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In a market driven by a Lévy martingale, we consider a claim ξ . We study the problem of minimal variance hedging and we give an explicit formula for the minimal variance portfolio in terms of Malliavin derivatives. We discuss two types of stochastic (Malliavin) derivatives for ξ : one based on the chaos expansion in terms of iterated integrals with respect to the power jump processes and one based on the chaos expansion in terms of iterated integrals with respect to the Wiener process and the Poisson random measure components. We study the relation between these two expansions, the corresponding two derivatives, and the corresponding versions of the Clark-HaussmannOcone theorem.

In this paper, we establish the existence of a stochastic flow of Sobolev diffeomorphismsThe above SDE is driven by a bounded measurable drift coefficientMore specifically, we show that the stochastic flow φs,t(·) of the SDE lives in the spacedenotes a weighted Sobolev space with weight w possessing a pth moment with respect to Lebesgue measure on R d . From the viewpoint of stochastic (and deterministic) dynamical systems, this is a striking result, since the dominant "culture" in these dynamical systems is that the flow "inherits" its spatial regularity from that of the driving vector fields.The spatial regularity of the stochastic flow yields existence and uniqueness of a Sobolev differentiable weak solution of the (Stratonovich) stochastic transport equationwhere b is bounded and measurable, u0 is C

This thesis has been funded by Center of Mathematics for Applications (CMA) and the Mathematical Institute both at the University of Oslo. During four years I have had the pleasure of being a part of the Stochastic Analysis group which has given me the opportunity to meet and work with a number of inspiring people. First and foremost, I would like to thank my primary supervisor, Frank Proske. It is difficult to explain how important his guidance has been for me. For me, there could not be a better supervisor. Our hundreds of talks over hundreds of cups of coffee has always left me motivated and inspired. I would also like to thank all the people at the CMA for creating such an inspiring environment. In particular I want to thank David Baños, Giulia di Nunno, Sven Haadem, Erlend Storrøsten and Bernt Øksendal for valuable discussions as well as my coauthors Franco Flandoli, Olivier Menokeu Pamen, Thilo Meyer-Brandis, Salah Mohammed and Tusheng Zhang. Also the administrative staff, in particular Biljana Dragisic, Robin Jacobsen and Elisabeth Seland deserves a thank you for making bureaucracy a less painful experience. Between January and July 2013 I spent my time at Humboldt University of Berlin. I would like to thank Peter Imkeller and his research group for including me in Berlin. In particular, I am grateful to Nicolas Perkowski for interesting discussions. Finally, I would like to thank the most important person in my life, Ellen, for always being loving and supportive.

We examine existence and uniqueness of strong solutions of multidimensional mean-field stochastic differential equations with irregular drift coefficients. Furthermore, we establish Malliavin differentiability of the solution and show regularity properties such as Sobolev differentiability in the initial data as well as Hölder continuity in time and the initial data. Using the Malliavin and Sobolev differentiability we formulate a Bismut-Elworthy-Li type formula for mean-field stochastic differential equations, i.e. a probabilistic representation of the first order derivative of an expectation functional with respect to the initial condition. Keywords.McKean-Vlasov equation · mean-field stochastic differential equation · weak solution · strong solution · uniqueness in law · pathwise uniqueness · singular coefficients · Malliavin derivative · Sobolev derivative · Hölder continuity · Bismut-Elworthy-Li formula.Date: December 13, 2019.