The passage from random walk to diffusion in quantum probability

1988 ◽  
Vol 25 (A) ◽  
pp. 151-166 ◽  
Author(s):  
K. R. Parthasarathy
Keyword(s):  
Random Walk ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Discrete Time ◽  
Continuous Time ◽  
Quantum Probability ◽  
Diffusion Limit ◽  
Quantum Random Walk

The notion of a quantum random walk in discrete time is formulated and the passage to a continuous time diffusion limit is established. The limiting diffusion is described in terms of solutions of certain quantum stochastic differential equations.

The passage from random walk to diffusion in quantum probability

1988 ◽  
Vol 25 (A) ◽  
pp. 151-166 ◽  
Author(s):  
K. R. Parthasarathy
Keyword(s):  
Random Walk ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Discrete Time ◽  
Continuous Time ◽  
Quantum Probability ◽  
Diffusion Limit ◽  
Quantum Random Walk

The notion of a quantum random walk in discrete time is formulated and the passage to a continuous time diffusion limit is established. The limiting diffusion is described in terms of solutions of certain quantum stochastic differential equations.

scholarly journals Some Applications of Lévy Processes to Stochastic Investment Models for Actuarial Use

1998 ◽  
Vol 28 (1) ◽  
pp. 77-93 ◽  
Author(s):  
Terence Chan
Keyword(s):  
Time Series ◽  
Brownian Motion ◽  
Differential Equations ◽  
White Noise ◽  
Stochastic Differential Equations ◽  
Continuous Time ◽  
Investment Model ◽  
Stable Processes ◽  
Gamma Processes ◽  
Investment Models

AbstractThis paper presents a continuous time version of a stochastic investment model originally due to Wilkie. The model is constructed via stochastic differential equations. Explicit distributions are obtained in the case where the SDEs are driven by Brownian motion, which is the continuous time analogue of the time series with white noise residuals considered by Wilkie. In addition, the cases where the driving “noise” are stable processes and Gamma processes are considered.

scholarly journals Dividend Moments in the Dual Risk Model: Exact and Approximate Approaches

2008 ◽  
Vol 38 (02) ◽  
pp. 399-422 ◽  
Author(s):  
Eric C.K. Cheung ◽  
Steve Drekic
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Size Distribution ◽  
Discrete Time ◽  
Continuous Time ◽  
Risk Model ◽  
Time Model ◽  
Jump Size ◽  
Time Of Ruin ◽  
The Time Of Ruin

In the classical compound Poisson risk model, it is assumed that a company (typically an insurance company) receives premium at a constant rate and pays incurred claims until ruin occurs. In contrast, for certain companies (typically those focusing on invention), it might be more appropriate to assume expenses are paid at a fixed rate and occasional random income is earned. In such cases, the surplus process of the company can be modelled as a dual of the classical compound Poisson model, as described in Avanzi et al. (2007). Assuming further that a barrier strategy is applied to such a model (i.e., any overshoot beyond a fixed level caused by an upward jump is paid out as a dividend until ruin occurs), we are able to derive integro-differential equations for the moments of the total discounted dividends as well as the Laplace transform of the time of ruin. These integro-differential equations can be solved explicitly assuming the jump size distribution has a rational Laplace transform. We also propose a discrete-time analogue of the continuous-time dual model and show that the corresponding quantities can be solved for explicitly leaving the discrete jump size distribution arbitrary. While the discrete-time model can be considered as a stand-alone model, it can also serve as an approximation to the continuous-time model. Finally, we consider a generalization of the so-called Dickson-Waters modification in optimal dividends problems by maximizing the difference between the expected value of discounted dividends and the present value of a fixed penalty applied at the time of ruin.

CONTINUOUS TIME RANDOM WALK AND TIME FRACTIONAL DIFFUSION: A NUMERICAL COMPARISON BETWEEN THE FUNDAMENTAL SOLUTIONS

2005 ◽  
Vol 05 (02) ◽  
pp. L291-L297 ◽  
Author(s):  
FRANCESCO MAINARDI ◽  
ALESSANDRO VIVOLI ◽  
RUDOLF GORENFLO
Keyword(s):  
Random Walk ◽  
Continuous Time ◽  
Fundamental Solutions ◽  
Fractional Diffusion ◽  
Continuous Time Random Walk ◽  
Diffusion Limit ◽  
Numerical Comparison ◽  
Time Fractional Diffusion Equation ◽  
Quality Of Approximation

We consider the basic models for anomalous transport provided by the integral equation for continuous time random walk (CTRW) and by the time fractional diffusion equation to which the previous equation is known to reduce in the diffusion limit. We compare the corresponding fundamental solutions of these equations, in order to investigate numerically the increasing quality of approximation with advancing time.

On effects of discretization on estimators of drift parameters for diffusion processes

1996 ◽  
Vol 33 (04) ◽  
pp. 1061-1076 ◽  
Author(s):  
P. E. Kloeden ◽  
E. Platen ◽  
H. Schurz ◽  
M. Sørensen
Keyword(s):  
Parameter Estimation ◽  
Maximum Likelihood ◽  
Numerical Methods ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Discrete Time ◽  
Diffusion Processes ◽  
Maximum Likelihood Estimators ◽  
Statistical Properties ◽  
Stochastic Integrals

In this paper statistical properties of estimators of drift parameters for diffusion processes are studied by modern numerical methods for stochastic differential equations. This is a particularly useful method for discrete time samples, where estimators can be constructed by making discrete time approximations to the stochastic integrals appearing in the maximum likelihood estimators for continuously observed diffusions. A review is given of the necessary theory for parameter estimation for diffusion processes and for simulation of diffusion processes. Three examples are studied.

Continuous Autoregressive Moving Average Models

2021 ◽  
pp. 118-141
Author(s):  
Yakup Ari
Keyword(s):  
Time Series ◽  
Parameter Estimation ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Discrete Time ◽  
Moving Average ◽  
Likelihood Estimation ◽  
Real Data ◽  
Autoregressive Moving Average ◽  
The Difference

The financial time series have a high frequency and the difference between their observations is not regular. Therefore, continuous models can be used instead of discrete-time series models. The purpose of this chapter is to define Lévy-driven continuous autoregressive moving average (CARMA) models and their applications. The CARMA model is an explicit solution to stochastic differential equations, and also, it is analogue to the discrete ARMA models. In order to form a basis for CARMA processes, the structures of discrete-time processes models are examined. Then stochastic differential equations, Lévy processes, compound Poisson processes, and variance gamma processes are defined. Finally, the parameter estimation of CARMA(2,1) is discussed as an example. The most common method for the parameter estimation of the CARMA process is the pseudo maximum likelihood estimation (PMLE) method by mapping the ARMA coefficients to the corresponding estimates of the CARMA coefficients. Furthermore, a simulation study and a real data application are given as examples.

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