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Martin Möhle - One of the best experts on this subject based on the ideXlab platform.
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On the Number of Jumps of random walks with a barrier
Advances in Applied Probability, 2008Co-Authors: Alex Iksanov, Martin MöhleAbstract:LetS0:= 0 andSk:=ξ1+ ··· +ξkfork∈ ℕ := {1, 2, …}, where {ξk:k∈ ℕ} are independent copies of a random variableξwith values in ℕ and distributionpk:= P{ξ=k},k∈ ℕ. We interpret the random walk {Sk:k= 0, 1, 2, …} as a particle jumping to the right through integer positions. Fixn∈ ℕ and modify the process by requiring that the particle is bumped back to its current state each time a jump would bring the particle to a state larger than or equal ton. This constraint defines an increasing Markov chain {Rk(n):k= 0, 1, 2, …} which never reaches the staten. We call this process a random walk with barriern. LetMndenote the Number of Jumps of the random walk with barriern. This paper focuses on the asymptotics ofMnasntends to ∞. A key observation is that, underp1> 0, {Mn:n∈ ℕ} satisfies the distributional recursionM1= 0 andforn= 2, 3, …, whereInis independent ofM2, …,Mn−1with distribution P{In=k} =pk/ (p1+ ··· +pn−1),k∈ {1, …,n− 1}. Depending on the tail behavior of the distribution ofξ, several scalings forMnand corresponding limiting distributions come into play, including stable distributions and distributions of exponential integrals of subordinators. The methods used in this paper are mainly probabilistic. The key tool is to compare (couple) the Number of Jumps,Mn, with the first time,Nn, when the unrestricted random walk {Sk:k= 0, 1, …} reaches a state larger than or equal ton. The results are applied to derive the asymptotics of the Number of collision events (that take place until there is just a single block) forβ(a,b)-coalescent processes with parameters 0 <a< 2 andb= 1.
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On the Number of Jumps of random walks with a barrier
Advances in Applied Probability, 2008Co-Authors: Alex Iksanov, Martin MöhleAbstract:Let So := 0 and S k := ξ 1 +... + ξk for k ∈N:= {1, 2,...}, where (ξ k : κ e N} are independent copies of a random variable ξ with values in N and distribution P κ := P{ξ = κ), k ∈ N. We interpret the random walk (S k k = 0, 1, 2,...} as a particle jumping to the right through integer positions. Fix n e N and modify the process by requiring that the particle is bumped back to its current state each time a jump would bring the particle to a state larger than or equal to n. This constraint defines an increasing Markov chain {R κ (n) : k = 0, 1, 2,...} which never reaches the state n. We call this process a random walk with barrier n. Let M n denote the Number of Jumps of the random walk with barrier n. This paper focuses on the asymptotics of M n as n tends to ∞. A key observation is that, under p 1 > 0, (M n : n ∈ N} satisfies the distributional recursion M 1 = 0 and M n = M n -I n + 1 for n = 2, 3 where I n is independent of M 2 M n-1 with distribution P(I n = k} = p k / (P1 +... + P n-1 ), k e {1,..., n - 1}. Depending on the tail behavior of the distribution of ξ, several scalings for M n and corresponding limiting distributions come into play, including stable distributions and distributions of exponential integrals of subordinators. The methods used in this paper are mainly probabilistic. The key tool is to compare (couple) the Number of Jumps, M n , with the first time, N n , when the unrestricted random walk (S k : k = 0, 1,...} reaches a state larger than or equal to n. The results are applied to derive the asymptotics of the Number of collision events (that take place until there is just a single block) for β(a, b)-coalescent processes with parameters 0 < a < 2 and b = 1.
P. Tardelli - One of the best experts on this subject based on the ideXlab platform.
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AN ESTIMATE of THE APPROXIMATION ERROR IN THE FILTERING of A DISCRETE JUMP PROCESS
Mathematical Models and Methods in Applied Sciences, 2001Co-Authors: Claudia Ceci, Anna Gerardi, P. TardelliAbstract:This paper deals with the filtering of a discrete Markov jump process when only the total Number of Jumps is observed. By using the recursive structure of the filtering equation, a technique to approximate the filter with a finite-dimensional filter is presented. An explicit bound for the approximation error, both almost surely and in L1, is given.
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An approximation method for controlled discrete jump processes under partial observations
IEEE Transactions on Automatic Control, 2001Co-Authors: Claudia Ceci, Anna Gerardi, P. TardelliAbstract:The paper is concerned with control and filtering of a discrete jump Markov process when only the total Number of Jumps is observed. A partially observable control problem is considered and equivalence between this problem and the separated one is proved. A technique to approximate the value function, under some suitable assumptions, is presented.
Claudia Ceci - One of the best experts on this subject based on the ideXlab platform.
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AN ESTIMATE of THE APPROXIMATION ERROR IN THE FILTERING of A DISCRETE JUMP PROCESS
Mathematical Models and Methods in Applied Sciences, 2001Co-Authors: Claudia Ceci, Anna Gerardi, P. TardelliAbstract:This paper deals with the filtering of a discrete Markov jump process when only the total Number of Jumps is observed. By using the recursive structure of the filtering equation, a technique to approximate the filter with a finite-dimensional filter is presented. An explicit bound for the approximation error, both almost surely and in L1, is given.
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Filtering of a Markov Jump Process with Counting Observations
Constructive Approximation, 2001Co-Authors: Claudia Ceci, Anna GerardiAbstract:This paper concerns the filtering of an R d -valued Markov pure jump process when only the total Number of Jumps are observed. Strong and weak uniqueness for the solutions of the filtering equations are discussed.
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An approximation method for controlled discrete jump processes under partial observations
IEEE Transactions on Automatic Control, 2001Co-Authors: Claudia Ceci, Anna Gerardi, P. TardelliAbstract:The paper is concerned with control and filtering of a discrete jump Markov process when only the total Number of Jumps is observed. A partially observable control problem is considered and equivalence between this problem and the separated one is proved. A technique to approximate the value function, under some suitable assumptions, is presented.
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Partially observed control of a Markov jump process with counting observations: equivalence with the separated problem
Stochastic Processes and their Applications, 1998Co-Authors: Claudia Ceci, Anna GerardiAbstract:This paper concerns a partially observable finite horizon control problem for -valued pure Markov jump process using the information given by the point process which counts the total Number of Jumps. Equivalence between the partially observable control problem and the separated control problem is discussed.
Robert J. Elliott - One of the best experts on this subject based on the ideXlab platform.
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Incomplete markets with Jumps and informed agents
Mathematical Methods of Operations Research (ZOR), 1999Co-Authors: Robert J. Elliott, Monique JeanblancAbstract:An asset is considered whose logarithmic price is the sum of a drift term, a Brownian motion and Jumps of a Poisson process. Various items of future information about the price process are considered available to an informed agent. The optimal attainable wealths of both informed and uninformed agents are compared in the case where the informed agent knows the total Number of Jumps.
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FINITE-DIMENSIONAL MODELS FOR HIDDEN MARKOV CHAINS
Advances in Applied Probability, 1995Co-Authors: Lakhdar Aggoun, Robert J. ElliottAbstract:A continuous-time, non-linear filtering problem is considered in which both signal and observation processes are Markov chains. New finite-dimensional filters and smoothers are obtained for the state of the signal, for the Number of Jumps from one state to another, for the occupation time in any state of the signal, and for joint occupation times of the two processes. These estimates are then used in the expectation maximization algorithm tO improve the parameters in the model. Consequently, our filters and model are adaptive, or self-tuning.
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Finite dimensional predictors for hidden Markov chains
Systems & Control Letters, 1992Co-Authors: Lakhdar Aggoun, Robert J. ElliottAbstract:Abstract A continuous time Markov chain is observed in Gaussian noise. Finite dimensional normalized and unnormalized (Zakai) predictors are obtained for the state of the chain, for the Number of Jumps from one state to another and for the occupation time in any state.
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Finite dimensional filters related to Markov chains
Stochastic Theory and Adaptive Control, 1Co-Authors: Robert J. ElliottAbstract:New finite dimensional filters and smoothers are obtained which are related to the Wonham filter of a noisily observed Markov chain. In particular, finite dimensional, recursive filters and smoothers are given for the Number of Jumps from state i to state j, for the occupation time of state i, and for a stochastic integral related to the drift in the observations. These filters allow easy application of the EM algorithm for the estimation of the parameters of the Markov chain and observation process.
Alex Iksanov - One of the best experts on this subject based on the ideXlab platform.
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On the Number of Jumps of random walks with a barrier
Advances in Applied Probability, 2008Co-Authors: Alex Iksanov, Martin MöhleAbstract:LetS0:= 0 andSk:=ξ1+ ··· +ξkfork∈ ℕ := {1, 2, …}, where {ξk:k∈ ℕ} are independent copies of a random variableξwith values in ℕ and distributionpk:= P{ξ=k},k∈ ℕ. We interpret the random walk {Sk:k= 0, 1, 2, …} as a particle jumping to the right through integer positions. Fixn∈ ℕ and modify the process by requiring that the particle is bumped back to its current state each time a jump would bring the particle to a state larger than or equal ton. This constraint defines an increasing Markov chain {Rk(n):k= 0, 1, 2, …} which never reaches the staten. We call this process a random walk with barriern. LetMndenote the Number of Jumps of the random walk with barriern. This paper focuses on the asymptotics ofMnasntends to ∞. A key observation is that, underp1> 0, {Mn:n∈ ℕ} satisfies the distributional recursionM1= 0 andforn= 2, 3, …, whereInis independent ofM2, …,Mn−1with distribution P{In=k} =pk/ (p1+ ··· +pn−1),k∈ {1, …,n− 1}. Depending on the tail behavior of the distribution ofξ, several scalings forMnand corresponding limiting distributions come into play, including stable distributions and distributions of exponential integrals of subordinators. The methods used in this paper are mainly probabilistic. The key tool is to compare (couple) the Number of Jumps,Mn, with the first time,Nn, when the unrestricted random walk {Sk:k= 0, 1, …} reaches a state larger than or equal ton. The results are applied to derive the asymptotics of the Number of collision events (that take place until there is just a single block) forβ(a,b)-coalescent processes with parameters 0 <a< 2 andb= 1.
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On the Number of Jumps of random walks with a barrier
Advances in Applied Probability, 2008Co-Authors: Alex Iksanov, Martin MöhleAbstract:Let So := 0 and S k := ξ 1 +... + ξk for k ∈N:= {1, 2,...}, where (ξ k : κ e N} are independent copies of a random variable ξ with values in N and distribution P κ := P{ξ = κ), k ∈ N. We interpret the random walk (S k k = 0, 1, 2,...} as a particle jumping to the right through integer positions. Fix n e N and modify the process by requiring that the particle is bumped back to its current state each time a jump would bring the particle to a state larger than or equal to n. This constraint defines an increasing Markov chain {R κ (n) : k = 0, 1, 2,...} which never reaches the state n. We call this process a random walk with barrier n. Let M n denote the Number of Jumps of the random walk with barrier n. This paper focuses on the asymptotics of M n as n tends to ∞. A key observation is that, under p 1 > 0, (M n : n ∈ N} satisfies the distributional recursion M 1 = 0 and M n = M n -I n + 1 for n = 2, 3 where I n is independent of M 2 M n-1 with distribution P(I n = k} = p k / (P1 +... + P n-1 ), k e {1,..., n - 1}. Depending on the tail behavior of the distribution of ξ, several scalings for M n and corresponding limiting distributions come into play, including stable distributions and distributions of exponential integrals of subordinators. The methods used in this paper are mainly probabilistic. The key tool is to compare (couple) the Number of Jumps, M n , with the first time, N n , when the unrestricted random walk (S k : k = 0, 1,...} reaches a state larger than or equal to n. The results are applied to derive the asymptotics of the Number of collision events (that take place until there is just a single block) for β(a, b)-coalescent processes with parameters 0 < a < 2 and b = 1.
