Use this URL to cite or link to this record in EThOS:  https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.753319 
Title:  An optimisationbased approach to FKPPtype equations  
Author:  Driver, David Philip 
ORCID:
0000000241598120
ISNI:
0000 0004 7426 4113


Awarding Body:  University of Cambridge  
Current Institution:  University of Cambridge  
Date of Award:  2018  
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Abstract:  
In this thesis, we study a class of reactiondiffusion equations of the form $\frac{\partial u}{\partial t} = \mathcal{L}u + \phi u  \tfrac{1}{k} u^{k+1}$ where $\mathcal{L}$ is the stochastic generator of a Markov process, $\phi$ is a function of the space variables and $k\in \mathbb{R}\backslash\{0\}$. An important example, in the case when $k > 0$, is equations of the FKPPtype. We also give an example from the theory of utility maximisation problems when such equations arise and in this case $k < 0$. We introduce a new representation, for the solution of the equation, as the optimal value of an optimal control problem. We also give a second representation which can be seen as a dual problem to the first optimisation problem. We note that this is a new type of dual problem and we compare it to the standard Lagrangian dual formulation. By choosing controls in the optimisation problems we obtain upper and lower bounds on the solution to the PDE. We use these bounds to study the speed of the wave front of the PDE in the case when $\mathcal{L}$ is the generator of a suitable Lévy process.


Supervisor:  Tehranchi, Michael  Sponsor:  EPSRC/CCA  
Qualification Name:  Thesis (Ph.D.)  Qualification Level:  Doctoral  
EThOS ID:  uk.bl.ethos.753319  DOI:  
Keywords:  KPP ; FKPP ; ReactionDiffusion Equations ; Branching processes ; Front Propagation ; HJB Equation ; Stochastic Optimisation ; Travelling Waves  
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