Differential Equations, November 1, 2021. http://dx.doi.org/10.1007/s10884-021-10095-1.

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AbstractWe provide a unified analytic approach to study the asymptotic dynamics of Young differential equations, using the framework of random dynamical systems and random attractors. Our method helps to generalize recent results (Duc et al. in J Differ Equ 264:1119–1145, 2018, SIAM J Control Optim 57(4):3046–3071, 2019; Garrido-Atienza et al. in Int J Bifurc Chaos 20(9):2761–2782, 2010) on the existence of the global pullback attractors for the generated random dynamical systems. We also prove sufficient conditions for the attractor to be a singleton, thus the pathwise convergence is in both pullback and forward senses.

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van Neerven, Jan, and Mark Veraar. "Maximal inequalities for stochastic convolutions and pathwise uniform convergence of time discretisation schemes." Stochastics and Partial Differential Equations: Analysis and Computations, July 10, 2021. http://dx.doi.org/10.1007/s40072-021-00204-y.

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AbstractWe prove a new Burkholder–Rosenthal type inequality for discrete-time processes taking values in a 2-smooth Banach space. As a first application we prove that if $$(S(t,s))_{0\leqslant s\le t\leqslant T}$$ ( S ( t , s ) ) 0 ⩽ s ≤ t ⩽ T is a $$C_0$$ C 0 -evolution family of contractions on a 2-smooth Banach space X and $$(W_t)_{t\in [0,T]}$$ ( W t ) t ∈ [ 0 , T ] is a cylindrical Brownian motion on a probability space $$(\Omega ,{\mathbb {P}})$$ ( Ω , P ) adapted to some given filtration, then for every $$0<p<\infty $$ 0 < p < ∞ there exists a constant $$C_{p,X}$$ C p , X such that for all progressively measurable processes $$g: [0,T]\times \Omega \rightarrow X$$ g : [ 0 , T ] × Ω → X the process $$(\int _0^t S(t,s)g_s\,\mathrm{d} W_s)_{t\in [0,T]}$$ ( ∫ 0 t S ( t , s ) g s d W s ) t ∈ [ 0 , T ] has a continuous modification and $$\begin{aligned} {\mathbb {E}}\sup _{t\in [0,T]}\Big \Vert \int _0^t S(t,s)g_s\,\mathrm{d} W_s \Big \Vert ^p\leqslant C_{p,X}^p {\mathbb {E}} \Bigl (\int _0^T \Vert g_t\Vert ^2_{\gamma (H,X)}\,\mathrm{d} t\Bigr )^{p/2}. \end{aligned}$$ E sup t ∈ [ 0 , T ] ‖ ∫ 0 t S ( t , s ) g s d W s ‖ p ⩽ C p , X p E ( ∫ 0 T ‖ g t ‖ γ ( H , X ) 2 d t ) p / 2 . Moreover, for $$2\leqslant p<\infty $$ 2 ⩽ p < ∞ one may take $$C_{p,X} = 10 D \sqrt{p},$$ C p , X = 10 D p , where D is the constant in the definition of 2-smoothness for X. The order $$O(\sqrt{p})$$ O ( p ) coincides with that of Burkholder’s inequality and is therefore optimal as $$p\rightarrow \infty $$ p → ∞ . Our result improves and unifies several existing maximal estimates and is even new in case X is a Hilbert space. Similar results are obtained if the driving martingale $$g_t\,\mathrm{d} W_t$$ g t d W t is replaced by more general X-valued martingales $$\,\mathrm{d} M_t$$ d M t . Moreover, our methods allow for random evolution systems, a setting which appears to be completely new as far as maximal inequalities are concerned. As a second application, for a large class of time discretisation schemes (including splitting, implicit Euler, Crank-Nicholson, and other rational schemes) we obtain stability and pathwise uniform convergence of time discretisation schemes for solutions of linear SPDEs $$\begin{aligned} \,\mathrm{d} u_t = A(t)u_t\,\mathrm{d} t + g_t\,\mathrm{d} W_t, \quad u_0 = 0, \end{aligned}$$ d u t = A ( t ) u t d t + g t d W t , u 0 = 0 , where the family $$(A(t))_{t\in [0,T]}$$ ( A ( t ) ) t ∈ [ 0 , T ] is assumed to generate a $$C_0$$ C 0 -evolution family $$(S(t,s))_{0\leqslant s\leqslant t\leqslant T}$$ ( S ( t , s ) ) 0 ⩽ s ⩽ t ⩽ T of contractions on a 2-smooth Banach spaces X. Under spatial smoothness assumptions on the inhomogeneity g, contractivity is not needed and explicit decay rates are obtained. In the parabolic setting this sharpens several know estimates in the literature; beyond the parabolic setting this seems to provide the first systematic approach to pathwise uniform convergence to time discretisation schemes.

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Journal articles on the topic 'Pathwise approach'

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