The fundamental theorem of asset pricing, the hedging problem and maximal claims in financial markets with short sales prohibitions

This paper consists of two parts. In the first part we prove the fundamental theorem of asset pricing under short sales prohibitions in continuous-time financial models where asset prices are driven by nonnegative, locally bounded semimartingales. A key step in this proof is an extension of a well-known result of Ansel and Stricker. In the second part we study the hedging problem in these models and connect it to a properly defined property of "maximality" of contingent claims.Comment: Published in at http://dx.doi.org/10.1214/12-AAP914 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Density of the set of probability measures with the martingale representation property

Let $\psi$ be a multi-dimensional random variable. We show that the set of probability measures $\mathbb{Q}$ such that the $\mathbb{Q}$-martingale $S^{\mathbb{Q}}_t=\mathbb{E}^{\mathbb{Q}}\left[\psi\lvert\mathcal{F}_{t}\right]$ has the Martingale Representation Property (MRP) is either empty or dense in $\mathcal{L}_\infty$-norm. The proof is based on a related result involving analytic fields of terminal conditions $(\psi(x))_{x\in U}$ and probability measures $(\mathbb{Q}(x))_{x\in U}$ over an open set $U$. Namely, we show that the set of points $x\in U$ such that $S_t(x) = \mathbb{E}^{\mathbb{Q}(x)}\left[\psi(x)\lvert\mathcal{F}_{t}\right]$ does not have the MRP, either coincides with $U$ or has Lebesgue measure zero. Our study is motivated by the problem of endogenous completeness in financial economics.Comment: 24 pages, forthcoming in Annals of Probabilit

Polynomial diffusions on compact quadric sets

Polynomial processes are defined by the property that conditional expectations of polynomial functions of the process are again polynomials of the same or lower degree. Many fundamental stochastic processes, including affine processes, are polynomial, and their tractable structure makes them important in applications. In this paper we study polynomial diffusions whose state space is a compact quadric set. Necessary and sufficient conditions for existence, uniqueness, and boundary attainment are given. The existence of a convenient parameterization of the generator is shown to be closely related to the classical problem of expressing nonnegative polynomials---specifically, biquadratic forms vanishing on the diagonal---as a sum of squares. We prove that in dimension $d\le 4$ every such biquadratic form is a sum of squares, while for $d\ge6$ there are counterexamples. The case $d=5$ remains open. An equivalent probabilistic description of the sum of squares property is provided, and we show how it can be used to obtain results on pathwise uniqueness and existence of smooth densities.Comment: Forthcoming in Stochastic Processes and their Application

A system of quadratic BSDEs arising in a price impact model

We consider a financial model where the prices of risky assets are quoted by a representative market maker who takes into account an exogenous demand. We characterize these prices in terms of a system of BSDEs with quadratic growth. We show that this system admits a unique solution for every bounded demand if and only if the market maker's risk-aversion is sufficiently small. The uniqueness is established in the natural class of solutions, without any additional norm restrictions. To the best of our knowledge, this is the first study that proves such (global) uniqueness result for a system of fully coupled quadratic BSDEs.Comment: Published at http://dx.doi.org/10.1214/15-AAP1103 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Stability and analytic expansions of local solutions of systems of quadratic BSDEs with applications to a price impact model

We obtain stability estimates and derive analytic expansions for local solutions of multi-dimensional quadratic BSDEs. We apply these results to a financial model where the prices of risky assets are quoted by a representative dealer in such a way that it is optimal to meet an exogenous demand. We show that the prices are stable under the demand process and derive their analytic expansions for small risk aversion coefficients of the dealer.Comment: Final version, 28 page

Markov cubature rules for polynomial processes

We study discretizations of polynomial processes using finite state Markov processes satisfying suitable moment matching conditions. The states of these Markov processes together with their transition probabilities can be interpreted as Markov cubature rules. The polynomial property allows us to study such rules using algebraic techniques. Markov cubature rules aid the tractability of path-dependent tasks such as American option pricing in models where the underlying factors are polynomial processes.Comment: 29 pages, 6 Figures, 2 Tables; forthcoming in Stochastic Processes and their Application

Affine Volterra processes

We introduce affine Volterra processes, defined as solutions of certain stochastic convolution equations with affine coefficients. Classical affine diffusions constitute a special case, but affine Volterra processes are neither semimartingales, nor Markov processes in general. We provide explicit exponential-affine representations of the Fourier-Laplace functional in terms of the solution of an associated system of deterministic integral equations of convolution type, extending well-known formulas for classical affine diffusions. For specific state spaces, we prove existence, uniqueness, and invariance properties of solutions of the corresponding stochastic convolution equations. Our arguments avoid infinite-dimensional stochastic analysis as well as stochastic integration with respect to non-semimartingales, relying instead on tools from the theory of finite-dimensional deterministic convolution equations. Our findings generalize and clarify recent results in the literature on rough volatility models in finance

EXPLoRA-web: linkage analysis of quantitative trait loci using bulk segregant analysis

Identification of genomic regions associated with a phenotype of interest is a fundamental step toward solving questions in biology and improving industrial research. Bulk segregant analysis (BSA) combined with high-throughput sequencing is a technique to efficiently identify these genomic regions associated with a trait of interest. However, distinguishing true from spuriously linked genomic regions and accurately delineating the genomic positions of these truly linked regions requires the use of complex statistical models currently implemented in software tools that are generally difficult to operate for non-expert users. To facilitate the exploration and analysis of data generated by bulked segregant analysis, we present EXPLoRA-web, a web service wrapped around our previously published algorithm EXPLoRA, which exploits linkage disequilibrium to increase the power and accuracy of quantitative trait loci identification in BSA analysis. EXPLoRA-web provides a user friendly interface that enables easy data upload and parallel processing of different parameter configurations. Results are provided graphically and as BED file and/or text file and the input is expected in widely used formats, enabling straightforward BSA data analysis. The web server is available at http://bioinformatics.intec.ugent.be/explora-web/

Exploiting natural selection to study adaptive behavior

The research presented in this dissertation explores different computational and modeling techniques that combined with predictions from evolution by natural selection leads to the analysis of the adaptive behavior of populations under selective pressure. For this thesis three computational methods were developed: EXPLoRA, EVORhA and SSA-ME. EXPLoRA finds genomic regions associated with a trait of interests (QTL) by explicitly modeling the expected linkage disequilibrium of a population of sergeants under selection. Data from BSA experiments was analyzed to find genomic loci associated with ethanol tolerance. EVORhA explores the interplay between driving and hitchhiking mutations during evolution to reconstruct the subpopulation structure of clonal bacterial populations based on deep sequencing data. Data from mixed infections and evolution experiments of E. Coli was used and their population structure reconstructed. SSA-ME uses mutual exclusivity in cancer to prioritize cancer driver genes. TCGA data of breast cancer tumor samples were analyzed.status: publishe