A note on absorption probabilities in one-dimensional random walk via complex-valued martingales

Abstract In this note we study the probability and the mean time for absorption for discrete time Markov chains. In particular, we are interested in estimating the mean time for absorption when absorption is not certain and connect it with some other known results. Computing a suitable probability generating function, we are able to estimate the mean time for absorption when absorption is not certain giving some applications concerning the random walk. Furthermore, we investigate the probability for a Markov chain to reach a set A before reach B generalizing this result for a sequence of sets A 1 , A 2 , … , A k {A_{1},A_{2},\dots,A_{k}} .

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Limit diffusion process for a non-homogeneous random walk on a one-dimensional lattice

Russian Mathematical Surveys ◽

10.1070/rm1997v052n02abeh001778 ◽

1997 ◽

Vol 52 (2) ◽

pp. 327-340 ◽

Cited By ~ 2

Author(s):

R A Minlos ◽

E A Zhizhina

Keyword(s):

Random Walk ◽

Diffusion Process ◽

Dimensional Lattice ◽

One Dimensional ◽

Limit Diffusion

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On coupling of random walks and renewal processes

Journal of Applied Probability ◽

10.2307/3215269 ◽

1996 ◽

Vol 33 (1) ◽

pp. 122-126

Author(s):

Torgny Lindvall ◽

L. C. G. Rogers

Keyword(s):

Random Walk ◽

Random Walks ◽

Simple Random Walk ◽

Renewal Processes ◽

Renewal Theorem ◽

One Dimensional ◽

Continuous State Space ◽

Continuous State ◽

Efficient Coupling ◽

Blackwell's Renewal Theorem

The use of Mineka coupling is extended to a case with a continuous state space: an efficient coupling of random walks S and S' in can be made such that S' — S is virtually a one-dimensional simple random walk. This insight settles a zero-two law of ergodicity. One more proof of Blackwell's renewal theorem is also presented.

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Optimal Strategies for Prudent Investors

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024998000254 ◽

1998 ◽

Vol 01 (04) ◽

pp. 473-486 ◽

Cited By ~ 10

Author(s):

Roberto Baviera ◽

Michele Pasquini ◽

Maurizio Serva ◽

Angelo Vulpiani

Keyword(s):

Growth Rate ◽

Random Walk ◽

Exponential Growth ◽

Optimal Strategies ◽

Maximum Amount ◽

Exponential Growth Rate ◽

One Dimensional ◽

Economic Level ◽

Random Walk With Drift ◽

Analytical Expressions

We consider a stochastic model of investment on an asset in a stock market for a prudent investor. she decides to buy permanent goods with a fraction α of the maximum amount of money owned in her life in order that her economic level never decreases. The optimal strategy is obtained by maximizing the exponential growth rate for a fixed α. We derive analytical expressions for the typical exponential growth rate of the capital and its fluctuations by solving an one-dimensional random walk with drift.

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Quantum walks and elliptic integrals

Mathematical Structures in Computer Science ◽

10.1017/s0960129510000393 ◽

2010 ◽

Vol 20 (6) ◽

pp. 1091-1098 ◽

Cited By ~ 3

Author(s):

NORIO KONNO

Keyword(s):

Random Walk ◽

Generating Function ◽

Elliptic Integral ◽

Quantum Walk ◽

Quantum Walks ◽

Elliptic Integrals ◽

Two Dimensional ◽

One Dimensional ◽

Similar Expression ◽

Return Probability

Pólya showed in his 1921 paper that the generating function of the return probability for a two-dimensional random walk can be written in terms of an elliptic integral. In this paper we present a similar expression for a one-dimensional quantum walk.

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One dimensional quantum walks with memory

Quantum Information and Computation ◽

10.26421/qic10.5-6-9 ◽

2010 ◽

Vol 10 (5&6) ◽

pp. 509-524

Author(s):

M. Mc Gettrick

Keyword(s):

Markov Chain ◽

Random Walk ◽

Random Walks ◽

Quantum Walks ◽

One Dimensional ◽

With Memory ◽

Previous Step

We investigate the quantum versions of a one-dimensional random walk, whose corresponding Markov Chain is of order 2. This corresponds to the walk having a memory of one previous step. We derive the amplitudes and probabilities for these walks, and point out how they differ from both classical random walks, and quantum walks without memory.

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On the spectral properties of the Schrödinger operator with a periodic PT-symmetric potential

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887817500657 ◽

2017 ◽

Vol 14 (05) ◽

pp. 1750065 ◽

Cited By ~ 3

Author(s):

Oktay Veliev

Keyword(s):

Schrödinger Operator ◽

Spectral Properties ◽

Schrodinger Operator ◽

Symmetric Potential ◽

One Dimensional ◽

Symmetric Complex ◽

The One ◽

Complex Valued

In this paper, we investigate the spectrum and spectrality of the one-dimensional Schrödinger operator with a periodic PT-symmetric complex-valued potential.

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