Stochastic Differential Equations with Jumps

Platen, Eckhard; Bruti‐Liberati, Nicola

doi:10.1007/978-3-642-13694-8_1

Cited by 25 publications

(21 citation statements)

References 0 publications

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“…We begin by deriving conditional moments of the Langevin equation (1) for different orders of the time interval t d (see appendix A and [20]). We find the following expressions for the conditional moments of orders Î { } m 2, 4, 6 , if we consider terms up to the order of the first non-vanishing power in ( ) t d 2 for m=2 and m=4, respectively ( ) t d 3 for m=6 (here we omit the x-and t-dependence of a and b to enhance readability) where the subscript 'd' denotes diffusion, and where ¢ a and ¢ b denote the first and a″ and b″ the second derivatives with respect to state variable x.…”

Section: Higher-order Conditional Moments Of Continuous Linear and Nmentioning

confidence: 99%

Characterizing abrupt transitions in stochastic dynamics

2018

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Data sampled at discrete times appears as a succession of discontinuous jumps, even if the underlying trajectory is continuous. We analytically derive a criterion that allows one to check whether for a given, even noisy time series the underlying process has a continuous (diffusion) trajectory or has jump discontinuities. This enables one to detect and characterize abrupt changes (jump events) in given time series. The proposed criterion is validated numerically using synthetic continuous and discontinuous time series. We demonstrate applicability of our criterion to distinguish diffusive and jumpy behavior by a data-driven inference of higher-order conditional moments from empirical observations.

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Section: Higher-order Conditional Moments Of Continuous Linear and Nmentioning

confidence: 99%

Characterizing abrupt transitions in stochastic dynamics

2018

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“…For the sample for (30), we have to add sample points of the normal distribution of observation errors with the corresponding parameters w x as well as consider output transformation, as in equation (25). Finally, to draw realizations of rainfall intensities, e.g., to numerically quantify input uncertainty, we simply need to generate random paths of n and transform those into time series of x via h. This Monte Carlo method is similar to drawing samples from a desired distribution by drawing from a standard uniform distribution and then transforming those samples via an inverse distribution function [Platen and Bruti-Liberati, 2010;Kroese et al, 2011]. Our method, however, additionally aims to capture the autocorrelation structure of the rainfall.…”

Section: Predictions With Sipmentioning

confidence: 99%

“…. The bias correction B here follows an Ornstein-Uhlenbeck (OU) dynamics [e.g.,Platen and Bruti-Liberati, 2010; Kroese et al, 2011, and references therein] …”

mentioning

confidence: 99%

Describing the catchment‐averaged precipitation as a stochastic process improves parameter and input estimation

Giudice

Albert

Rieckermann

et al. 2016

Water Resources Research

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Rainfall input uncertainty is one of the major concerns in hydrological modeling. Unfortunately, during inference, input errors are usually neglected, which can lead to biased parameters and implausible predictions. Rainfall multipliers can reduce this problem but still fail when the observed input (precipitation) has a different temporal pattern from the true one or if the true nonzero input is not detected. In this study, we propose an improved input error model which is able to overcome these challenges and to assess and reduce input uncertainty. We formulate the average precipitation over the watershed as a stochastic input process (SIP) and, together with a model of the hydrosystem, include it in the likelihood function. During statistical inference, we use ''noisy'' input (rainfall) and output (runoff) data to learn about the ''true'' rainfall, model parameters, and runoff. We test the methodology with the rainfall-discharge dynamics of a small urban catchment. To assess its advantages, we compare SIP with simpler methods of describing uncertainty within statistical inference: (i) standard least squares (LS), (ii) bias description (BD), and (iii) rainfall multipliers (RM). We also compare two scenarios: accurate versus inaccurate forcing data. Results show that when inferring the input with SIP and using inaccurate forcing data, the whole-catchment precipitation can still be realistically estimated and thus physical parameters can be ''protected'' from the corrupting impact of input errors. While correcting the output rather than the input, BD inferred similarly unbiased parameters. This is not the case with LS and RM. During validation, SIP also delivers realistic uncertainty intervals for both rainfall and runoff. Thus, the technique presented is a significant step toward better quantifying input uncertainty in hydrological inference. As a next step, SIP will have to be combined with a technique addressing model structure uncertainty.

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“…In order to obtain the relevant transport characteristics we have to resort to comprehensive numerical simulations of driven Langevin dynamics. We integrated (19) by employing a weak version of the stochastic second order predictor corrector algorithm [21] with a time step typically set to about 10 −3 · 2π/ω. Since (19) is a secondorder differential equation, we have to specify two initial conditions x(0) andẋ(0).…”

Section: Deterministic Dynamicsmentioning

confidence: 99%

Josephson junction ratchet: The impact of finite capacitances

2014

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We study transport in an asymmetric SQUID which is composed of a loop with three capacitively and resistively shunted Josephson junctions: two in series in one arm and the remaining one in the other arm. The loop is threaded by an external magnetic flux and the system is subjected to both a time-periodic and a constant current. We formulate the deterministic and, as well, the stochastic dynamics of the SQUID in terms of the Stewart-McCumber model and derive an equation for the phase difference across one arm, in which an effective periodic potential is of the ratchet type, i.e. its reflection symmetry is broken. In doing so, we extend and generalize earlier study by Zapata et al. [Phys. Rev. Lett. 77, 2292(1996] and analyze directed transport in wide parameter regimes: covering the over-damped to moderate damping regime up to its fully under-damped regime. As a result we detect the intriguing features of a negative (differential) conductance, repeated voltage reversals, noise induced voltage reversals and solely thermal noise-induced ratchet currents. We identify a set of parameters for which the ratchet effect is most pronounced and show how the direction of transport can be controlled by tailoring the external magnetic flux.

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Stochastic Differential Equations with Jumps

Cited by 25 publications

References 0 publications

Characterizing abrupt transitions in stochastic dynamics

Characterizing abrupt transitions in stochastic dynamics

Describing the catchment‐averaged precipitation as a stochastic process improves parameter and input estimation

Josephson junction ratchet: The impact of finite capacitances

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