Backward stochastic differential equations and Feynman-Kac formula in the presence of jump processes

Download
2013
İncegül Yücetürk, Cansu
Backward Stochastic Differential Equations (BSDEs) appear as a new class of stochastic differential equations, with a given value at the terminal time T. The application area of the BSDEs is conceptually wide which is known only for forty years. In financial mathematics, El Karoui, Peng and Quenez have a fundamental and significant article called “Backward Stochastic Differential Equations in Finance” (1997) which is taken as a groundwork for this thesis. In this thesis we follow the following steps: Firstly, the principal theorems of BSDEs driven by Brownian motion are proved. Later, an application to partial differential equations (PDEs) is presented i.e. generalization of Feynman-Kac formula. Moreover, the studies of Situ in 1997 and his book entitled with “Theory of Stochastic Differential Equations with Jumps and Applications” provide us a framework to prove explicitly the main theorems of BSDEs in the presence of jumps. Afterward, Feynman-Kac formula for general Lévy processes is proven. Lastly, the results are concluded by some applications in financial mathematics.

Suggestions

Continuity problem for singular BSDE with random terminal time
Samuel, Sharoy Augustine; Popier, Alexandre; Sezer, Ali Devin (2022-1-01)
All Rights Reserved.We study a class of non-linear Backward stochastic differential equations (BSDE) with a superlinear driver process f adapted to a filtration F and over a random time interval [[0, S]] where S is a stopping time of F. The terminal condition ξ is allowed to take the value +∞, i.e., singular. We call a stopping time S solvable with respect to a given BSDE and filtration if the BSDE has a minimal supersolution with terminal value 1 at terminal time S. Our goal is to show existence of solutio...
Least squares differential quadrature time integration scheme in the dual reciprocity boundary element method solution of convection-diffusion problems
Bozkaya, Canan (2005-03-18)
The least squares differential quadrature method (DQM) is used for solving the ordinary differential equations in time, obtained from the application of the dual reciprocity boundary element method (DRBEM) for the spatial partial derivatives in convection-diffusion type problems. The DRBEM enables us to use the fundamental solution of the Laplace equation which is easy to implement computationally. The time derivative and the convection terms are considered as the nonhomogeneity in the equation which are ap...
On principles of b-smooth discontinuous flows
Akalın, Ebru Çiğdem; Akhmet, Marat; Department of Mathematics (2004)
Discontinuous dynamical system defined by impulsive autonomous differential equation is a field that has actually been considered rarely. Also, the properties of such systems have not been discussed thoroughly in the course of mathematical researches so far. This thesis comprises two parts, elaborated with a number of examples. In the first part, some results of the previous studies on the classical dynamical system are exposed. In the second part, the definition of discontinuous dynamical system defined by...
Differential equations with discontinuities and population dynamics
Aruğaslan Çinçin, Duygu; Akhmet, Marat; Department of Mathematics (2009)
In this thesis, both theoretical and application oriented results are obtained for differential equations with discontinuities of different types: impulsive differential equations, differential equations with piecewise constant argument of generalized type and differential equations with discontinuous right-hand sides. Several qualitative problems such as stability, Hopf bifurcation, center manifold reduction, permanence and persistence are addressed for these equations and also for Lotka-Volterra predator-...
Inverse problems for parabolic equations
Baysal, Arzu; Çelebi, Okay; Department of Mathematics (2004)
In this thesis, we study inverse problems of restoration of the unknown function in a boundary condition, where on the boundary of the domain there is a convective heat exchange with the environment. Besides the temperature of the domain, we seek either the temperature of the environment in Problem I and II, or the coefficient of external boundary heat emission in Problem III and IV. An additional information is given, which is the overdetermination condition, either on the boundary of the domain (in Proble...
Citation Formats
C. İncegül Yücetürk, “Backward stochastic differential equations and Feynman-Kac formula in the presence of jump processes,” M.S. - Master of Science, Middle East Technical University, 2013.