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Dissertation_Kumar_Somnath_2022.pdf (6.11 MB)

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Exact Simulation Methods for Functionals of Constrained Brownian Motion Processes and Stochastic Differential Equations

Somnath, Kumar
http://rave.ohiolink.edu/etdc/view?acc_num=osu1641489719104867

Abstract Details

2022, Doctor of Philosophy, Ohio State University, Statistics.
Stochastic differential equations (SDEs) are used to model random phenomena which evolve over time and space. Applications of SDEs range from finance (modelling the evolution of stock prices, interest rates and complex derivatives) to biology (modelling growth of populations, spread of disease) to weather systems (modelling changes in temperature and precipitation over a region) and many others. Solutions of stochastic differential equations are stochastic processes. The realizations of such stochastic processes are referred to as sample paths. A closed form solution of a given SDE is generally not available and in general, SDEs are not solvable. However, the existence of a solution is guaranteed when the SDE satisfies certain conditions which are easily verifiable. Numerical methods which provide approximate solutions for a given SDE have been widely used as a compromise due to the lack of a closed form solution or existence of exact simulation methods. Such methods have a tremendous advantage due to the simplicity of their implementation. However, the inexact nature of approximate solutions introduce unknown bias and corrective measures come at a larger computational cost. Exact simulation methods for a class of one-dimensional time-homogeneous SDEs emerged in mid 2000s. The classic rejection sampling method for random variables could be extended to random elements whose realizations are continuous functions over an interval. Sample paths are proposed from the Wiener measure and accepted or rejected without requiring knowledge about the entire sample path. The simplicity of the early exact methods led to a thrust in developing new algorithms for a wider class of SDEs. In this context I began my research to develop exact simulation methods for a large class of one-dimensional time-homogeneous SDEs. In pursuit of this objective, I first developed new algorithms to perform exact joint simulation of the maximum and the time of maximum for constrained Brownian motion processes such as Brownian Excursion, Restricted Brownian Meander, Brownian Meander and Reflected Brownian Bridge. Next, I built new algorithms to perform interpolation for a Brownian motion process restricted to an interval. Using these tools I am able to devise a novel method for exact simulation of sample paths for a large class of stochastic differential equations. Later, I combine all of these methods to determine whether SDE sample paths exit a given non-constant double boundary before a fixed time point.
Radu Herbei (Advisor)
153 p.
Statistics
Exact Simulation Methods, Stochastic Differential Equations, Non-Constant Boundary, Constrained Brownian Motion Processes

Recommended Citations

Citations

  • Somnath, K. (2022). Exact Simulation Methods for Functionals of Constrained Brownian Motion Processes and Stochastic Differential Equations [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1641489719104867

    APA Style (7th edition)

  • Somnath, Kumar. Exact Simulation Methods for Functionals of Constrained Brownian Motion Processes and Stochastic Differential Equations. 2022. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1641489719104867.

    MLA Style (8th edition)

  • Somnath, Kumar. "Exact Simulation Methods for Functionals of Constrained Brownian Motion Processes and Stochastic Differential Equations." Doctoral dissertation, Ohio State University, 2022. http://rave.ohiolink.edu/etdc/view?acc_num=osu1641489719104867

    Chicago Manual of Style (17th edition)

osu1641489719104867
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