Large Deviations and Importance Sampling for Systems of Slow-Fast Motion

In this paper we develop the large deviations principle and a rigorous mathematical framework for asymptotically efficient importance sampling schemes for general, fully dependent systems of stochastic differential equations of slow and fast motion with small noise in the slow component. We assume periodicity with respect to the fast component. Depending on the interaction of the fast scale with the smallness of the noise, we get different behavior. We examine how one range of interaction differs from the other one both for the large deviations and for the importance sampling. We use the large deviations results to identify asymptotically optimal importance sampling schemes in each case. Standard Monte Carlo schemes perform poorly in the small noise limit. In the presence of multiscale aspects one faces additional difficulties and straightforward adaptation of importance sampling schemes for standard small noise diffusions will not produce efficient schemes. It turns out that one has to consider the so called cell problem from the homogenization theory for Hamilton-Jacobi-Bellman equations in order to guarantee asymptotic optimality. We use stochastic control arguments.Comment: More detailed proofs. Differences from the published version are editorial and typographica

On large deviations for small noise It\^o processes

The large deviation principle in the small noise limit is derived for solutions of possibly degenerate It\^o stochastic differential equations with predictable coefficients, which may depend also on the large deviation parameter. The result is established under mild assumptions using the Dupuis-Ellis weak convergence approach. Applications to certain systems with memory and to positive diffusions with square-root-like dispersion coefficient are included.Comment: 30 page

Hedging of American Options under Transaction Costs

International audienceWe consider a continuous-time model of financial market with proportional transaction costs. Our result is a dual description of the set of initial endowments of self-financing portfolios super replicating American - type contingent claim. The latter is a right-continuous adapted vector process describing the number of assets to be delivered at the exercise date. We introduce a specific class of price systems, called coherent, and show that the hedging endowments are those whose 'values' are larger than the expected weighted 'values' of the pay-off process for every coherent price system used for the 'evaluation' of the assets

Finite State N-player and Mean Field Games

Mean field games represent limit models for symmetric non-zero sum dynamic games when the number N of players tends to infinity. In this thesis, we study mean field games and corresponding N- player games in continuous time over a finite time horizon where the position of each agent belongs to a finite state space. As opposed to previous works on finite statemean field games, we use a probabilistic representation of the system dynamics in terms of stochastic differential equations driven by Poisson random measures. Firstly, under mild assumptions, we prove existence of solutions to the mean field game in relaxed open-loop as well as relaxed feedback controls. Relying on the probabilistic representation and a coupling argument, we show that mean field game solutions provide symmetric εN- Nash equilibria for the N-player game, both in open-loop and in feedback strategies (not relaxed), with εN≤ constant √N. Under stronger assumptions, we also find solutions of the mean field game in ordinary feedback controls and prove uniqueness either in case of a small time horizon or under monotonicity. Then, assuming that players control just their transition rates from state to state, we show the convergence, as N tends to infinity, of the N-player game to a limiting dynamics given by a finite state mean field game system made of two coupled forward-backward ODEs. We exploit the so-called master equation, which in this finite-dimensional framework is a first order PDE in the simplex of probability measures. If the master equation possesses a unique regular solution, then such solution can be used to prove the convergence of the value functions of the N players and of the feedback Nash equilibria, and a propagation of chaos property for the associated optimal trajectories. A sufficient condition for the required regularity of the master equation is given by the monotonicity assumptions. Further, we employ the convergence results to establish a Central Limit Theorem and a Large Deviation Principle for the evolution of the N-player optimal empirical measures. Finally, we analyze an example with as state space and anti-monotonous cost,and show that the mean field game has exactly three solutions. The Nash equilibrium is always unique and we prove that the N-player game always admits a limit: it selects one mean field game solution, except in one critical case, so there is propagation of chaos. The value functions also converge and the limit is the entropy solution to the master equation, which for two state models can be written as a scalar conservation law. Moreover, viewing the mean field game system as the necessary conditions for optimality of a deterministic control problem, we show that the N-player game selects the optimum of this problem when it is unique

Nonparametric modelling of interest rates

We approximate interest rate financial data by homogeneous diffusion process with piecewise constant coefficients. Both drift and diffusion terms are estimated nonparametrically. For estimating of the drift term we use the taut string method which minimizes the numbers of peaks but for the diffusion term we minimize the number of intervals of constancy. For both estimators consistency is proved and convergence rate is calculated. Also the efficacy of the methods is demonstrated using simulated data

Mathematics of Quantitative Finance

The workshop on Mathematics of Quantitative Finance, organised at the Mathematisches Forschungsinstitut Oberwolfach from 26 February to 4 March 2017, focused on cutting edge areas of mathematical finance, with an emphasis on the applicability of the new techniques and models presented by the participants

Moderate deviations for systems of slow-fast stochastic reaction-diffusion equations

The goal of this paper is to study the Moderate Deviation Principle (MDP) for a system of stochastic reaction-diffusion equations with a time-scale separation in slow and fast components and small noise in the slow component. Based on weak convergence methods in infinite dimensions and related stochastic control arguments, we obtain an exact form for the moderate deviations rate function in different regimes as the small noise and time-scale separation parameters vanish. Many issues that come up due to the infinite dimensionality of the problem are completely absent in their finite-dimensional counterpart. In comparison to corresponding Large Deviation Principles, the moderate deviation scaling necessitates a more delicate approach to establishing tightness and properly identifying the limiting behavior of the underlying controlled problem. The latter involves regularity properties of a solution of an associated elliptic Kolmogorov equation on Hilbert space along with a finite-dimensional approximation argument.First author draf