Title:

On the existence of a certain class of nonlinear stochastic processes

In this thesis, we investigate a class of stochastic processes whose definition can be achieved by formulating a nonlinear martingale problem and subsequently proving its wellposedness. This class includes socalled nonlinear Markov processes, such as McKeanVlasov processes and nonlinear diffusions, but also nonMarkovian versions of those. Roughly speaking, these processes are characterised by the fact that the evolution of their realisations depends on a particular finite dimensional distribution of the process itself. To formalise our idea we need to specify three components: (1) a collection of delay points which determines the finite dimensional distributions to be considered in the nonlinearity; (2) a family of operators which describes the evolution of the marginal probability distributions of the process; and (3) an initial condition which characterises the process on an initial period of time defined by the collection of delay points. Given these three elements we are able to formulate rigorously a nonlinear martingale problem and investigate its wellposedness. Our main results, which can be found in Chapter 4, provide sufficient conditions to guarantee the existence of a unique solution to the nonlinear martingale problem. The proof consists of three parts: constructing an approximating sequence of “standard” stochastic processes – together with a sequence of related curves of probability measures – proving its convergence, and finally demonstrating that its limit satisfies the martingale problem. To accomplish the proof we require a decomposition akin to the one provided by Ito’s formula. The reason why the classical Ito’s formula cannot be applied is that we need a decomposition for functions depending on the process at a finite number of nonanticipating times and not just on the process at the current time. To overcome this difficulty we establish an appropriate Itotype formula by using Skorohod integration theory. The material related to this formula can be found in Chapter 3. In addition, in Chapter 5 we prove the existence of solutions of a class of nonlinear SDEs with unbounded coefficients by using a different approach which was proposed in Kolokoltsov, 2010 and allows to investigate a class of nonlinear stochastic processes. Finally, we present two examples of nonlinear SDEs in Chapter 6. The purpose of such examples is twofold, first illustrate that the conditions for existence of solutions are sufficient but not necessary; and second to show potential applications. The idea is to propose stochastic volatility models with nonlinear dependence. In particular, we set two models via SDEs.
