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Brück F. Exact simulation of continuous max-id processes with applications to exchangeable max-id sequences. J MULTIVARIATE ANAL 2023. [DOI: 10.1016/j.jmva.2022.105117] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/05/2022]
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Mai JF. The infinite extendibility problem for exchangeable real-valued random vectors. PROBABILITY SURVEYS 2020. [DOI: 10.1214/19-ps336] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Scherer M, Sloot H. Exogenous shock models: analytical characterization and probabilistic construction. METRIKA 2019. [DOI: 10.1007/s00184-019-00715-8] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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Mai JF, Schenk S, Scherer M. Analyzing model robustness via a distortion of the stochastic root: A Dirichlet prior approach. STATISTICS & RISK MODELING 2016. [DOI: 10.1515/strm-2015-0009] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
Abstract
Abstract
It is standard in quantitative risk management to model a random vector
𝐗
:
=
{
X
t
k
}
k
=
1
,
...
,
d
${\mathbf {X}:=\lbrace X_{t_k}\rbrace _{k=1,\ldots ,d}}$
of consecutive log-returns to ultimately analyze the probability law of the accumulated return
X
t
1
+
⋯
+
X
t
d
${X_{t_1}+\cdots +X_{t_d}}$
.
By the Markov regression representation (see [25]), any stochastic model for
𝐗
${\mathbf {X}}$
can be represented as
X
t
k
=
f
k
(
X
t
1
,
...
,
X
t
k
-
1
,
U
k
)
${X_{t_k}=f_k(X_{t_1},\ldots ,X_{t_{k-1}},U_k)}$
,
k
=
1
,
...
,
d
${k=1,\ldots ,d}$
, yielding a decomposition into a vector
𝐔
:
=
{
U
k
}
k
=
1
,
...
,
d
${\mathbf {U}:=\lbrace U_{k}\rbrace _{k=1,\ldots ,d}}$
of i.i.d. random variables accounting for the randomness in the model, and a function
f
:
=
{
f
k
}
k
=
1
,
...
,
d
${f:=\lbrace f_k\rbrace _{k=1,\ldots ,d}}$
representing the economic reasoning behind. For most models, f is known explicitly and Uk
may be interpreted as an exogenous risk factor affecting the return X
t
k
in time step k. While existing literature addresses model uncertainty by manipulating the function f, we introduce a new philosophy by distorting the source of randomness
𝐔
${\mathbf {U}}$
and interpret this as an analysis of the model's robustness. We impose consistency conditions for a reasonable distortion and present a suitable probability law and a stochastic representation for
𝐔
${\mathbf {U}}$
based on a Dirichlet prior. The resulting framework has one parameter
c
∈
[
0
,
∞
]
${c\in [0,\infty ]}$
tuning the severity of the imposed distortion. The universal nature of the methodology is illustrated by means of a case study comparing the effect of the distortion to different models for
𝐗
${\mathbf {X}}$
. As a mathematical byproduct, the consistency conditions of the suggested distortion function reveal interesting insights into the dependence structure between samples from a Dirichlet prior.
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Affiliation(s)
| | - Steffen Schenk
- Lehrstuhl für Finanzmathematik, Technische Universität München, Parkring 11, 85748 Garching-Hochbrück, Germany
| | - Matthias Scherer
- Lehrstuhl für Finanzmathematik, Technische Universität München, Parkring 11, 85748 Garching-Hochbrück, Germany
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Mai JF, Schenk S, Scherer M. Two Novel Characterizations of Self-Decomposability on the Half-Line. J THEOR PROBAB 2015. [DOI: 10.1007/s10959-015-0644-6] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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