Integration by parts formula for locally smooth laws and applications to sensitivity computations

We consider random variables of the form $F=f(V_1,...,V_n)$, where $f$ is a smooth function and $V_i,i\in\mathbb{N}$, are random variables with absolutely continuous law $p_i(y) dy$. We assume that $p_i$, $i=1,...,n$, are piecewise differentiable and we develop a differential calculus of Malliavin type based on $\partial\ln p_i$. This allows us to establish an integration by parts formula $E(\partial_i\phi(F)G)=E(\phi(F)H_i(F,G))$, where $H_i(F,G)$ is a random variable constructed using the differential operators acting on $F$ and $G.$ We use this formula in order to give numerical algorithms for sensitivity computations in a model driven by a L\'{e}vy process.Comment: Published at http://dx.doi.org/10.1214/105051606000000592 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Integration by parts formula for locally smooth laws and applications to sensitivity computations

We consider random variables of the form $F=f(V_1,...,V_n)$ where $f$ is a smooth function and V_i,i\in\mathbbN are random variables with absolutely continuous law $p_i(y).$ We assume that $p_i,i=1,...,n$ are piecewise differentiable and we develop a differential calculus of Malliavin type based on \partial\lnp_i. This allows us to establish an integration by parts formula $E(\partial_i\phi(F)G)=E(\phi(F)H_i(F,G))$ where $H_i(F,G)$ is a random variable constructed using the differential operators acting on $F$ and $G.$ We use this formula in order to give numerical algorithms for sensitivity computations in a model driven by a Lévy process

Contrôle optimal et calcul de Malliavin appliqués à la finance

La première partie est consacrée au contrôle optimal stochastique et impulsionnel. Nous proposons deux algorithmes pour résoudre numériquement des inéquations Quasi Variationnelles qui apparaissant dans un problème de gestion de portefeuille avec coûts de transaction fixes et proportionnels. Dans la deuxième partie nous appliquons le calcul de Malliavin au calcul des sensibilités. Nous étudions des processus de sauts purs et nous établissons des formules d intégration par partie à l aide des densités des amplitudes de sauts que nous supposons différentiables. Ensuite nous affaiblissons l hypothèse sur les densités en les supposant différentiables par morceaux. Ainsi nous utilisons la densité des temps de sauts pour établir des formules d IPP. Nous étudions aussi des modèles de diffusons continues à plusieurs facteurs. L ellipticité de la diffusion est nécessaire pour l approche classique du calcul de Malliavin. Pour les options européennes nous établissons plusieurs IPP indépendamment de l ellipticité de la diffusion, à l aide d autres variables qui agrègent la diffusion multidimensionnelle et qui réduisent la dimension de la matrice covariance de Malliavin. Dans le dernier chapitre nous étudions le calibrage de la volatilité locale par minimisation de l entropie relative. Il s agit de résoudre un problème de contrôle stochastique. Nous proposons des améliorations aux algorithmes déjà existants.In the first part we study the optimal stochastic and impulse control. We give two algorithms to solve numerically the Quasi Variationnel Inequalities that appear in a portfolio management optimization with fixed and proportional transaction cost. In the second part we use the Malliavin calculus to compute the sensitivities. We study a pure jump process and use a differentiable density of jump amplitude to establish an Integration By Parts formula. Next we use the density of the jump time which is not continuously differentiable and establish also an IBP formula. We apply also the Malliavin calculus for continuous multidimensional diffusions in a multi factors models. The classical approach is based on the ellipticity of the diffusion. This condition is not always satisfied in the interest rate frame. For European option we establish some IBP formula based on other variables that aggregate the multidimensional diffusion and reduce the dimension of the Malliavin covariance matrix. In the last chapter we comment the method of calibrating the local volatility by using an entropy minimization. We have to solve a stochastic control problem. We give numerical improvement for the existingPARIS-DAUPHINE-BU (751162101) / SudocSudocFranceF

Computation of Greeks using Malliavin's calculus in jump type market models

We use the Malliavin calculus for Poisson processes in order to compute sensitivities for European options with underlying following a jump type diffusion. The main point is to settle an integration by parts formula (similar to the one in the Malliavin calculus) for a general multidimensional random variable which has an absolutely continuous law with differentiable density. We give an explicit expression of the differential operators involved in this formula and this permits to simulate them and consequently to run a Monte Carlo algorithm

Integration by parts formula for locally smooth laws and applications to sensitivity computations

We consider random variables of the form $F=f(V_1,...,V_n)$ where $f$ is a smooth function and V_i,i\in\mathbbN are random variables with absolutely continuous law $p_i(y).$ We assume that $p_i,i=1,...,n$ are piecewise differentiable and we develop a differential calculus of Malliavin type based on \partial\lnp_i. This allows us to establish an integration by parts formula $E(\partial_i\phi(F)G)=E(\phi(F)H_i(F,G))$ where $H_i(F,G)$ is a random variable constructed using the differential operators acting on $F$ and $G.$ We use this formula in order to give numerical algorithms for sensitivity computations in a model driven by a Lévy process

High Frequency of Enterocytozoon bieneusi Genotype WL12 Occurrence among Immunocompromised Patients with Intestinal Microsporidiosis

International audienceMicrosporidiosis is an emerging opportunistic infection causing severe digestive disorders in immunocompromised patients. The aim of this study was to investigate the prevalence of intestinal microsporidia carriage among immunocompromised patients hospitalized at a major hospital complex in the Tunis capital area, Tunisia (North Africa), and perform molecular epidemiology and population structure analyses of Enterocytozoon bieneusi, which is an emerging fungal pathogen. We screened 250 stool samples for the presence of intestinal microsporidia from 171 patients, including 81 organ transplant recipients, 73 Human Immunodeficiency Virus (HIV)-positive patients, and 17 patients with unspecified immunodeficiency. Using a nested PCR-based diagnostic approach for the detection of E. bieneusi and Encephalitozoon spp., we identified 18 microsporidia-positive patients out of 171 (10.5%), among which 17 were infected with E. bieneusi. Microsporidia-positive cases displayed chronic diarrhea (17 out of 18), which was associated more with HIV rather than with immunosuppression other than HIV (12 out of 73 versus 6 out of 98, respectively, p = 0.02) and correlated with extended hospital stays compared to microsporidia-negative cases (60 versus 19 days on average, respectively; p = 0.001). Strikingly, internal transcribed spacer (ITS)-based genotyping of E. bieneusi strains revealed high-frequency occurrence of ITS sequences that were identical (n = 10) or similar (with one single polymorphic site, n = 3) to rare genotype WL12. Minimum-spanning tree analyses segregated the 17 E. bieneusi infection cases into four distinct genotypic clusters and confirmed the high prevalence of genotype WL12 in our patient population. Phylogenetic analyses allowed the mapping of all 17 E. bieneusi strains to zoonotic group 1 (subgroups 1a and 1b/1c), indicating loose host specificity and raising public health concern. Our study suggests a probable common source of E. bieneusi genotype WL12 transmission and prompts the implementation of a wider epidemiological investigation