1,721,112 research outputs found
Stochastic taylor expansions for functionals of martingales
No full textWe use the Functional Ito Calculus to develop a stochastic Taylor formula and a chaos
expansion for functionals of a continuous square-integrable martingale. Given a continuous
square-integrable martingale X, we define Sobolev spaces of non-anticipative functionals using
the concept of weak vertical and horizontal derivatives developed in the functional Itˆo calculus.
We then show that any functional in these Sobolev spaces may be expanded as a sum of
multiple Ito integrals, with integrands expressed in terms of horizontal and vertical derivatives
of the process. This result extends the well-known Wiener-Ito decomposition beyond the
Gaussian setting, to functionals of a continuous square-integrable martingale, with the n-th
homogeneous chaos replaced by a space of n-fold iterated Ito integrals with respect to X.Open Acces
Arbitrage Strategy
No full textA (riskless) arbitrage strategy allows a financial agent to make certain profit out of nothing, that is, out of zero initial investment. This has to be disallowed on economic basis if the market is in equilibrium state, as opportunities for riskless profit would result in an instantaneous movement of prices of certain financial instruments. The principle of not allowing for arbitrage opportunities in financial markets has far-reaching consequences, most notably the option-pricing and hedging formulas in complete markets
Duration models
No full textDuration models appear when studying the moment in time that certain events occur. Next to general applications in economics and medical sciences, their financial applications are in the more specific field of market microstructure (transaction times of assets or derivatives in a given market), and also in corporate governance (tenure of management)
Robustness and sensitivity analysis of risk measurement procedures
Full text linkMeasuring the risk of a financial portfolio involves two steps: estimating the loss distribution of the portfolio from available observations and computing a 'risk measure' that summarizes the risk of the portfolio. We define the notion of 'risk measurement procedure', which includes both of these steps, and introduce a rigorous framework for studying the robustness of risk measurement procedures and their sensitivity to changes in the data set. Our results point to a conflict between the subadditivity and robustness of risk measurement procedures and show that the same risk measure may exhibit quite different sensitivities depending on the estimation procedure used. Our results illustrate, in particular, that using recently proposed risk measures such as CVaR/expected shortfall leads to a less robust risk measurement procedure than historical Value-at-Risk. We also propose alternative risk measurement procedures that possess the robustness property.Risk management, Risk measurement, Coherent risk measures, Law invariant risk measures, Value-at-Risk, Expected shortfall,
Asymptotic analysis of deep learning algorithms
Full text linkWe investigate the asymptotic properties of deep residual networks as the number of layers increases. We first show the existence of scaling regimes for trained weights markedly different from those implicitly assumed in the neural ODE literature. We study the convergence of the hidden state dynamics in these scaling regimes, showing that one may obtain an ODE, a stochastic differential equation (SDE) or neither. Furthermore, we derive the corresponding scaling limits for the backpropagation dynamics. Finally, we prove that in the case of a smooth activation function, the scaling regime arises as a consequence of using gradient descent. In particular, we prove linear convergence of gradient descent to a global minimum for the training of deep residual networks. We also show that if the trained weights, as a function of the layer index, admit a scaling limit as the depth increases, then the limit has finite 2-variation.
This work also investigate the mean-field limit of path-homogeneous neural architectures. We prove convergence of the Wasserstein gradient flow to a global minimum, and we derive a generalization bound based on the stability of the optimization algorithm for 2-layer neural networks with ReLU activation
Free Lunch
No full textThe concept of absence of opportunities for free lunches is one of the pillars in the economic theory of financial markets. This natural assumption has proved very fruitful and has lead to great mathematical, as well as economical, insights in quantitative finance. Formulating rigorously, the exact definition of absence of opportunities for riskless profit turned out to be a highly nontrivial fact that troubled mathematicians and economists for at least two decades. The purpose of this article is to give a quick (and, hence necessarily, incomplete) account of the recent work aimed at providing a simple and intuitive no-free-lunch assumption that would suffice in formulating a version of the celebrated fundamental theorem of asset pricing
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