scholarly journals Approximation of the invariant distribution for a class of ergodic jump diffusions

2020 ◽  
Vol 24 ◽  
pp. 883-913
Author(s):  
A. Gloter ◽  
I. Honoré ◽  
D. Loukianova
Keyword(s):  
Empirical Distribution ◽  
Compound Poisson Process ◽  
Jump Diffusion ◽  
Invariant Distribution ◽  
Brownian Diffusion ◽  
Test Functions ◽  
Time Step ◽  
Jump Diffusions ◽  
The Difference ◽  
Lévy Jumps

In this article, we approximate the invariant distribution ν of an ergodic Jump Diffusion driven by the sum of a Brownian motion and a Compound Poisson process with sub-Gaussian jumps. We first construct an Euler discretization scheme with decreasing time steps. This scheme is similar to those introduced in Lamberton and Pagès Bernoulli 8 (2002) 367-405. for a Brownian diffusion and extended in F. Panloup, Ann. Appl. Probab. 18 (2008) 379-426. to a diffusion with Lévy jumps. We obtain a non-asymptotic quasi Gaussian (asymptotically Gaussian) concentration bound for the difference between the invariant distribution and the empirical distribution computed with the scheme of decreasing time step along appropriate test functions f such that f − ν(f) is a coboundary of the infinitesimal generator.

scholarly journals Jump Diffusion Models for Risky Debts: Quality Spread Differentials

2003 ◽  
Vol 06 (06) ◽  
pp. 655-662 ◽  
Author(s):  
Hoi Ying Wong ◽  
Yue Kuen Kwok
Keyword(s):  
Diffusion Process ◽  
Closed Form ◽  
Firm Value ◽  
Jump Diffusion ◽  
Brownian Diffusion ◽  
Fixed Rate ◽  
Floating Rate ◽  
The Difference ◽  
The Impact ◽  
Risky Debt

The quality spread differential is defined to be the difference between the default premiums demanded for fixed rate and floating rate risky debts. The risky debt model based on Merton's firm value approach is used to examine the behaviors of the quality spread differential of fixed rate and floating rate debts. We extend earlier result by adopting Geometric Brownian diffusion process with jumps for the underlying firm value process of the debt issuer. Closed form formulas are obtained for the default premiums for risky debts. The impact of the jumps on the fixed-floating spread differential is examined.

scholarly journals Recursive computation of the invariant distributions of Feller processes: Revisited examples and new applications

2019 ◽  
Vol 25 (1) ◽  
pp. 1-36 ◽  
Author(s):  
Gilles Pagès ◽  
Clément Rey
Keyword(s):  
Diffusion Processes ◽  
Invariant Distribution ◽  
Brownian Diffusion ◽  
Euler Scheme ◽  
Test Functions ◽  
Milstein Scheme ◽  
Invariant Distributions ◽  
Recursive Computation ◽  
New Applications ◽  
Abstract Framework

Abstract In this paper, we show that the abstract framework developed in [G. Pagès and C. Rey, Recursive computation of the invariant distribution of Markov and Feller processes, preprint 2017, https://arxiv.org/abs/1703.04557] and inspired by [D. Lamberton and G. Pagès, Recursive computation of the invariant distribution of a diffusion, Bernoulli 8 2002, 3, 367–405] can be used to build invariant distributions for Brownian diffusion processes using the Milstein scheme and for diffusion processes with censored jump using the Euler scheme. Both studies rely on a weakly mean-reverting setting for both cases. For the Milstein scheme we prove the convergence for test functions with polynomial (Wasserstein convergence) and exponential growth. For the Euler scheme of diffusion processes with censored jump we prove the convergence for test functions with polynomial growth.

PRICING VULNERABLE AMERICAN PUT OPTIONS UNDER JUMP–DIFFUSION PROCESSES

2016 ◽  
Vol 31 (2) ◽  
pp. 121-138 ◽  
Author(s):  
Guanying Wang ◽  
Xingchun Wang ◽  
Zhongyi Liu
Keyword(s):  
Default Risk ◽  
Diffusion Processes ◽  
Compound Poisson Process ◽  
Jump Diffusion ◽  
Option Prices ◽  
Jump Diffusions ◽  
Put Options ◽  
Jump Diffusion Processes ◽  
American Put Options ◽  
American Put

This paper evaluates vulnerable American put options under jump–diffusion assumptions on the underlying asset and the assets of the counterparty. Sudden shocks on the asset prices are described as a compound Poisson process. Analytical pricing formulae of vulnerable European put options and vulnerable twice-exercisable European put options are derived. Employing the two-point Geske and Johnson method, we derive an approximate analytical pricing formula of vulnerable American put options under jump–diffusions. Numerical simulations are performed for investigating the impacts of jumps and default risk on option prices.

CREDIT MODELING UNDER JUMP DIFFUSIONS WITH EXPONENTIALLY DISTRIBUTED JUMPS — STABLE CALIBRATION, DYNAMICS AND GAP RISK

2013 ◽  
Vol 16 (04) ◽  
pp. 1350021 ◽  
Author(s):  
MARTIN HELLMICH ◽  
STEFAN KASSBERGER ◽  
WOLFGANG M. SCHMIDT
Keyword(s):  
Credit Default Swap ◽  
Jump Diffusion ◽  
Credit Spreads ◽  
Jump Diffusions ◽  
Time Dynamics ◽  
Credit Default ◽  
Gap Risk ◽  
Default Swap ◽  
Swap Spreads ◽  
Quantities Of Interest

This paper investigates a structural credit default model that is based on a hyper-exponential jump diffusion process for the value of the firm. For credit default swap prices and other quantities of interest, explicit expressions for the corresponding Laplace transforms are derived. The time-dynamics of the model are studied, particularly the jumps in credit spreads, the understanding of which is crucial e.g. for the pricing of gap risk. As an application of our findings, the model is calibrated to credit default swap spreads observed in the market.

scholarly journals A data-driven method to learn a jump diffusion process from aggregate biological gene expression data

2021 ◽  
Author(s):  
Jia-Xing Gao ◽  
Zhen-Yi Wang ◽  
Michael Q. Zhang ◽  
Min-Ping Qian ◽  
Da-Quan Jiang
Keyword(s):  
Gene Expression ◽  
Diffusion Process ◽  
Gene Expression Data ◽  
Dynamic Models ◽  
Empirical Distribution ◽  
Jump Diffusion ◽  
Parametric Models ◽  
Expression Data ◽  
Instantaneous Rate ◽  
Jump Diffusion Process

AbstractDynamic models of gene expression are urgently required. Different from trajectory inference and RNA velocity, our method reveals gene dynamics by learning a jump diffusion process for modeling the biological process directly. The algorithm needs aggregate gene expression data as input and outputs the parameters of the jump diffusion process. The learned jump diffusion process can predict population distributions of gene expression at any developmental stage, achieve long-time trajectories for individual cells, and offer a novel approach to computing RNA velocity. Moreover, it studies biological systems from a stochastic dynamics perspective. Gene expression data at a time point, which is a snapshot of a cellular process, is treated as an empirical marginal distribution of a stochastic process. The Wasserstein distance between the empirical distribution and predicted distribution by the jump diffusion process is minimized to learn the dynamics. For the learned jump diffusion equation, its trajectories correspond to the development process of cells and stochasticity determines the heterogeneity of cells. Its instantaneous rate of state change can be taken as “RNA velocity”, and the changes in scales and orientations of clusters can be noticed too. We demonstrate that our method can recover the underlying nonlinear dynamics better compared to parametric models and diffusion processes driven by Brownian motion for both synthetic and real world datasets. Our method is also robust to perturbations of data because it only involves population expectations.

scholarly journals The Robust Consensus of a Noisy Deffuant-Weisbuch Model

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Jiangbo Zhang ◽  
Yiyi Zhao
Keyword(s):  
Discrete Time ◽  
Model Parameters ◽  
Positive Probability ◽  
Noise Strength ◽  
Opinion Formation ◽  
Time Step ◽  
Absolute Value ◽  
The Difference ◽  
Robust Consensus

We construct a new opinion formation of the Deffuant-Weisbuch model with the interference of the outer noise, where there are finite n agents and the evolution is discrete-time. The opinion interaction occurs by one randomly chosen pair at each time step. The difference to the original Deffuant-Weisbuch model is that communications of any selected pairs will be affected by noises. The aim of this paper is to study the robust consensus of this noisy Deffuant-Weisbuch model. We first define the noise strength as the maximum noise absolute value. We will then show that when the noise strength is less than a certain threshold, this noisy model will achieve T-robust consensus when t is sufficiently large; next we prove that the noisy model achieves robust consensus with a positive probability; finally, we demonstrate these results and provide numerical relations among the noise strength and some model parameters.

Concentration inequalities for the empirical distribution of discrete distributions: beyond the method of types

2019 ◽  
Vol 9 (4) ◽  
pp. 813-850 ◽  
Author(s):  
Jay Mardia ◽  
Jiantao Jiao ◽  
Ervin Tánczos ◽  
Robert D Nowak ◽  
Tsachy Weissman
Keyword(s):  
Sample Size ◽  
Empirical Distribution ◽  
Discrete Distributions ◽  
Concentration Inequalities ◽  
Sample Sizes ◽  
Alphabet Size ◽  
Large Sample ◽  
Kl Divergence ◽  
The Difference ◽  
True Distribution

Abstract We study concentration inequalities for the Kullback–Leibler (KL) divergence between the empirical distribution and the true distribution. Applying a recursion technique, we improve over the method of types bound uniformly in all regimes of sample size $n$ and alphabet size $k$, and the improvement becomes more significant when $k$ is large. We discuss the applications of our results in obtaining tighter concentration inequalities for $L_1$ deviations of the empirical distribution from the true distribution, and the difference between concentration around the expectation or zero. We also obtain asymptotically tight bounds on the variance of the KL divergence between the empirical and true distribution, and demonstrate their quantitatively different behaviours between small and large sample sizes compared to the alphabet size.

scholarly journals ESTIMATION OF VOLATILITY FUNCTIONS IN JUMP DIFFUSIONS USING TRUNCATED BIPOWER INCREMENTS

2020 ◽  
pp. 1-33
Author(s):  
Jihyun Kim ◽  
Joon Y. Park ◽  
Bin Wang
Keyword(s):  
Kernel Estimation ◽  
Jump Diffusion ◽  
Sampling Interval ◽  
Diffusion Models ◽  
Time Span ◽  
Estimation Technique ◽  
Jump Diffusions ◽  
Empirical Analyses

In this article, we introduce and analyze a new methodology to estimate the volatility functions of jump diffusion models. Our methodology relies on the standard kernel estimation technique using truncated bipower increments. The relevant asymptotics are fully developed, allowing for the time span to increase as well as the sampling interval to decrease, and accommodate both stationary and nonstationary recurrent processes. We evaluate the performance of our estimators by simulation and provide some illustrative empirical analyses.

scholarly journals Numerical Investigation on the Water Entry of Several Different Bow-Flared Sections

2020 ◽  
Vol 10 (22) ◽  
pp. 7952
Author(s):  
Qiang Wang ◽  
Boran Zhang ◽  
Pengyao Yu ◽  
Guangzhao Li ◽  
Zhijiang Yuan
Keyword(s):  
Stokes Equations ◽  
Water Entry ◽  
Navier Stokes ◽  
Hydrodynamic Characteristics ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Surface Evolution ◽  
Mesh Method ◽  
The Difference ◽  
Dynamics Method

The bow-flared section may be simplified in the prediction of slamming loads and whipping responses of ships. However, the difference of hydrodynamic characteristics between the water entry of the simplified sections and that of the original section has not been well documented. In this study, the water entry of several different bow-flared sections was numerically investigated using the computational fluid dynamics method based on Reynolds-averaged Navier–Stokes equations. The motion of the grid around the section was realized using the overset mesh method. Reasonable grid size and time step were determined through convergence studies. The application of the numerical method in the water entry of bow-flared sections was validated by comparing the present predictions with previous numerical and experimental results. Through a comparative study on the water entry of one original section and three simplified sections, the influences of simplification of the bow-flared section on hydrodynamic characteristics, free surface evolution, pressure field, and impact force were investigated and are discussed here.

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