Joint densities of hitting times for finite state Markov processes

Tomasz R. BIELECKI; Monique JEANBLANC; Ali Devin SEZER

Makale detayları

Joint densities of hitting times for finite state Markov processes

Yazarlar:

Tomasz R. BIELECKI | Monique JEANBLANC | Ali Devin SEZER

Yayıncı:

TUBITAK

Dergi:

Turkish Journal of Mathematics 2018 Vol.42 Issue 2, pp.586-608

Belge Tipi:

Makale

Yayın Tarihi:

3/2018

Belge Dili:

Türkçe

Document ID (Org No.):

2000745918

Sıklık:

İki Aylık

eISSN:

1303-6149

pISSN:

1300-0098

Özet:

 

For a finite state Markov process XXX and a finite collection {Γk,k∈K}{Γk,k∈K}\{  \Gamma_k, k \in K \} of subsets of its state space, let τkτk\tau_k  be the first time the process visits the set ΓkΓk\Gamma_k. In  general, XXX may enter some of the ΓkΓk\Gamma_k at the same time and  therefore the vector \bmτ:=(τk,k∈K)\bmτ:=(τk,k∈K)\bm\tau :=(\tau_k, k \in K) may put nonzero  mass over lower dimensional regions of R|K|+R+|K|{\mathbb R}_+^{|K|};  these regions are of the form Rs={\bmt∈R|K|+:ti=tj,  i,j∈s(1)}∩⋂|s|l=2{\bmt:tm<ti=tj,  i,j∈s(l),m∈s(l−1)}Rs={\bmt∈R+|K|:ti=tj,  i,j∈s(1)}∩⋂l=2|s|{\bmt:tm<ti=tj,  i,j∈s(l),m∈s(l−1)}R_s=\{{\bm t} \in {\mathbb R}_+^{|K|}: t_i = t_j, ~~i,j \in  s(1) \} \cap \bigcap_{l=2}^{|s|} \{{\bm t}:t_m < t_i = t_j,~~ i,j  \in s(l), m \in s(l-1)  \} where sss is any ordered partition of  the set KKK and s(j)s(j)s(j) denotes the jthjthj^{th} subset of KKK in the  partition sss. When |s|<|K||s|<|K||s| < |K|, the density of the law of  \bmτ\bmτ\bm\tau over these regions is said to be singular'' because it  is with respect to the |s||s||s|-dimensional Lebesgue measure over the  region Rs.Rs.R_s. We derive explicit/recursive and simple to compute  formulas for these singular densities and their corresponding tail  probabilities over all RsRsR_s as sss ranges over ordered partitions  of KKK. We give a numerical example and indicate the relevance of  our results to credit risk modeling.

Makale detayları

Joint densities of hitting times for finite state Markov processes

Yazarlar: Tomasz R. BIELECKI | Monique JEANBLANC | Ali Devin SEZER

Yayıncı: TUBITAK

Dergi: Turkish Journal of Mathematics 2018 Vol.42 Issue 2, pp.586-608

Belge Tipi: Makale

Yayın Tarihi: 3/2018

Belge Dili: Türkçe

Document ID (Org No.): 2000745918

Sıklık: İki Aylık

eISSN: 1303-6149

pISSN: 1300-0098

Sayı

Yazarlar:

Tomasz R. BIELECKI | Monique JEANBLANC | Ali Devin SEZER

Yayıncı:

TUBITAK

Dergi:

Turkish Journal of Mathematics 2018 Vol.42 Issue 2, pp.586-608

Belge Tipi:

Makale

Yayın Tarihi:

3/2018

Belge Dili:

Türkçe

Document ID (Org No.):

2000745918

Sıklık:

İki Aylık

eISSN:

1303-6149

pISSN:

1300-0098