Off-Critical SLE(2) and SLE(4): a Field Theory Approach

Using their relationship with the free boson and the free symplectic fermion, we study the off-critical perturbation of SLE(4) and SLE(2) obtained by adding a mass term to the action. We compute the off-critical statistics of the source in the Loewner equation describing the two dimensional interfaces. In these two cases we show that ratios of massive by massless partition functions, expressible as ratios of regularised determinants of massive and massless Laplacians, are (local) martingales for the massless interfaces. The off-critical drifts in the stochastic source of the Loewner equation are proportional to the logarithmic derivative of these ratios. We also show that massive correlation functions are (local) martingales for the massive interfaces. In the case of massive SLE(4), we use this property to prove a factorisation of the free boson measure.Comment: 30 pages, 1 figures, Published versio

Arbitrage-free regularization, geometric learning, and non-Euclidean filtering in finance

This thesis brings together elements of differential geometry, machine learning, and pathwise stochastic analysis to answer problems in mathematical finance. The overarching theme is the development of new stochastic machine learning algorithms which incorporate arbitrage-free and geometric features into their estimation procedures in order to give more accurate forecasts and preserve the geometric and financial structure in the data. This thesis is divided into three part. The first part introduces the non-Euclidean upgrading (NEU) meta-algorithm which builds the universal reconfiguration and universal approximation properties into any objective learning algorithm. These properties state that a procedure can reproduce any dataset exactly and approximate any function to arbitrary precision, respectively. This is done through an unsupervised learning procedure which identifies a geometry optimizing the relationship between a dataset and the objective learning algorithm used to explain it. The effectiveness of this procedure is supported both theoretically and numerically. The numerical implementations find that NEU-ordinary least squares outperforms leading regularized regression algorithms and that NEU-PCA explains more variance with one NEU-principal component than PCA does with four classical principal components. The second part of the thesis introduces a computationally efficient characterization of intrinsic conditional expectation for Cartan-Hadamard manifolds. This alternative characterization provides an explicit way of computing non-Euclidean conditional expectation by using geometric transformations of specific Euclidean conditional expectations. This reduces many non-convex intrinsic estimation problems to transformations of well studied Euclidean conditional expectations. As a consequence, computationally tractable non-Euclidean filtering equations are derived and used to successfully forecast efficient portfolios by exploiting their geometry. The third and final part of this thesis introduces a flexible modeling framework and a stochastic learning methodology for incorporating arbitrage-free features into many asset price models. The procedure works by minimally deforming the structure of a model until the objective measure acts as a martingale measure for that model. Reformulations of classical no-arbitrage results such as NFLVR, the minimal martingale measure, and the arbitrage-free Nelson-Siegel correction of the Nelson-Siegel model are all derived as solutions to specific arbitrage-free regularization problems. The flexibility and generality of this framework allows classical no-arbitrage pricing theory to be extended to models that admit arbitrage opportunities but are deformable into arbitrage-free models. Numerical implications are investigated in each of the three parts making up this thesis

Some applications of optimal stopping and control in finance and economics

In this thesis, we consider some applications of optimal stopping and control problems in specific scenarios. In Chapter 1, a review of the established general results is provided. In Chapter 2, we study a mathematical model capturing the support/ resistance line method (a technique in technical analysis) where the underlying stock price transitions between two states of nature in a path-dependent manner. For optimal stopping problems with respect to a general class of reward functions and dynamics, using probabilistic methods, we show that the value function is C1 and solves a general free boundary problem. Moreover, for a wide range of utilities, we prove that the best time to buy and sell the stock is obtained by solving free boundary problems corresponding to two linked optimal stopping problems. We use this to numerically compute optimal trading strategies and compare the strategies with the standard trading rule to investigate the viability of this form of technical analysis. In Chapter 3, the model studied in Chapter 2 is extended by adding a partial reflection boundary and an additional regime (the 0 regime). In Chapter 4, we study a two dimensional continuous-time infinite horizon singular control problem related with the optimal management of inventory and production. The primary source of production is modeled as an uncontrolled one-dimensional diffusion process with general dynamics. By controlling the accumulated secondary source of production and output, which are both finite variation processes, we aim to optimise the inventory process under a general concave running reward function and maximise the profit generated from the production. By solving the associated Dynkin game, we obtain two non-intersecting bounded and monotone free-boundaries where one is directly computable and the other is characterised by a free-boundary problem with smooth-pasting conditions. By restricting the volatility term of the diffusion to linear functions with no intercepts, desired smoothness of the value function is obtained by utilising its viscosity property. This leads to the verification of the proposed candidate optimal control that keeps the state process within the inaction set by reflecting the inventory process at the free-boundaries with the minimum effort

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

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