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


Other title
Martingale Deflators
Stochastic Basis
Optional Martingales
Stochastic Equations
Optional Processes
Comparison Theorem
Default Time
Gronwall Lemma
Optional Semimartingales
Type of item
Degree grantor
University of Alberta
Author or creator
Abdelghani, Mohamed Nabeel
Supervisor and department
Alexander Melnikov
Examining committee member and department
Chris Frei, Mathematical and Statistical Sciences University of Alberta
Vakhtang Putkaradze, Mathematical and Statistical Sciences University of Alberta
Alexandru Badescu, Department of Mathematics and Statistics University of Calgary
Byron Schmuland, Mathematical and Statistical Sciences University of Alberta
Alexander Litvak, Mathematical and Statistical Sciences University of Alberta
Department of Mathematical and Statistical Sciences
Mathematical Finance
Date accepted
Graduation date
Doctor of Philosophy
Degree level
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.
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