Existence of Law Density of a Pair (d-dimensional Diffusion X, First Component Running Maximum M) with Coefficients Depending on (X, M), d > 1

Balauze Téo; Pontier Monique

doi:doi:10.11648/j.ajam.20261402.13

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Existence of Law Density of a Pair (d-dimensional Diffusion X, First Component Running Maximum M) with Coefficients Depending on (X, M), d > 1

Balauze Téo^*

, Pontier Monique

Published in American Journal of Applied Mathematics (Volume 14, Issue 2)

Received: 2 February 2026 Accepted: 24 February 2026 Published: 18 March 2026

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Abstract

The purpose of this paper is to consider a stochastic differential equation guiding X a continuous d-dimensional diffusion process, the coefficients of which depending on the pair (X, M) , where M is the running supremum of the first component X¹. As an application of our work, we could think of a firm the activity of which is characterized by a set of processes (X¹,···, X^d). But one of them, for instance X¹, could be linked to an alarm. Such a (d + 1)- dimensional process (X, M) could present a crucial interest in this case where M could be an alarm. Indeed, the possibility of an alarm at time t namely the event {∃s ≤ t : X¹_s > u} is identical to the event {M_t> u} > when the specific (and dangerous) threshold u is exceeded. This means that the law of M is closely linked to the law of the hitting time when X¹ reaches such a dangerous level u. Here is proved that, for all positive real number t; the law of the (d+1)-dimensional random vector (X_t, M_t) admits a density with respect to the Lebesgue measure. The solution of such a stochastic differential equation is built using a recursion method. The existence of the density of the law of (X, M) is based on Malliavin calculus. This density is solution of a partial differential equation in a weak sense. Moreover, such a recursive construction will allow to build simulated solutions. So finally such a tool could allow us to build an alarm system to detect the hitting time when the alarm could occur.

Published in	American Journal of Applied Mathematics (Volume 14, Issue 2)
DOI	10.11648/j.ajam.20261402.13
Page(s)	53-65
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Running Supremum Process, Joint Law Density, Malliavin Calculus

References

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APA Style

Téo, B., Monique, P. (2026). Existence of Law Density of a Pair (d-dimensional Diffusion X, First Component Running Maximum M) with Coefficients Depending on (X, M), d > 1. American Journal of Applied Mathematics, 14(2), 53-65. https://doi.org/10.11648/j.ajam.20261402.13

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ACS Style

Téo, B.; Monique, P. Existence of Law Density of a Pair (d-dimensional Diffusion X, First Component Running Maximum M) with Coefficients Depending on (X, M), d > 1. Am. J. Appl. Math. 2026, 14(2), 53-65. doi: 10.11648/j.ajam.20261402.13

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AMA Style

Téo B, Monique P. Existence of Law Density of a Pair (d-dimensional Diffusion X, First Component Running Maximum M) with Coefficients Depending on (X, M), d > 1. Am J Appl Math. 2026;14(2):53-65. doi: 10.11648/j.ajam.20261402.13

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@article{10.11648/j.ajam.20261402.13,
  author = {Balauze Téo and Pontier Monique},
  title = {Existence of Law Density of a Pair (d-dimensional Diffusion X, First Component Running Maximum M) with Coefficients Depending on (X, M), d > 1
},
  journal = {American Journal of Applied Mathematics},
  volume = {14},
  number = {2},
  pages = {53-65},
  doi = {10.11648/j.ajam.20261402.13},
  url = {https://doi.org/10.11648/j.ajam.20261402.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20261402.13},
  abstract = {The purpose of this paper is to consider a stochastic differential equation guiding X a continuous d-dimensional diffusion process, the coefficients of which depending on the pair (X, M) , where M is the running supremum of the first component X1. As an application of our work, we could think of a firm the activity of which is characterized by a set of processes (X1,···, Xd). But one of them, for instance X1, could be linked to an alarm. Such a (d + 1)- dimensional process (X, M) could present a crucial interest in this case where M could be an alarm. Indeed, the possibility of an alarm at time t namely the event {∃s ≤ t : X1s > u} is identical to the event {Mt> u} > when the specific (and dangerous) threshold u is exceeded. This means that the law of M is closely linked to the law of the hitting time when X1 reaches such a dangerous level u. Here is proved that, for all positive real number t; the law of the (d+1)-dimensional random vector (Xt, Mt) admits a density with respect to the Lebesgue measure. The solution of such a stochastic differential equation is built using a recursion method. The existence of the density of the law of (X, M) is based on Malliavin calculus. This density is solution of a partial differential equation in a weak sense. Moreover, such a recursive construction will allow to build simulated solutions. So finally such a tool could allow us to build an alarm system to detect the hitting time when the alarm could occur.},
 year = {2026}
}

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TY  - JOUR
T1  - Existence of Law Density of a Pair (d-dimensional Diffusion X, First Component Running Maximum M) with Coefficients Depending on (X, M), d > 1

AU  - Balauze Téo
AU  - Pontier Monique
Y1  - 2026/03/18
PY  - 2026
N1  - https://doi.org/10.11648/j.ajam.20261402.13
DO  - 10.11648/j.ajam.20261402.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 53
EP  - 65
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20261402.13
AB  - The purpose of this paper is to consider a stochastic differential equation guiding X a continuous d-dimensional diffusion process, the coefficients of which depending on the pair (X, M) , where M is the running supremum of the first component X1. As an application of our work, we could think of a firm the activity of which is characterized by a set of processes (X1,···, Xd). But one of them, for instance X1, could be linked to an alarm. Such a (d + 1)- dimensional process (X, M) could present a crucial interest in this case where M could be an alarm. Indeed, the possibility of an alarm at time t namely the event {∃s ≤ t : X1s > u} is identical to the event {Mt> u} > when the specific (and dangerous) threshold u is exceeded. This means that the law of M is closely linked to the law of the hitting time when X1 reaches such a dangerous level u. Here is proved that, for all positive real number t; the law of the (d+1)-dimensional random vector (Xt, Mt) admits a density with respect to the Lebesgue measure. The solution of such a stochastic differential equation is built using a recursion method. The existence of the density of the law of (X, M) is based on Malliavin calculus. This density is solution of a partial differential equation in a weak sense. Moreover, such a recursive construction will allow to build simulated solutions. So finally such a tool could allow us to build an alarm system to detect the hitting time when the alarm could occur.
VL  - 14
IS  - 2
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Author Information

Balauze Téo

Téo Balauze, Blaise Pascal Mathematics Laboratory , University Clermont Auvergne, Clermont-Ferrand, France

Contact Email

http://orcid.org/0000-0002-0381-7353
Pontier Monique

Monique Pontier, Toulouse Mathematics Institute , University Paul Sabatier, Toulouse, France

Contact Email

http://orcid.org/0009-0007-1662-7608

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Téo, B., Monique, P. (2026). Existence of Law Density of a Pair (d-dimensional Diffusion X, First Component Running Maximum M) with Coefficients Depending on (X, M), d > 1. American Journal of Applied Mathematics, 14(2), 53-65. https://doi.org/10.11648/j.ajam.20261402.13

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ACS Style

Téo, B.; Monique, P. Existence of Law Density of a Pair (d-dimensional Diffusion X, First Component Running Maximum M) with Coefficients Depending on (X, M), d > 1. Am. J. Appl. Math. 2026, 14(2), 53-65. doi: 10.11648/j.ajam.20261402.13

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AMA Style

Téo B, Monique P. Existence of Law Density of a Pair (d-dimensional Diffusion X, First Component Running Maximum M) with Coefficients Depending on (X, M), d > 1. Am J Appl Math. 2026;14(2):53-65. doi: 10.11648/j.ajam.20261402.13

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@article{10.11648/j.ajam.20261402.13,
  author = {Balauze Téo and Pontier Monique},
  title = {Existence of Law Density of a Pair (d-dimensional Diffusion X, First Component Running Maximum M) with Coefficients Depending on (X, M), d > 1
},
  journal = {American Journal of Applied Mathematics},
  volume = {14},
  number = {2},
  pages = {53-65},
  doi = {10.11648/j.ajam.20261402.13},
  url = {https://doi.org/10.11648/j.ajam.20261402.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20261402.13},
  abstract = {The purpose of this paper is to consider a stochastic differential equation guiding X a continuous d-dimensional diffusion process, the coefficients of which depending on the pair (X, M) , where M is the running supremum of the first component X1. As an application of our work, we could think of a firm the activity of which is characterized by a set of processes (X1,···, Xd). But one of them, for instance X1, could be linked to an alarm. Such a (d + 1)- dimensional process (X, M) could present a crucial interest in this case where M could be an alarm. Indeed, the possibility of an alarm at time t namely the event {∃s ≤ t : X1s > u} is identical to the event {Mt> u} > when the specific (and dangerous) threshold u is exceeded. This means that the law of M is closely linked to the law of the hitting time when X1 reaches such a dangerous level u. Here is proved that, for all positive real number t; the law of the (d+1)-dimensional random vector (Xt, Mt) admits a density with respect to the Lebesgue measure. The solution of such a stochastic differential equation is built using a recursion method. The existence of the density of the law of (X, M) is based on Malliavin calculus. This density is solution of a partial differential equation in a weak sense. Moreover, such a recursive construction will allow to build simulated solutions. So finally such a tool could allow us to build an alarm system to detect the hitting time when the alarm could occur.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - Existence of Law Density of a Pair (d-dimensional Diffusion X, First Component Running Maximum M) with Coefficients Depending on (X, M), d > 1

AU  - Balauze Téo
AU  - Pontier Monique
Y1  - 2026/03/18
PY  - 2026
N1  - https://doi.org/10.11648/j.ajam.20261402.13
DO  - 10.11648/j.ajam.20261402.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 53
EP  - 65
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20261402.13
AB  - The purpose of this paper is to consider a stochastic differential equation guiding X a continuous d-dimensional diffusion process, the coefficients of which depending on the pair (X, M) , where M is the running supremum of the first component X1. As an application of our work, we could think of a firm the activity of which is characterized by a set of processes (X1,···, Xd). But one of them, for instance X1, could be linked to an alarm. Such a (d + 1)- dimensional process (X, M) could present a crucial interest in this case where M could be an alarm. Indeed, the possibility of an alarm at time t namely the event {∃s ≤ t : X1s > u} is identical to the event {Mt> u} > when the specific (and dangerous) threshold u is exceeded. This means that the law of M is closely linked to the law of the hitting time when X1 reaches such a dangerous level u. Here is proved that, for all positive real number t; the law of the (d+1)-dimensional random vector (Xt, Mt) admits a density with respect to the Lebesgue measure. The solution of such a stochastic differential equation is built using a recursion method. The existence of the density of the law of (X, M) is based on Malliavin calculus. This density is solution of a partial differential equation in a weak sense. Moreover, such a recursive construction will allow to build simulated solutions. So finally such a tool could allow us to build an alarm system to detect the hitting time when the alarm could occur.
VL  - 14
IS  - 2
ER  -

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