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An explicit solution of a recurrence differential equationand its application in determining the conditional momentsof quadratic variance diffusion processes |
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| รหัสดีโอไอ | |
| Creator | 1. Kiattisak Prathom 2. Sanae Rujivan |
| Title | An explicit solution of a recurrence differential equationand its application in determining the conditional momentsof quadratic variance diffusion processes |
| Publisher | Research and Development Office, Prince of Songkla University |
| Publication Year | 2021 |
| Journal Title | Songklanakarin Journal of Science and Technology (SJST) |
| Journal Vol. | 43 |
| Journal No. | 4 |
| Page no. | 936-947 |
| Keyword | recurrence differential equation, explicit solution, conditional moments, quadratic variance diffusion processes |
| URL Website | https://rdo.psu.ac.th/sjstweb/index.php |
| ISSN | 0125-3395 |
| Abstract | This paper investigates solutions of a recurrence differential equation (RDE) of the form:'1 1 1'2 2 2 2 1'2 2 2 2 1 2( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ), k k k k k k kA t a A tA t a A t b A tA t a A t b A t c A t+ + + + + +== += + +'1 1 1'2 2 2 2 1'2 2 2 2 1 2( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ), k k k k k k kA t a A tA t a A t b A tA t a A t b A t c A t+ + + + + +== += + +'1 1 1'2 2 2 2 1'2 2 2 2 1 2( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ), k k k k k k kA t a A tA t a A t b A tA t a A t b A t c A t+ + + + + +== += + +fork K ? ? 1,2,..., 2and any positive integerK ? 3subject to the initial conditions(0) i i A R = ? Rfori K = 1,2,...,where, , i i i b c a ? ? C Randi j a a ?fori j ? . Firstly, we apply Laplace transform to the RDE to obtain a difference equation inLaplace space. Our success in performing Laplace inverse transform leads to an explicit solution of the RDE. Finally, we presentan application of our results by deriving closed-form formulas for the conditional moment, variance, covariance, and correlationof quadratic variance diffusion processes which are commonly used for studying model variance or interest rate processes infinancial engineering. |
