Exponential functionals of Markov additive processes Anita Behme - - PowerPoint PPT Presentation
Exponential functionals of Markov additive processes Anita Behme joint work in progress with Apostolos Sideris May 24th 2019 Probability and Analysis Overview Exponential functionals of L evy processes From L evy processes to
Anita Behme joint work in progress with Apostolos Sideris May 24th 2019 Probability and Analysis
◮ Exponential functionals of L´
evy processes
◮ From L´
evy processes to MAPs
◮ Exponential functionals of MAPs:
◮ Definition ◮ An example ◮ Main results and methodology ◮ Discussion and more results ◮ Open questions Anita Behme P&A 2019, 2
Let (ξt, ηt)t≥0 be a bivariate L´ evy process. The generalized Ornstein-Uhlenbeck (GOU) process (Vt)t≥0 driven by (ξ, η) is given by Vt = e−ξt
eξs−dηs
t ≥ 0, where V0 is a finite random variable, independent of (ξ, η).
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Definition: A L´ evy process in Rd on a probability space (Ω, F, P) is a stochastic process X = (Xt)t≥0, Xt : Ω → Rd satisfying the following properties:
◮ X0 = 0 a.s. ◮ X has independent increments, i.e. for all 0 ≤ t0 ≤ t1 ≤ . . . ≤ tn the
random variables Xt0, Xt1 − Xt0, . . . , Xtn − Xtn−1 are independent.
◮ X has stationary increments, i.e. for all s, t ≥ 0 it holds
Xs+t − Xs
d
= Xt.
◮ X has a.s. c`
adl` ag paths, i.e. for P-a.e. ω ∈ Ω the path t → Xt(ω) is right-continuous in t ≥ 0 and has left limits in t > 0.
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Definition: A L´ evy process in Rd on a probability space (Ω, F, P) is a stochastic process X = (Xt)t≥0, Xt : Ω → Rd satisfying the following properties:
◮ X0 = 0 a.s. ◮ X has independent increments, i.e. for all 0 ≤ t0 ≤ t1 ≤ . . . ≤ tn the
random variables Xt0, Xt1 − Xt0, . . . , Xtn − Xtn−1 are independent.
◮ X has stationary increments, i.e. for all s, t ≥ 0 it holds
Xs+t − Xs
d
= Xt.
◮ X has a.s. c`
adl` ag paths, i.e. for P-a.e. ω ∈ Ω the path t → Xt(ω) is right-continuous in t ≥ 0 and has left limits in t > 0. A L´ evy process X is uniquely determined by its characteristic triplet (γX, σX, νX).
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Theorem (Lindner, Maller ’05): The GOU process Vt = e−ξt
t eξs−dηs
t ≥ 0, solving dVt = Vt−dUt + dLt, with
◮ ξt = − log(E(U)t) ◮ (U, L) (or similarly (ξ, η)) bivariate L´
evy processes
◮ V0 starting random variable independent of (ξ, η)
has a (nontrivial) stationary distribution if and only if the integral V∞ := ∞ e−ξt−dLt converges a.s.
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Theorem (Lindner, Maller ’05): The GOU process Vt = e−ξt
t eξs−dηs
t ≥ 0, solving dVt = Vt−dUt + dLt, with
◮ ξt = − log(E(U)t) ◮ (U, L) (or similarly (ξ, η)) bivariate L´
evy processes
◮ V0 starting random variable independent of (ξ, η)
has a (nontrivial) stationary distribution if and only if the integral V∞ := ∞ e−ξt−dLt converges a.s. The stationary distribution is given by the law of the so-called exponential functional V∞.
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Erickson and Maller (2005) proposed necessary and sufficient conditions for convergence of V∞ := ∞ e−ξt−dηt. Mainly one needs:
◮ ξ tends to infinity ◮ η has a finite log+-moment
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Precisely, they stated Theorem (Erickson, Maller ’05): V∞ exists as a.s. limit as t → ∞ of t
0 e−ξs−dηs if and only if
lim
t→∞ ξt = ∞ a.s. and
Iξ,η =
Aξ(log y)
νη(dy)| < ∞, where Aξ(x) = γξ + ¯ ν+
ξ (1) +
x
1
¯ ν+
ξ (y)dy,
with ¯ ν+
ξ (x) = νξ((x, ∞)),
¯ ν−
ξ (x) = νξ((−∞, −x)),
¯ νξ(x) = ¯ ν+
ξ (x)+¯
ν−
ξ (x),
and ¯ ν+
η , ¯
ν−
η and ¯
νη defined likewise. Hereby a > 0 is chosen such that Aξ(x) > 0 for all x > 0 and its existence is guaranteed whenever limt→∞ ξt = ∞ a.s.
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Further: Theorem (Erickson, Maller ’05, continued): If limt→∞ ξt = ∞ a.s. but Iξ,η = ∞, then
e−ξs−dηs
− → ∞, (1) while for limt→∞ ξt = −∞ or oscillating ξ either (1) holds, or there exists some k ∈ R \ {0} such that t e−ξs−dηs = k(1 − e−ξt) for all t > 0 a.s.
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(ξt, ηt, Jt)t≥0 is a (bivariate) MAP with
◮ Markovian component (Jt)t≥0: Right-continuous, ergodic,
continuous time Markov chain with countable state space S, intensity matrix Q and stationary law π.
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(ξt, ηt, Jt)t≥0 is a (bivariate) MAP with
◮ Markovian component (Jt)t≥0: Right-continuous, ergodic,
continuous time Markov chain with countable state space S, intensity matrix Q and stationary law π.
◮ Additive component (ξt, ηt)t≥0:
(ξt, ηt) := (X (1)
t
, Y (1)
t
) + (X (2)
t
, Y (2)
t
), t ≥ 0.
◮ (X (1)
t
, Y (1)
t
) behaves in law like a bivariate L´ evy process (ξ(j)
t , η(j) t ) whenever Jt = j,
◮ (X (2)
t
, Y (2)
t
) is a pure jump process given by (X (2)
t
, Y (2)
t
) =
Z (i,j)
n
1{JTn−=i,JTn =j,Tn≤t}, for i.i.d. random variables Z (i,j)
n
in R2.
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(ξt, ηt, Jt)t≥0 is a (bivariate) MAP with
◮ Markovian component (Jt)t≥0: Right-continuous, ergodic,
continuous time Markov chain with countable state space S, intensity matrix Q and stationary law π.
◮ Additive component (ξt, ηt)t≥0:
(ξt, ηt) := (X (1)
t
, Y (1)
t
) + (X (2)
t
, Y (2)
t
), t ≥ 0.
◮ (X (1)
t
, Y (1)
t
) behaves in law like a bivariate L´ evy process (ξ(j)
t , η(j) t ) whenever Jt = j,
◮ (X (2)
t
, Y (2)
t
) is a pure jump process given by (X (2)
t
, Y (2)
t
) =
Z (i,j)
n
1{JTn−=i,JTn =j,Tn≤t}, for i.i.d. random variables Z (i,j)
n
in R2. As starting value under Pj we use (ξ0, η0, J0) = (0, 0, j). Throughout neither ξ nor η is degenerate constantly equal to 0.
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◮ S = {0}: (bivariate) L´
evy process
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◮ S = {0}: (bivariate) L´
evy process
◮ (X (1)
t
, Y (1)
t
) ≡ 0: (bivariate) continuous-time Markov chain in R2
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◮ S = {0}: (bivariate) L´
evy process
◮ (X (1)
t
, Y (1)
t
) ≡ 0: (bivariate) continuous-time Markov chain in R2
◮ (X (2)
t
, Y (2)
t
) ≡ 0: no common jumps of (Jt)t≥0 and (ξt, ηt)t≥0:
2 4 6 8 10 0.0 0.5 1.0 1.5 t X_t(omega)
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Given a bivariate Markov additive process (ξt, ηt, Jt)t≥0 with Markovian component (Jt)t≥0, we denote E(t) := E(ξ,η)(t) :=
e−ξs−dηs, 0 < t < ∞. Literature: Some recent results on E for ηt = t on arXiv (Salminen et al./Stephenson)
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◮ S = N0 ◮ (Jt)t≥0 continuous time petal flower
Markov chain with intensity matrix Q = (qi,j)i,j∈N0 = −q q0,1 q0,2 . . . q −q . . . q −q . . . . . . ... for q > 0 fixed and q0,j = qp0,j j ∈ N. (Jt)t≥0 is irreducible, recurrent with stationary distribution π0 = 1 2, and πj = p0,j 2 = q0,j 2q , j ∈ N
1Thanks to Gerold Alsmeyer for the picture! Anita Behme P&A 2019, 12
Now choose ξ and η to be conditionally independent with ξt = X (2)
t
=
Z (i,j)
n
1{JTn−=i,JTn =j,Tn≤t}, Z (i,j)
n
:= −p−1
0,j ,
i = 0, 2 + p−1
0,i ,
j = 0, 0, else. Then ξτn(0) = 2n → ∞ P0-a.s. but lim inf
t→∞ ξt = −∞.
Thus ξ is oscillating.
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Now choose ξ and η to be conditionally independent with ξt = X (2)
t
=
Z (i,j)
n
1{JTn−=i,JTn =j,Tn≤t}, Z (i,j)
n
:= −p−1
0,j ,
i = 0, 2 + p−1
0,i ,
j = 0, 0, else. Then ξτn(0) = 2n → ∞ P0-a.s. but lim inf
t→∞ ξt = −∞.
Thus ξ is oscillating. Choosing ηt =
j = 0, 0,
we observe under P0
e−ξs−dηs =
e−ξs−γη(Js )ds =
e−Ns−1{Js=0}ds.
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Now choose ξ and η to be conditionally independent with ξt = X (2)
t
=
Z (i,j)
n
1{JTn−=i,JTn =j,Tn≤t}, Z (i,j)
n
:= −p−1
0,j ,
i = 0, 2 + p−1
0,i ,
j = 0, 0, else. Then ξτn(0) = 2n → ∞ P0-a.s. but lim inf
t→∞ ξt = −∞.
Thus ξ is oscillating. Choosing ηt =
j = 0, 0,
we observe under P0
e−ξs−dηs =
e−ξs−γη(Js )ds =
e−Ns−1{Js=0}ds. I.e. the exponential functional converges P0-a.s. as t → ∞.
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Now choose ξ and η to be conditionally independent with ξt = X (2)
t
=
Z (i,j)
n
1{JTn−=i,JTn =j,Tn≤t}, Z (i,j)
n
:= −p−1
0,j ,
i = 0, 2 + p−1
0,i ,
j = 0, 0, else. Then ξτn(0) = 2n → ∞ P0-a.s. but lim inf
t→∞ ξt = −∞.
Thus ξ is oscillating. Choosing ηt =
j = 0, 0,
we observe under P0
e−ξs−dηs =
e−ξs−γη(Js )ds =
e−Ns−1{Js=0}ds. I.e. the exponential functional converges P0-a.s. as t → ∞.
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L´ evy case (Erickson & Maller ’05): The exponential functional
e−ξt−dηt, can be discretized for any h > 0 as:
e−ξs−dηs =
n−1
e−ξs−dηs =
n−1
i−1
e−(ξ(j+1)h−ξjh)
e−(ξs−−ξih)dηs, Thus: Convergence of the integral is strongly connected to convergence
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Alsmeyer & Buckmann (2017) generalized the results from Goldie & Maller (2000) to a Markovian environment:
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Alsmeyer & Buckmann (2017) generalized the results from Goldie & Maller (2000) to a Markovian environment: They provide necessary and sufficient conditions for convergence as n → ∞ of Zn :=
n
i−1
Aj Bi, where
◮ (An, Bn)n∈N, sequence of random vectors in R2, modulated by ◮ (Mn)n∈N0, ergodic Markov chain with countable state space S and
stationary law π.
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Alsmeyer & Buckmann (2017) generalized the results from Goldie & Maller (2000) to a Markovian environment: They provide necessary and sufficient conditions for convergence as n → ∞ of Zn :=
n
i−1
Aj Bi, where
◮ (An, Bn)n∈N, sequence of random vectors in R2, modulated by ◮ (Mn)n∈N0, ergodic Markov chain with countable state space S and
stationary law π. ”Modulated” in the sense that
◮ conditionally on Mn = in ∈ S, n = 0, 1, 2, . . . the random vectors
(A1, B1), (A2, B2), . . . are independent, and
◮ for all n ∈ N the conditional law of (An, Bn) is temporally
homogeneous and depends only on (in−1, in) ∈ S2.
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We shall
◮ find ”good” discretizations of Eξ,η(t) and ◮ apply/extend Alsmeyer & Buckmanns results/techniques,
to obtain necessary and/or sufficient conditions for convergence of Eξ,η(t) as t → ∞.
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Define Tj := {t ≥ 0 : Jt = t}, and τ1(j) := first return time to j, τ −
1 (j) := first exit time of j.
Main Theorem (B&Sideris, ’19):
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Define Tj := {t ≥ 0 : Jt = t}, and τ1(j) := first return time to j, τ −
1 (j) := first exit time of j.
Main Theorem (B&Sideris, ’19): (i) Assume that limt∈Tj,t→∞ ξt = ∞ for some/all j ∈ S and that
log q
0≤t≤τ1(j)
e−ξs−dηs
for some/all j ∈ S, then E(ξ,η)(t) → E∞
(ξ,η) Pπ-a.s. as t → ∞ for some
random variable E∞
(ξ,η).
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Define Tj := {t ≥ 0 : Jt = t}, and τ1(j) := first return time to j, τ −
1 (j) := first exit time of j.
Main Theorem (B&Sideris, ’19): (ii) If limt∈Tj,t→∞ ξt = ∞ for some j ∈ S and
log q
1 (j),τ1(j)]
e−ξs−dηs + Y b,η
τ−
1 (j)
(2)
then there exists a probability measure Qj = Pj(E∞
(ξ,η) ∈ ·) on R, such that
E(ξ,η)(t) → E∞
(ξ,η) in Pj-probability.
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Define Tj := {t ≥ 0 : Jt = t}, and τ1(j) := first return time to j, τ −
1 (j) := first exit time of j.
Main Theorem (B&Sideris, ’19): (iii) If lim inft∈Tj,t→∞ ξt < ∞ for some j ∈ S , then either there exists a (unique) sequence {ci, i ∈ S} in R such that E(ξ,η)(t) =
e−ξs−dηs = cJ0 − cJte−ξt Pπ-a.s. for all t ≥ 0, or |E(ξ,η)(t)| → ∞ in Pπ-probability.
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Define Tj := {t ≥ 0 : Jt = t}, and τ1(j) := first return time to j, τ −
1 (j) := first exit time of j.
Main Theorem (B&Sideris, ’19): (iii) If lim inft∈Tj,t→∞ ξt < ∞ for some j ∈ S (or if (2) fails(?)), then either there exists a (unique) sequence {ci, i ∈ S} in R such that E(ξ,η)(t) =
e−ξs−dηs = cJ0 − cJte−ξt Pπ-a.s. for all t ≥ 0, or |E(ξ,η)(t)| → ∞ in Pπ-probability.
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Recall: Degeneracy in the L´ evy case S = {1} is characterized by P t e−ξs−dηs = k(1 − e−ξt) for all t > 0
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Recall: Degeneracy in the L´ evy case S = {1} is characterized by P t e−ξs−dηs = k(1 − e−ξt) for all t > 0
For Eξ,η we observe: Eξ,η is degenerate iff there exists a (unique) sequence {ci, i ∈ S} in R such that E(ξ,η)(t) =
e−ξs−dηs = cJ0 − cJte−ξt Pπ-a.s. for all t ≥ 0. (3)
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Recall: Degeneracy in the L´ evy case S = {1} is characterized by P t e−ξs−dηs = k(1 − e−ξt) for all t > 0
For Eξ,η we observe: Eξ,η is degenerate iff there exists a (unique) sequence {ci, i ∈ S} in R such that E(ξ,η)(t) =
e−ξs−dηs = cJ0 − cJte−ξt Pπ-a.s. for all t ≥ 0. (3) Further: Proposition (B&Sideris, ’19+) Assume (3) holds for all t ≥ 0 and some sequence {ci, i ∈ S}. Then ηt = −
cJs−dUs −
dcJs, t ≥ 0, Pπ-a.s. (4) where (Ut)t≥0 = (Log (e−ξt))t≥0. Conversely, if (4) holds for some sequence {ci, i ∈ S}, then (3) is fulfilled for all t ≥ 0.
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ηt = −
cJs−dUs −
dcJs, t ≥ 0, Pπ-a.s. implies
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ηt = −
cJs−dUs −
dcJs, t ≥ 0, Pπ-a.s. implies
◮ If (X (2)
t
, Y (2)
t
) ≡ 0 (that is ∆ξTn = 0 = ∆ηTn Pπ-a.s. for all n), then ci ≡ c, i.e. ηt = −cUt and E(ξ,η)(t) =
e−ξs−dηs = c(1 − e−ξt) Pπ-a.s. for all t ≥ 0.
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ηt = −
cJs−dUs −
dcJs, t ≥ 0, Pπ-a.s. implies
◮ If (X (2)
t
, Y (2)
t
) ≡ 0 (that is ∆ξTn = 0 = ∆ηTn Pπ-a.s. for all n), then ci ≡ c, i.e. ηt = −cUt and E(ξ,η)(t) =
e−ξs−dηs = c(1 − e−ξt) Pπ-a.s. for all t ≥ 0.
◮ If ∆ξTn = 0 Pπ-a.s. for all n, then Z (i,j)
η
= ci − cj.
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ηt = −
cJs−dUs −
dcJs, t ≥ 0, Pπ-a.s. implies
◮ If (X (2)
t
, Y (2)
t
) ≡ 0 (that is ∆ξTn = 0 = ∆ηTn Pπ-a.s. for all n), then ci ≡ c, i.e. ηt = −cUt and E(ξ,η)(t) =
e−ξs−dηs = c(1 − e−ξt) Pπ-a.s. for all t ≥ 0.
◮ If ∆ξTn = 0 Pπ-a.s. for all n, then Z (i,j)
η
= ci − cj.
◮ If ∆ηTn = 0 Pπ-a.s. for all n, then Z (i,j)
ξ
= log cj
ci .
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Although we can state necessary and sufficient conditions for convergence of E(ξ,η)(t), these are hardly applicable.
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To obtain easier (sufficient) conditions, we split the exponential functional in two pieces: E(ξ,η)(t) =
e−ξs−d(γη
s + W η s + Y b,η s
+ Y s,η
s
+ Y (2)
s
) =
e−ξs−d( γη
s
+ W η
s + Y s,η s
) +
e−ξs−d(Y b,η
s
+ Y (2)
s
) =: E(1)(t) + E(2)(t).
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To obtain easier (sufficient) conditions, we split the exponential functional in two pieces: E(ξ,η)(t) =
e−ξs−d(γη
s + W η s + Y b,η s
+ Y s,η
s
+ Y (2)
s
) =
e−ξs−d( γη
s
+ W η
s + Y s,η s
) +
e−ξs−d(Y b,η
s
+ Y (2)
s
) =: E(1)(t) + E(2)(t).
◮ E(1)(t) converges a.s. under rather weak conditions.
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To obtain easier (sufficient) conditions, we split the exponential functional in two pieces: E(ξ,η)(t) =
e−ξs−d(γη
s + W η s + Y b,η s
+ Y s,η
s
+ Y (2)
s
) =
e−ξs−d( γη
s
+ W η
s + Y s,η s
) +
e−ξs−d(Y b,η
s
+ Y (2)
s
) =: E(1)(t) + E(2)(t).
◮ E(1)(t) converges a.s. under rather weak conditions. ◮ E(2)(t) can be embedded in a discrete time setting and directly
corresponds to a Markov modulated perpetuity.
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Recall: L´ evy processes in R either drift to ±∞ or oscillate.
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Recall: L´ evy processes in R either drift to ±∞ or oscillate. MAPs often show a similar behaviour: Define the long-term mean (here for the MAP (ξt, Jt)t≥0) κξ :=
πj
xνξ(j)(dx)
i=j
πiqi,j
xF (i,j)
ξ
(dx). Whenever S is finite, κξ fully determines the long-term behaviour of ξ: κξ > 0 ⇒ lim
t→∞ ξt = ∞ Pπ-a.s.,
κξ < 0 ⇒ lim
t→∞ ξt = −∞ Pπ-a.s.,
while κξ = 0 and Pπ
lim supt→∞ ξt = ∞ and lim inft→∞ ξt = −∞
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Proposition (B&Sideris, ’19): Assume that 0 < κξ < ∞ and sup
j∈S
η(j) +
x2νη(j)(dx)
(5) Then E(1)(t) converges Pπ-a.s. to a finite random variable as t → ∞.
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Proposition (B&Sideris, ’19): Assume that 0 < κξ < ∞ and sup
j∈S
η(j) +
x2νη(j)(dx)
(5) Then E(1)(t) converges Pπ-a.s. to a finite random variable as t → ∞. Note that
◮ For S finite, (5) is always fulfilled.
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Proposition (B&Sideris, ’19): Assume that 0 < κξ < ∞ and sup
j∈S
η(j) +
x2νη(j)(dx)
(5) Then E(1)(t) converges Pπ-a.s. to a finite random variable as t → ∞. Note that
◮ For S finite, (5) is always fulfilled. ◮ In general (5) is necessary.
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Proposition (B&Sideris, ’19): Assume 0 < κξ < ∞.
and only if for some/all j ∈ S
log q Pj
0<t≤τ1(j)
e−ξt−|∆(Y b,η
t
+ Y (2)
t
)| ∈ dq
∞ for
all j ∈ S, if and only if for some/all j ∈ S
log q Pj
e−ξt−d(Y b,η
t
+ Y (2)
t
)
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◮ Fluctuation theory for MAPs with countable state space?
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◮ Fluctuation theory for MAPs with countable state space? ◮ E as stationary distribution of a Markov modulated generalized
Ornstein-Uhlenbeck process?
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◮ K. B. Erickson and R. A. Maller (2005)
Generalised Ornstein-Uhlenbeck processes and the convergence of L´ evy integrals. S´ eminaire de Probabilit´ es XXXVIII.
◮ G. Alsmeyer and F. Buckmann (2017)
Stability of perpetuities in Markovian environment.
◮ A. Behme and A. Sideris (2019+)
Exponential functionals of Markov additive processes (working title). To be submitted soon.
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