On The General Theory of Optional Stochastic Processes and Financial Markets Modeling

• Author / Creator

  Abdelghani, Mohamed Nabeel

• Optional processes including optional semimartingales are not necessarily right or left continuous. However, optional semimartingales have right and left limits. Moreover, optional processes may exist on ”un-usual” stochastic basis where the increasing information-filtrations are not complete or right continuous. Elements of the stochastic calculus of optional processes is reviewed. The linear stochastic differential equations with respect to optional semimartingales is solved. A solution of the nonhomogeneous linear stochastic differential equation and a proof of Gronwall inequality are given in this framework. Existence and uniqueness of solutions of stochastic equations of optional semimartingales under monotonicity condition is derived. Comparison theorem of solutions of stochastic equations of optional semimartingale under Yamada conditions is presented with a useful application to mathematical finance. Furthermore, a financial market model based on optional semimartingales is proposed and a method for finding local martingale deflators is given. Several examples of financial applications are given: a laglad jump diffusion model, Optimal debit repayment and a defaultable bond with a stock portfolio. Also, a pricing and hedging theory of a contingent claims for these markets is treated with optional semimartingale calculus. Finally, a new theory of defaultable markets on ”un-usual” probability spaces is presented. In this theory, default times are treated as stopping times in the broad sense where no enlargement of filtration and invariance principles are required. However, default process, in this context, become optional processes of finite variation and defaultable cash-flows become optional positive semimartingales.

• Subjects / Keywords

  • Stochastic Equations
  • Monotonicity
  • Optional Martingales
  • Martingale Deflators
  • Gronwall Lemma
  • Comparison Theorem
  • Default Time
  • Optional Processes
  • Optional Semimartingales
  • Stochastic Basis

• Graduation date

  Spring 2016

• Type of Item

  Thesis

• Degree

  Doctor of Philosophy

• DOI

  https://doi.org/10.7939/R3R49GF5H

• License

  This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.