scholarly journals Ruin probability for discrete risk processes

2019 ◽  
Vol 56 (4) ◽  
pp. 420-439
Author(s):  
Ivana Geček Tuden
Keyword(s):  
Random Walk ◽  
Discrete Time ◽  
Continuous Time ◽  
Ruin Probability ◽  
Risk Process ◽  
Dual Process ◽  
Type Formula ◽  
Fixed Level ◽  
Risk Processes

Abstract We study the discrete time risk process modelled by the skip-free random walk and derive results connected to the ruin probability and crossing a fixed level for this type of process. We use the method relying on the classical ballot theorems to derive the results for crossing a fixed level and compare them to the results known for the continuous time version of the risk process. We generalize this model by adding a perturbation and, still relying on the skip-free structure of that process, we generalize the previous results on crossing the fixed level for the generalized discrete time risk process. We further derive the famous Pollaczek-Khinchine type formula for this generalized process, using the decomposition of the supremum of the dual process at some special instants of time.

scholarly journals Ruin probability with certain stationary stable claims generated by conservative flows

2007 ◽  
Vol 39 (02) ◽  
pp. 360-384 ◽  
Author(s):  
Uğur Tuncay Alparslan ◽  
Gennady Samorodnitsky
Keyword(s):  
Markov Chains ◽  
Discrete Time ◽  
Continuous Time ◽  
Ruin Probability ◽  
Stable Process ◽  
Independent Interest ◽  
Stable Processes ◽  
Auxiliary Result ◽  
Initial Capital ◽  
Order Of Magnitude

We study the ruin probability where the claim sizes are modeled by a stationary ergodic symmetric α-stable process. We exploit the flow representation of such processes, and we consider the processes generated by conservative flows. We focus on two classes of conservative α-stable processes (one discrete-time and one continuous-time), and give results for the order of magnitude of the ruin probability as the initial capital goes to infinity. We also prove a solidarity property for null-recurrent Markov chains as an auxiliary result, which might be of independent interest.

Amplitudes of Composition Fluctuations in the Simplest Chemical Reaction Equilibrium

2004 ◽  
Vol 218 (9) ◽  
pp. 1033-1040 ◽  
Author(s):  
M. Šolc ◽  
J. Hostomský
Keyword(s):  
Random Walk ◽  
Discrete Time ◽  
Continuous Time ◽  
Time Scale ◽  
Numerical Study ◽  
Mean Amplitude ◽  
Time Interval ◽  
The Mean ◽  
Composition Fluctuations ◽  
Number Of Particles

AbstractWe present a numerical study of equilibrium composition fluctuations in a system where the reaction X1 ⇔ X2 having the equilibrium constant equal to 1 takes place. The total number of reacting particles is N. On a discrete time scale, the amplitude of a fluctuation having the lifetime 2r reaction events is defined as the difference between the number of particles X1 in the microstate most distant from the microstate N/2 visited at least once during the fluctuation lifetime, and the equilibrium number of particles X1, N/2. On the discrete time scale, the mean value of this amplitude, m̅(r̅), is calculated in the random walk approximation. On a continuous time scale, the average amplitude of fluctuations chosen randomly and regardless of their lifetime from an ensemble of fluctuations occurring within the time interval (0,z), z → ∞, tends with increasing N to ~1.243 N0.25. Introducing a fraction of fluctuation lifetime during which the composition of the system spends below the mean amplitude m̅(r̅), we obtain a value of the mean amplitude of equilibrium fluctuations on the continuous time scale equal to ~1.19√N. The results suggest that using the random walk value m̅(r̅) and taking into account a) the exponential density of fluctuations lifetimes and b) the fact that the time sequence of reaction events represents the Poisson process, we obtain values of fluctuations amplitudes which differ only slightly from those derived for the Ehrenfest model.

On a Conjecture of Bollobás and Brightwell Concerning Random Walks on Product Graphs

1998 ◽  
Vol 7 (4) ◽  
pp. 397-401 ◽  
Author(s):  
OLLE HÄGGSTRÖM
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Discrete Time ◽  
Complete Graph ◽  
Continuous Time ◽  
Product Graph ◽  
Product Graphs ◽  
Continuous Time Random Walks ◽  
Discrete Time Case

We consider continuous time random walks on a product graph G×H, where G is arbitrary and H consists of two vertices x and y linked by an edge. For any t>0 and any a, b∈V(G), we show that the random walk starting at (a, x) is more likely to have hit (b, x) than (b, y) by time t. This contrasts with the discrete time case and proves a conjecture of Bollobás and Brightwell. We also generalize the result to cases where H is either a complete graph on n vertices or a cycle on n vertices.

scholarly journals Limiting stochastic processes of shift-periodic dynamical systems

2019 ◽  
Vol 6 (11) ◽  
pp. 191423
Author(s):  
Julia Stadlmann ◽  
Radek Erban
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Dynamical Systems ◽  
Stochastic Processes ◽  
Discrete Time ◽  
Real Line ◽  
Continuous Time ◽  
Initial Distribution ◽  
Dynamical Behaviour ◽  
One Dimensional

A shift-periodic map is a one-dimensional map from the real line to itself which is periodic up to a linear translation and allowed to have singularities. It is shown that iterative sequences x n +1 = F ( x n ) generated by such maps display rich dynamical behaviour. The integer parts ⌊ x n ⌋ give a discrete-time random walk for a suitable initial distribution of x 0 and converge in certain limits to Brownian motion or more general Lévy processes. Furthermore, for certain shift-periodic maps with small holes on [0,1], convergence of trajectories to a continuous-time random walk is shown in a limit.

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.

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.

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.

scholarly journals Leaving an interval in limited playing time

1988 ◽  
Vol 20 (3) ◽  
pp. 635-645 ◽  
Author(s):  
David Heath ◽  
Robert P. Kertz
Keyword(s):  
Random Walk ◽  
Positive Integer ◽  
Discrete Time ◽  
Continuous Time ◽  
Value Function ◽  
Time Instant ◽  
Diffusion Parameter ◽  
Time Problem ◽  
Playing Time ◽  
The Value Function

A player starts at x in (-G, G) and attempts to leave the interval in a limited playing time. In the discrete-time problem, G is a positive integer and the position is described by a random walk starting at integer x, with mean increments zero, and variance increment chosen by the player from [0, 1] at each integer playing time. In the continuous-time problem, the player's position is described by an Ito diffusion process with infinitesimal mean parameter zero and infinitesimal diffusion parameter chosen by the player from [0, 1] at each time instant of play. To maximize the probability of leaving the interval (–G, G) in a limited playing time, the player should play boldly by always choosing largest possible variance increment in the discrete-time setting and largest possible diffusion parameter in the continuous-time setting, until the player leaves the interval. In the discrete-time setting, this result affirms a conjecture of Spencer. In the continuous-time setting, the value function of play is identified.

scholarly journals Leaving an interval in limited playing time

1988 ◽  
Vol 20 (03) ◽  
pp. 635-645 ◽  
Author(s):  
David Heath ◽  
Robert P. Kertz
Keyword(s):  
Random Walk ◽  
Positive Integer ◽  
Discrete Time ◽  
Continuous Time ◽  
Value Function ◽  
Time Instant ◽  
Diffusion Parameter ◽  
Time Problem ◽  
Playing Time ◽  
The Value Function

A player starts at x in (-G, G) and attempts to leave the interval in a limited playing time. In the discrete-time problem, G is a positive integer and the position is described by a random walk starting at integer x, with mean increments zero, and variance increment chosen by the player from [0, 1] at each integer playing time. In the continuous-time problem, the player's position is described by an Ito diffusion process with infinitesimal mean parameter zero and infinitesimal diffusion parameter chosen by the player from [0, 1] at each time instant of play. To maximize the probability of leaving the interval (–G, G) in a limited playing time, the player should play boldly by always choosing largest possible variance increment in the discrete-time setting and largest possible diffusion parameter in the continuous-time setting, until the player leaves the interval. In the discrete-time setting, this result affirms a conjecture of Spencer. In the continuous-time setting, the value function of play is identified.

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