Asymptotic expansions and distribution properties for diffusion processes
For this thesis, we derive new applications and theoretical results for some multidimensional mean-reverting stochastic processes via series expansion. We use Malliavin calculus and moment cumulant formula to specify the conditions on the stochastic process, under which, we are able to obtain som...
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| Format: | Thesis |
| Language: | English |
| Published: |
2017
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| Online Access: | http://hdl.handle.net/10356/69543 |
| _version_ | 1811689019320303616 |
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| author | She, Qihao |
| author2 | Nicolas Privault |
| author_facet | Nicolas Privault She, Qihao |
| author_sort | She, Qihao |
| collection | NTU |
| description | For this thesis, we derive new applications and theoretical results for some multidimensional mean-reverting stochastic processes via series expansion.
We use Malliavin calculus and moment cumulant formula to specify the conditions on the stochastic process, under which, we are able to obtain some conditional Gaussian identities for Brownian stochastic integrals. In particular, if we have the matrix condition ATA2=0, then from a characterization of Yor, we have the identity for the quadratic Brownian integral.
Afterwhich, we derive a conditional Edgeworth-type expansions. We then consider multiple stochastic integrals, where we obtain the Stein approximation bounds using the derived conditional Edgeworth-type expansions.
For application, we derive a closed-form analytical approximations formula in terms of series expansion for the option prices, implied volatility and delta under the 2-Hypergeometric stochastic volatility model with correlated Brownian motions. Most notably, the approximation formula we derive for the implied volatility is capable of recovering the well-known smiles and skew phenomenon on implied volatility surfaces, depending on the correlation.
Lastly, we consider a multidimensional ergodic Ornstein-Uhlenbeck process, X and let Y be a multidimensional stochastic process such that its stochastic differential equation is written as a drift-perturbation of X and µY be the stationary distribution of Y. We derive a first and second order expansion of µY in terms of X and their respective error estimates. Thereafter, we turn these approximations into a simulation scheme to sample µY approximately. |
| first_indexed | 2024-10-01T05:41:27Z |
| format | Thesis |
| id | ntu-10356/69543 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T05:41:27Z |
| publishDate | 2017 |
| record_format | dspace |
| spelling | ntu-10356/695432023-02-28T23:46:49Z Asymptotic expansions and distribution properties for diffusion processes She, Qihao Nicolas Privault School of Physical and Mathematical Sciences DRNTU::Engineering::Mathematics and analysis::Simulations DRNTU::Engineering::Computer science and engineering::Mathematics of computing::Probability and statistics For this thesis, we derive new applications and theoretical results for some multidimensional mean-reverting stochastic processes via series expansion. We use Malliavin calculus and moment cumulant formula to specify the conditions on the stochastic process, under which, we are able to obtain some conditional Gaussian identities for Brownian stochastic integrals. In particular, if we have the matrix condition ATA2=0, then from a characterization of Yor, we have the identity for the quadratic Brownian integral. Afterwhich, we derive a conditional Edgeworth-type expansions. We then consider multiple stochastic integrals, where we obtain the Stein approximation bounds using the derived conditional Edgeworth-type expansions. For application, we derive a closed-form analytical approximations formula in terms of series expansion for the option prices, implied volatility and delta under the 2-Hypergeometric stochastic volatility model with correlated Brownian motions. Most notably, the approximation formula we derive for the implied volatility is capable of recovering the well-known smiles and skew phenomenon on implied volatility surfaces, depending on the correlation. Lastly, we consider a multidimensional ergodic Ornstein-Uhlenbeck process, X and let Y be a multidimensional stochastic process such that its stochastic differential equation is written as a drift-perturbation of X and µY be the stationary distribution of Y. We derive a first and second order expansion of µY in terms of X and their respective error estimates. Thereafter, we turn these approximations into a simulation scheme to sample µY approximately. Doctor of Philosophy (SPMS) 2017-02-06T03:36:41Z 2017-02-06T03:36:41Z 2017 Thesis She, Q. (2017). Asymptotic expansions and distribution properties for diffusion processes. Doctoral thesis, Nanyang Technological University, Singapore. http://hdl.handle.net/10356/69543 10.32657/10356/69543 en 136 p. application/pdf |
| spellingShingle | DRNTU::Engineering::Mathematics and analysis::Simulations DRNTU::Engineering::Computer science and engineering::Mathematics of computing::Probability and statistics She, Qihao Asymptotic expansions and distribution properties for diffusion processes |
| title | Asymptotic expansions and distribution properties for diffusion processes |
| title_full | Asymptotic expansions and distribution properties for diffusion processes |
| title_fullStr | Asymptotic expansions and distribution properties for diffusion processes |
| title_full_unstemmed | Asymptotic expansions and distribution properties for diffusion processes |
| title_short | Asymptotic expansions and distribution properties for diffusion processes |
| title_sort | asymptotic expansions and distribution properties for diffusion processes |
| topic | DRNTU::Engineering::Mathematics and analysis::Simulations DRNTU::Engineering::Computer science and engineering::Mathematics of computing::Probability and statistics |
| url | http://hdl.handle.net/10356/69543 |
| work_keys_str_mv | AT sheqihao asymptoticexpansionsanddistributionpropertiesfordiffusionprocesses |