Drift parameter estimation in stochastic differential equation with multiplicative stochastic volatility
We consider a stochastic differential equation of the form \[ dX_{t}=\theta a(t,X_{t})\hspace{0.1667em}dt+\sigma _{1}(t,X_{t})\sigma _{2}(t,Y_{t})\hspace{0.1667em}dW_{t}\] with multiplicative stochastic volatility, where Y is some adapted stochastic process. We prove existence–uniqueness results for...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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VTeX
2016-12-01
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| Series: | Modern Stochastics: Theory and Applications |
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| Online Access: | https://vmsta.vtex.vmt/doi/10.15559/16-VMSTA66 |
| _version_ | 1818493322043326464 |
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| author | Meriem Bel Hadj Khlifa Yuliya Mishura Kostiantyn Ralchenko Mounir Zili |
| author_facet | Meriem Bel Hadj Khlifa Yuliya Mishura Kostiantyn Ralchenko Mounir Zili |
| author_sort | Meriem Bel Hadj Khlifa |
| collection | DOAJ |
| description | We consider a stochastic differential equation of the form \[ dX_{t}=\theta a(t,X_{t})\hspace{0.1667em}dt+\sigma _{1}(t,X_{t})\sigma _{2}(t,Y_{t})\hspace{0.1667em}dW_{t}\] with multiplicative stochastic volatility, where Y is some adapted stochastic process. We prove existence–uniqueness results for weak and strong solutions of this equation under various conditions on the process Y and the coefficients a, $\sigma _{1}$, and $\sigma _{2}$. Also, we study the strong consistency of the maximum likelihood estimator for the unknown parameter θ. We suppose that Y is in turn a solution of some diffusion SDE. Several examples of the main equation and of the process Y are provided supplying the strong consistency. |
| first_indexed | 2024-12-10T17:53:21Z |
| format | Article |
| id | doaj.art-541b0bbf26b3470387dfaea915e9c888 |
| institution | Directory Open Access Journal |
| issn | 2351-6046 2351-6054 |
| language | English |
| last_indexed | 2024-12-10T17:53:21Z |
| publishDate | 2016-12-01 |
| publisher | VTeX |
| record_format | Article |
| series | Modern Stochastics: Theory and Applications |
| spelling | doaj.art-541b0bbf26b3470387dfaea915e9c8882022-12-22T01:39:00ZengVTeXModern Stochastics: Theory and Applications2351-60462351-60542016-12-013426928510.15559/16-VMSTA66Drift parameter estimation in stochastic differential equation with multiplicative stochastic volatilityMeriem Bel Hadj Khlifa0Yuliya Mishura1Kostiantyn Ralchenko2Mounir Zili3University of Monastir, Faculty of Sciences of Monastir, Department of Mathematics, Avenue de l’Environnement, 5000, Monastir, TunisiaDepartment of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, 64 Volodymyrska, 01601 Kyiv, UkraineDepartment of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, 64 Volodymyrska, 01601 Kyiv, UkraineUniversity of Monastir, Faculty of Sciences of Monastir, Department of Mathematics, Avenue de l’Environnement, 5000, Monastir, TunisiaWe consider a stochastic differential equation of the form \[ dX_{t}=\theta a(t,X_{t})\hspace{0.1667em}dt+\sigma _{1}(t,X_{t})\sigma _{2}(t,Y_{t})\hspace{0.1667em}dW_{t}\] with multiplicative stochastic volatility, where Y is some adapted stochastic process. We prove existence–uniqueness results for weak and strong solutions of this equation under various conditions on the process Y and the coefficients a, $\sigma _{1}$, and $\sigma _{2}$. Also, we study the strong consistency of the maximum likelihood estimator for the unknown parameter θ. We suppose that Y is in turn a solution of some diffusion SDE. Several examples of the main equation and of the process Y are provided supplying the strong consistency.https://vmsta.vtex.vmt/doi/10.15559/16-VMSTA66Stochastic differential equationweak and strong solutionsstochastic volatilitydrift parameter estimationmaximum likelihood estimatorstrong consistency |
| spellingShingle | Meriem Bel Hadj Khlifa Yuliya Mishura Kostiantyn Ralchenko Mounir Zili Drift parameter estimation in stochastic differential equation with multiplicative stochastic volatility Modern Stochastics: Theory and Applications Stochastic differential equation weak and strong solutions stochastic volatility drift parameter estimation maximum likelihood estimator strong consistency |
| title | Drift parameter estimation in stochastic differential equation with multiplicative stochastic volatility |
| title_full | Drift parameter estimation in stochastic differential equation with multiplicative stochastic volatility |
| title_fullStr | Drift parameter estimation in stochastic differential equation with multiplicative stochastic volatility |
| title_full_unstemmed | Drift parameter estimation in stochastic differential equation with multiplicative stochastic volatility |
| title_short | Drift parameter estimation in stochastic differential equation with multiplicative stochastic volatility |
| title_sort | drift parameter estimation in stochastic differential equation with multiplicative stochastic volatility |
| topic | Stochastic differential equation weak and strong solutions stochastic volatility drift parameter estimation maximum likelihood estimator strong consistency |
| url | https://vmsta.vtex.vmt/doi/10.15559/16-VMSTA66 |
| work_keys_str_mv | AT meriembelhadjkhlifa driftparameterestimationinstochasticdifferentialequationwithmultiplicativestochasticvolatility AT yuliyamishura driftparameterestimationinstochasticdifferentialequationwithmultiplicativestochasticvolatility AT kostiantynralchenko driftparameterestimationinstochasticdifferentialequationwithmultiplicativestochasticvolatility AT mounirzili driftparameterestimationinstochasticdifferentialequationwithmultiplicativestochasticvolatility |