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- Dealing with partial hedging or risk management constraints via BSDEs with weak reflections | Applied Financial Mathematics

the super replication price of a given eventually non Markovian claim rewrites in terms of the minimal super solution of a well chosen Backward stochastic differential equation BSDE The price obtained is often numerically very high and hereby useless in practice In order to lower the price one must accept to take some risk and this can be formalized via the use of quantile hedging type objectives where the agent only wishes to upper hedge the claim of interest with a given a priori probability of success p Adding the probability of success process as a new forward process we are able to revisit quantile hedging type problems in a dynamically consistant manner This leads to the introduction of so called BSDEs with weak terminal condition where the terminal condition of the BSDE is not known explicitely but must only satisfy a given constraint written in terms of expectation We also introduce extensions of such BSDEs by considering BSDEs with weak or mean reflection where such type of constraint is imposed on any date t smaller than T Connections with 2nd order BSDES will be highlighted This presentation is based on joint works with Bruno Bouchard Philippe Briand Ying Hu

Original URL path: http://horst.qfl-berlin.de/tba-17 (2016-04-24)

Open archived version from archive - Convex duality in continuous-time stochastic optimization | Applied Financial Mathematics

optimization problems over spaces of predictable stochastic processes This is done by combining the conjugate duality theory of Rockafellar with some stochastic analysis Various duality relations in stochastic control and mathematical finance are obtained as special cases Besides classical models of financial markets the general framework allows for e g illiquidity effects and portfolio constraints This is joint work with Ari Pekka Perkkiö News Computer klüger als der Mensch Das

Original URL path: http://horst.qfl-berlin.de/convex-duality-continuous-time-stochastic-optimization (2016-04-24)

Open archived version from archive - Martingale Optimal Transport | Applied Financial Mathematics

also minimizes a given cost functional Kantorovich relaxed this problem by considering a measure whose marginals agree with given two measures instead of a bijection This generalization linearizes the problem Hence allows for an easy existence result and enables one to identify its convex dual In robust hedging problems we are also given two measures Namely the initial and the final distributions of a stock process We then construct an optimal connection In general however the cost functional depends on the whole path of this connection and not simply on the final value Hence one needs to consider processes instead of simply the transport maps The probability distribution of this process has prescribed marginals at final and initial times Thus it is in direct analogy with the Kantorovich measure But financial considerations restrict the process to be a martingale Interestingly the dual also has a financial interpretation as a robust hedging super replication problem In this talk we prove an analogue of Kantorovich duality the minimal super replication cost in the robust setting is given as the supremum of the expectations of the contingent claim over all martingale measures with a given marginal at the maturity This joint work with

Original URL path: http://horst.qfl-berlin.de/tba-16 (2016-04-24)

Open archived version from archive - A primal-dual algorithm for backward SDEs | Applied Financial Mathematics

second step this dynamic program has to be solved numerically This second step requires to approximate high order nestings of conditional expectations which is a challenging problem in particular when the BSDE is driven by a high dimensional Brownian motion In this talk we present a method to construct confidence intervals on the value of the dynamic program and hence on the solution of the time discretized BSDE This method generalizes the primal dual approach which is popular and well studied for Bermudan option pricing problems In a nutshell the idea is to derive a maximization problem and a minimization problem such that the value process of both problems coincides with the solution of the dynamic program and such that optimizers can be represented in terms of the solution of the dynamic program Using an approximate solution to the dynamic program which can be precomputed by any algorithm then leads to close to optimal controls for these optimization problems and to tight lower and upper bounds for the time discretized BSDE provided that the algorithm for constructing the approximate solution was successful We illustrate the method numerically for several nonlinear option pricing problems which can be formulated in terms of

Original URL path: http://horst.qfl-berlin.de/primal-dual-algorithm-backward-sdes (2016-04-24)

Open archived version from archive - Time Homogeneous Processes with Given Marginal | Applied Financial Mathematics

Homogeneous Processes with Given Marginal Time Homogeneous Processes with Given Marginal 12 November 2014 Kategorie Research Seminars Rudower Chaussee 25 Room 1 115 6 p m John M Noble University of Warsaw In this talk I consider the following problem given a probability measure mu over R with well defined expected value and given deterministic time does there exist a gap diffusion with the prescribed law at the prescribed time This is answered in the affirmative and it is shown that at least for an atomised space that a diffusion satisfying the property may be approximated by solutions to fixed point problems The introduction of drift b and killing k is considered and conditions under which there is a function a such that a 1 2 d 2 dx 2 b d dx k is the infinitesimal generator of a process with the given marginal at the given prescribed time t 0 News Computer klüger als der Mensch Das vergessene Werkzeug der Ökonomie Research Projects funded under HU s strategic partnership programs with Princeton University and NUS Young researcher worksho p d fine job opportunities d fine continuously offers job opportunities for students and university graduates both internships and permanent

Original URL path: http://horst.qfl-berlin.de/time-homogeneous-processes-given-marginal (2016-04-24)

Open archived version from archive - BSDEs of Counterparty Risk and Invariant Times | Applied Financial Mathematics

to generalize the classical credit risk reduced form modeling approach for counterparty risk applications We relax the basic immersion conditions of the classical approach by modeling the default time as an invariant time such that local martingales with respect to a reduced filtration and a possibly changed probability measure once stopped right before that time stay local martingales with respect to the original model filtration and probability measure Specifically we study a BSDE with random terminal time that appears in the modeling of counterparty risk in finance We proceed by reduction of the original BSDE into a simpler BSDE posed with respect to a subfiltration and a changed probability measure This is done under a relaxation of the classical immersion hypothesis stated in terms of the changed probability measure of which we determine the Radon Nikodym derivative We provide an Azema supermartingale characterization of invariant times and use it for establishing the equivalence between the original and the reduced BSDE This allows proving well posedness of the original nonstandard BSDE by well posedness of the reduced BSDE which holds under classical assumptions This is a joint work with Shiqi Song News Computer klüger als der Mensch Das vergessene Werkzeug der

Original URL path: http://horst.qfl-berlin.de/tba-11 (2016-04-24)

Open archived version from archive - Diffusive limits for stochastic kinetic equations | Applied Financial Mathematics

kinetic equations 29 October 2014 Kategorie Research Seminars Rudower Chaussee 25 Room 1 115 5 p m Arnoud Debussche ENS Rennes In this talk we consider kinetic equations containing random terms The kinetic models contain a small parameter and it is well known that after scaling when this parameter goes to zero the limit problem is a diffusion equation in the PDE sense i e a parabolic equation of second order A smooth noise is added accounting for external perturbation It scales also with the small parameter It is expected that the limit equation is then a stochastic parabolic equation where the noise is in Stratonovitch form Our aim is to justify in this way several SPDEs commonly used We first treat linear equations with multiplicative noise Then show how to extend the methods to nonlinear equations or to the more physical case of a random forcing term The results have been obtained jointly with S De Moor and J Vovelle News Computer klüger als der Mensch Das vergessene Werkzeug der Ökonomie Research Projects funded under HU s strategic partnership programs with Princeton University and NUS Young researcher worksho p d fine job opportunities d fine continuously offers job opportunities

Original URL path: http://horst.qfl-berlin.de/diffusive-limits-stochastic-kinetic-equations (2016-04-24)

Open archived version from archive - Anomalous random walks and their scaling limits: From fractals to random media | Applied Financial Mathematics

From fractals to random media Anomalous random walks and their scaling limits From fractals to random media 29 October 2014 Kategorie Research Seminars Rudower Chaussee 25 Room 1 115 6 p m Takashi Kumagai Kyoto University In this talk I present results concerning the behavior of random walks and diffusions on disordered media Examples treated include fractals and various models of random graphs such as percolation clusters trees generated by branching processes Erdos Rényi random graphs and uniform spanning trees As a consequence of the inhomogeneity of the underlying spaces we observe anomalous behavior of the corresponding random walks and diffusions The main focus is to estimate the long time behavior of the heat kernel and to obtain a scaling limit of the random walk I will overview the research in these areas chronologically and describe how the techniques have developed from those introduced for exactly self similar fractals to the more robust arguments required for random graphs News Computer klüger als der Mensch Das vergessene Werkzeug der Ökonomie Research Projects funded under HU s strategic partnership programs with Princeton University and NUS Young researcher worksho p d fine job opportunities d fine continuously offers job opportunities for students and

Original URL path: http://horst.qfl-berlin.de/anomalous-random-walks-and-their-scaling-limits-fractals-random-media (2016-04-24)

Open archived version from archive