scholarly journals Stochastic stability of some state-dependent growth-collapse processes

2007 ◽  
Vol 39 (1) ◽  
pp. 189-220 ◽  
Author(s):  
Christian Y. Robert
Keyword(s):  
Random Walk ◽  
Discrete Time ◽  
Power Law ◽  
Stochastic Stability ◽  
Long Range Dependence ◽  
Time Process ◽  
Power Law Distribution ◽  
State Dependent ◽  
Dependent Growth ◽  
Collapse Process

In this paper we consider a discrete-time process which grows according to a random walk with nonnegative increments between crash times at which it collapses to 0. We assume that the probability of crashing depends on the level of the process. We study the stochastic stability of this growth-collapse process. Special emphasis is given to the case in which the probability of crashing tends to 0 as the level of the process increases. In particular, we show that the process may exhibit long-range dependence and that the crash sizes may have a power law distribution.

scholarly journals Stochastic stability of some state-dependent growth-collapse processes

2007 ◽  
Vol 39 (01) ◽  
pp. 189-220
Author(s):  
Christian Y. Robert
Keyword(s):  
Random Walk ◽  
Discrete Time ◽  
Power Law ◽  
Stochastic Stability ◽  
Long Range Dependence ◽  
Time Process ◽  
Power Law Distribution ◽  
State Dependent ◽  
Dependent Growth ◽  
Collapse Process

In this paper we consider a discrete-time process which grows according to a random walk with nonnegative increments between crash times at which it collapses to 0. We assume that the probability of crashing depends on the level of the process. We study the stochastic stability of this growth-collapse process. Special emphasis is given to the case in which the probability of crashing tends to 0 as the level of the process increases. In particular, we show that the process may exhibit long-range dependence and that the crash sizes may have a power law distribution.

HOW LONG CAN YOU ENJOY BLACKJACK WITH 100 CHIPS?

2011 ◽  
Vol 22 (10) ◽  
pp. 1161-1171
Author(s):  
TAO YANG ◽  
QIANQIAN LI ◽  
XINGANG XIA ◽  
ERBO ZHAO ◽  
GUO LIU ◽  
...  
Keyword(s):  
Decision Making ◽  
Random Walk ◽  
Power Law ◽  
Risk Taking ◽  
Real World ◽  
Power Law Distribution ◽  
Related Research ◽  
Fat Tail ◽  
Playing Time ◽  
Pathologic Gambling

Gambling-related research has implications in financial area understandings and applications. Researches in this area usually focus on pathology, risk-taking, decision-making and addiction. Few works have been done to demonstrate the distribution of the playing time before players go bankrupt. One problem is that it is difficult to get statistics in real world gambling. In this paper, we do simulations in a Blackjack game with a selected strategy. We find the distribution of playing time before players lose a certain amount of money as a power law distribution, indicating the existence of very long playing time players. We also find that double is the most important factor that causes the fat tail. Comparison shows that when removing double, split and three to two payoff, Blackjack goes back to a random walk. The increase of the number of decks somewhat decreases the average playing time. Our results may have pathologic gambling intervention implications.

scholarly journals An Agent-Based Model of a Pricing Process with Power Law, Volatility Clustering, and Jumps

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Yu Shi ◽  
Qixuan Luo ◽  
Handong Li
Keyword(s):  
Random Walk ◽  
Power Law ◽  
Asset Prices ◽  
Structural Changes ◽  
Power Law Distribution ◽  
Volatility Clustering ◽  
Agent Based ◽  
Tail Distribution ◽  
Pricing Process ◽  
Security Price

In this paper, we propose a new model of security price dynamics in order to explain the stylized facts of the pricing process such as power law distribution, volatility clustering, jumps, and structural changes. We assume that there are two types of agents in the financial market: speculators and fundamental investors. Speculators use past prices to predict future prices and only buy assets whose prices are expected to rise. Fundamental investors attach a certain value to each asset and buy when the asset is undervalued by the market. When the expectations of agents are exogenously driven, that is, entirely shaped by exogenous news, then they can be modeled as following a random walk. We assume that the information related to the two types of agents in the model will arrive randomly with a certain probability distribution and change the viewpoint of the agents according to a certain percentage. Our simulated results show that this model can simulate well the random walk of asset prices and explain the power-law tail distribution of returns, volatility clustering, jumps, and structural changes of asset prices.

Simple random walk statistics. Part II: Continuous time results

1996 ◽  
Vol 33 (02) ◽  
pp. 331-339 ◽  
Author(s):  
W. Böhm ◽  
W. Panny
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Discrete Time ◽  
Continuous Time ◽  
Simple Random Walk ◽  
Laplace Transforms ◽  
Time Process ◽  
Basic Approach

In this paper various statistics for randomized random walks and their distributions are presented. The distributional results are derived by means of a limiting procedure applied to the pertaining discrete time process, which has been considered in part I of this work (Katzenbeisser and Panny 1996). This basic approach, originally due to Meisling (1958), seems to offer certain technical advantages, since it avoids the use of Laplace transforms and is even simpler than Feller's randomization technique.

A power-law model and other models for long-range dependence

1997 ◽  
Vol 34 (3) ◽  
pp. 657-670 ◽  
Author(s):  
R. J. Martin ◽  
A. M. Walker
Keyword(s):  
Long Range ◽  
Discrete Time ◽  
Power Law ◽  
Long Range Dependence ◽  
Stationary Processes ◽  
Power Law Model ◽  
Slow Decay ◽  
Exact Power ◽  
Correlation Structures ◽  
Discrete Time Models

It is becoming increasingly recognized that some long series of data can be adequately and parsimoniously modelled by stationary processes with long-range dependence. Some new discrete-time models for long-range dependence or slow decay, defined by their correlation structures, are discussed. The exact power-law correlation structure is examined in detail.

Simple random walk statistics. Part II: Continuous time results

1996 ◽  
Vol 33 (2) ◽  
pp. 331-339 ◽  
Author(s):  
W. Böhm ◽  
W. Panny
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Discrete Time ◽  
Continuous Time ◽  
Simple Random Walk ◽  
Laplace Transforms ◽  
Time Process ◽  
Basic Approach

In this paper various statistics for randomized random walks and their distributions are presented. The distributional results are derived by means of a limiting procedure applied to the pertaining discrete time process, which has been considered in part I of this work (Katzenbeisser and Panny 1996). This basic approach, originally due to Meisling (1958), seems to offer certain technical advantages, since it avoids the use of Laplace transforms and is even simpler than Feller's randomization technique.

A power-law model and other models for long-range dependence

1997 ◽  
Vol 34 (03) ◽  
pp. 657-670 ◽  
Author(s):  
R. J. Martin ◽  
A. M. Walker
Keyword(s):  
Long Range ◽  
Discrete Time ◽  
Power Law ◽  
Long Range Dependence ◽  
Stationary Processes ◽  
Power Law Model ◽  
Slow Decay ◽  
Exact Power ◽  
Correlation Structures ◽  
Discrete Time Models

It is becoming increasingly recognized that some long series of data can be adequately and parsimoniously modelled by stationary processes with long-range dependence. Some new discrete-time models for long-range dependence or slow decay, defined by their correlation structures, are discussed. The exact power-law correlation structure is examined in detail.

scholarly journals Transient and slim versus recurrent and fat: Random walks and the trees they grow

2019 ◽  
Vol 56 (3) ◽  
pp. 769-786
Author(s):  
Giulio Iacobelli ◽  
Daniel R. Figueiredo ◽  
Giovanni Neglia
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Exponential Distribution ◽  
Power Law ◽  
Random Network ◽  
Power Law Distribution ◽  
Current Position ◽  
Network Growth ◽  
Model Driven ◽  
Bounded Below

AbstractThe no restart random walk (NRRW) is a random network growth model driven by a random walk that builds the graph while moving on it, adding and connecting a new leaf node to the current position of the walker every s steps. We show a fundamental dichotomy in NRRW with respect to the parity of s: for ${s}=1$ we prove that the random walk is transient and non-leaf nodes have degrees bounded above by an exponential distribution; for s even we prove that the random walk is recurrent and non-leaf nodes have degrees bounded below by a power law distribution. These theoretical findings highlight and confirm the diverse and rich behaviour of NRRW observed empirically.

scholarly journals Should i stay or should i go? zero-size jumps in random walks for lévy flights

2021 ◽  
Vol 24 (1) ◽  
pp. 137-167
Author(s):  
Gianni Pagnini ◽  
Silvia Vitali
Keyword(s):  
Random Walk ◽  
Diffusion Equation ◽  
Power Law ◽  
Site Fidelity ◽  
Fractional Diffusion ◽  
Wild Animals ◽  
Power Law Distribution ◽  
Lévy Flights ◽  
Levy Flights ◽  
Modal Power

Abstract We study Markovian continuos-time random walk models for Lévy flights and we show an example in which the convergence to stable densities is not guaranteed when jumps follow a bi-modal power-law distribution that is equal to zero in zero. The significance of this result is two-fold: i) with regard to the probabilistic derivation of the fractional diffusion equation and also ii) with regard to the concept of site fidelity in the framework of Lévy-like motion for wild animals.

scholarly journals The structure of co-publications multilayer network

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Ghislain Romaric Meleu ◽  
Paulin Yonta Melatagia
Keyword(s):  
Power Law ◽  
Degree Distribution ◽  
Preferential Attachment ◽  
Small World ◽  
Generation Model ◽  
Small World Network ◽  
Power Law Distribution ◽  
Multilayer Networks ◽  
Multilayer Network ◽  
Scientific Papers

AbstractUsing the headers of scientific papers, we have built multilayer networks of entities involved in research namely: authors, laboratories, and institutions. We have analyzed some properties of such networks built from data extracted from the HAL archives and found that the network at each layer is a small-world network with power law distribution. In order to simulate such co-publication network, we propose a multilayer network generation model based on the formation of cliques at each layer and the affiliation of each new node to the higher layers. The clique is built from new and existing nodes selected using preferential attachment. We also show that, the degree distribution of generated layers follows a power law. From the simulations of our model, we show that the generated multilayer networks reproduce the studied properties of co-publication networks.

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