A Study of SDEs Driven by Brownian Motion and Fractional Brownian Motion

In this thesis, we mainly study some properties for certain stochastic di↵er-ential equations.The types of stochastic di↵erential equations we are interested in are (i) stochastic di↵erential equations driven by Brownian motion, (ii) stochastic functional di↵erential equations driven by fractional Brownian motion, (iii) McKean-Vlasov stochastic di↵erential equations driven by Brownian motion,(iv) McKean-Vlasov stochastic di↵erential equations driven by fractional Brownian motion.The properties we investigate include the weak approximation rate of Euler-Maruyama scheme, the central limit theorem and moderate deviation principle for McKean-Vlasov stochastic di↵erential equations. Additionally, we investigate the existence and uniqueness of solution to McKean-Vlasov stochastic di↵erential equations driven by fractional Brownian motion, and then the Bismut formula of Lion’s derivatives for this model is also obtained.The crucial method we utilised to establish the weak approximation rate of Euler-Maruyama scheme for stochastic equations with irregular drift is the Girsanov transformation. More precisely, giving a reference stochastic equa-tions, we construct the equivalent expressions between the aim stochastic equations and associated numerical stochastic equations in another proba-bility spaces in view of the Girsanov theorem.For the Mckean-Vlasov stochastic di↵erential equation model, we first construct the moderate deviation principle for the law of the approxima-tion stochastic di↵erential equation in view of the weak convergence method. Subsequently, we show that the approximation stochastic equations and the McKean-Vlasov stochastic di↵erential equations are in the same exponen-tially equivalent family, and then we establish the moderate deviation prin-ciple for this model.Based on the result of Well-posedness for Mckean-Vlasov stochastic di↵er-ential equation driven by fractional Brownian motion, by using the Malliavin analysis, we first establish a general result of the Bismut type formula for Lions derivative, and then we apply this result to the non-degenerate case of this model.

Space-distribution PDEs for path independent additive functionals of McKean–Vlasov SDEs

Let [Formula: see text] be the space of probability measures on [Formula: see text] with finite second moment. The path independence of additive functionals of McKean–Vlasov SDEs is characterized by PDEs on the product space [Formula: see text] equipped with the usual derivative in space variable and Lions’ derivative in distribution. These PDEs are solved by using probabilistic arguments developed from Ref. 2. As a consequence, the path independence of Girsanov transformations is identified with nonlinear PDEs on [Formula: see text] whose solutions are given by probabilistic arguments as well. In particular, the corresponding results on the Girsanov transformation killing the drift term derived earlier for the classical SDEs are recovered as special situations.

Exact simulation of first exit times for one-dimensional diffusion processes

The simulation of exit times for diffusion processes is a challenging task since it concerns many applications in different fields like mathematical finance, neuroscience, reliability… The usual procedure is to use discretization schemes which unfortunately introduce some error in the target distribution. Our aim is to present a new algorithm which simulates exactly the exit time for one-dimensional diffusions. This acceptance-rejection algorithm requires to simulate exactly the exit time of the Brownian motion on one side and the Brownian position at a given time, constrained not to have exit before, on the other side. Crucial tools in this study are the Girsanov transformation, the convergent series method for the simulation of random variables and the classical rejection sampling. The efficiency of the method is described through theoretical results and numerical examples.

Change of Numeraire

In this chapter we discuss how a suitable change of numeraire and the corresponding change of martingale measure, can simplify the computation of pricing formula for financial derivatives. We derive a general formula for the likelihood process related to an arbitrary numeraire, and we identify the corresponding Girsanov transformation. As an example, we compute the price of an exchange option. In particular we study the class of forward measures related to zero coupon bonds and we derive a general option pricing formula. As an application of the general theory we also study the so-called numeraire portfolio.