Necessary and sufficient conditions for the existence of the q-optimal measure

This paper presents the general form and essential properties of the q-optimal measure following the approach of Delbaen and Schachermayer (1996) and proves its existence under mild conditions. Most importantly, it states a necessary and sufficient condition for a candidate measure to be the q-optimal measure in the case even of signed measures. Finally, an updated characterization of the q-optimal measure for continuous asset price processes is presented in the light of the counterexample appearing in Cerny and Kallsen (2006) concerning Hobson's (2004) approach

Delay geometric Brownian motion in financial option valuation

Motivated by influential work on complete stochastic volatility models, such as Hobson and Rogers [11], we introduce a model driven by a delay geometric Brownian motion (DGBM) which is described by the stochastic delay differential equation dSðtÞ ¼ mðSðt 2tÞÞSðtÞdt þ VðSðt 2tÞÞSðtÞdWðtÞ. We show that the equation has a unique positive solution under a very general condition, namely that the volatility function V is a continuous mapping from Rþ to itself. Moreover, we show that the delay effect is not too sensitive to time lag changes. The desirable robustness of the delay effect is demonstrated on several important financial derivatives as well as on the value process of the underlying asset. Finally, we introduce an Euler–Maruyama numerical scheme for our proposed model and show that this numerical method approximates option prices very well. All these features show that the proposedDGBMserves as a rich alternative in modelling financial instruments in a complete market framework

A note on tamed Euler approximations

Strong convergence results on tamed Euler schemes, which approximate stochastic differential equations with superlinearly growing drift coefficients that are locally one-sided Lipschitz continuous, are presented in this article. The diffusion coefficients are assumed to be locally Lipschitz continuous and have at most linear growth. Furthermore, the classical rate of convergence, i.e. one--half, for such schemes is recovered when the local Lipschitz continuity assumptions are replaced by global and, in addition, it is assumed that the drift coefficients satisfy polynomial Lipschitz continuity.Comment: 10 page

Taming under isoperimetry

In this article we propose a novel taming Langevin-based scheme called $\mathbf{sTULA}$ to sample from distributions with superlinearly growing log-gradient which also satisfy a Log-Sobolev inequality. We derive non-asymptotic convergence bounds in $KL$ and consequently total variation and Wasserstein-$2$ distance from the target measure. Non-asymptotic convergence guarantees are provided for the performance of the new algorithm as an optimizer. Finally, some theoretical results on isoperimertic inequalities for distributions with superlinearly growing gradients are provided. Key findings are a Log-Sobolev inequality with constant independent of the dimension, in the presence of a higher order regularization and a Poincare inequality with constant independent of temperature and dimension under a novel non-convex theoretical framework.Comment: 50 page

A Strongly Monotonic Polygonal Euler Scheme

In recent years tamed schemes have become an important technique for simulating SDEs and SPDEs whose continuous coefficients display superlinear growth. The taming method, which involves curbing the growth of the coefficients as a function of stepsize, has so far however not been adapted to preserve the monotonicity of the coefficients. This has arisen as an issue particularly in \cite{articletam}, where the lack of a strongly monotonic tamed scheme forces strong conditions on the setting. In the present work we give a novel and explicit method for truncating monotonic functions in separable Hilbert spaces, and show how this can be used to define a polygonal (tamed) Euler scheme on finite dimensional space, preserving the monotonicity of the drift coefficient. This new method of truncation is well-defined with almost no assumptions and, unlike the well-known Moreau-Yosida regularisation, does not require an optimisation problem to be solved at each evaluation. Our construction is the first infinite dimensional method for truncating monotone functions that we are aware of, as well as the first explicit method in any number of dimensions

A fully data-driven approach to minimizing CVaR for portfolio of assets via SGLD with discontinuous updating

A new approach in stochastic optimization via the use of stochastic gradient Langevin dynamics (SGLD) algorithms, which is a variant of stochastic gradient decent (SGD) methods, allows us to efficiently approximate global minimizers of possibly complicated, high-dimensional landscapes. With this in mind, we extend here the non-asymptotic analysis of SGLD to the case of discontinuous stochastic gradients. We are thus able to provide theoretical guarantees for the algorithm's convergence in (standard) Wasserstein distances for both convex and non-convex objective functions. We also provide explicit upper estimates of the expected excess risk associated with the approximation of global minimizers of these objective functions. All these findings allow us to devise and present a fully data-driven approach for the optimal allocation of weights for the minimization of CVaR of portfolio of assets with complete theoretical guarantees for its performance. Numerical results illustrate our main findings.Comment: arXiv admin note: text overlap with arXiv:1910.0200