Mean square solutions of random linear models and computation of their probability density function

[EN] This thesis concerns the analysis of differential equations with uncertain input parameters, in the form of random variables or stochastic processes with any type of probability distributions. In modeling, the input coefficients are set from experimental data, which often involve uncertainties from measurement errors. Moreover, the behavior of the physical phenomenon under study does not follow strict deterministic laws. It is thus more realistic to consider mathematical models with randomness in their formulation. The solution, considered in the sample-path or the mean square sense, is a smooth stochastic process, whose uncertainty has to be quantified. Uncertainty quantification is usually performed by computing the main statistics (expectation and variance) and, if possible, the probability density function. In this dissertation, we study random linear models, based on ordinary differential equations with and without delay and on partial differential equations. The linear structure of the models makes it possible to seek for certain probabilistic solutions and even approximate their probability density functions, which is a difficult goal in general. A very important part of the dissertation is devoted to random second-order linear differential equations, where the coefficients of the equation are stochastic processes and the initial conditions are random variables. The study of this class of differential equations in the random setting is mainly motivated because of their important role in Mathematical Physics. We start by solving the randomized Legendre differential equation in the mean square sense, which allows the approximation of the expectation and the variance of the stochastic solution. The methodology is extended to general random second-order linear differential equations with analytic (expressible as random power series) coefficients, by means of the so-called Fröbenius method. A comparative case study is performed with spectral methods based on polynomial chaos expansions. On the other hand, the Fröbenius method together with Monte Carlo simulation are used to approximate the probability density function of the solution. Several variance reduction methods based on quadrature rules and multilevel strategies are proposed to speed up the Monte Carlo procedure. The last part on random second-order linear differential equations is devoted to a random diffusion-reaction Poisson-type problem, where the probability density function is approximated using a finite difference numerical scheme. The thesis also studies random ordinary differential equations with discrete constant delay. We study the linear autonomous case, when the coefficient of the non-delay component and the parameter of the delay term are both random variables while the initial condition is a stochastic process. It is proved that the deterministic solution constructed with the method of steps that involves the delayed exponential function is a probabilistic solution in the Lebesgue sense. Finally, the last chapter is devoted to the linear advection partial differential equation, subject to stochastic velocity field and initial condition. We solve the equation in the mean square sense and provide new expressions for the probability density function of the solution, even in the non-Gaussian velocity case.[ES] Esta tesis trata el análisis de ecuaciones diferenciales con parámetros de entrada aleatorios, en la forma de variables aleatorias o procesos estocásticos con cualquier tipo de distribución de probabilidad. En modelización, los coeficientes de entrada se fijan a partir de datos experimentales, los cuales suelen acarrear incertidumbre por los errores de medición. Además, el comportamiento del fenómeno físico bajo estudio no sigue patrones estrictamente deterministas. Es por tanto más realista trabajar con modelos matemáticos con aleatoriedad en su formulación. La solución, considerada en el sentido de caminos aleatorios o en el sentido de media cuadrática, es un proceso estocástico suave, cuya incertidumbre se tiene que cuantificar. La cuantificación de la incertidumbre es a menudo llevada a cabo calculando los principales estadísticos (esperanza y varianza) y, si es posible, la función de densidad de probabilidad. En este trabajo, estudiamos modelos aleatorios lineales, basados en ecuaciones diferenciales ordinarias con y sin retardo, y en ecuaciones en derivadas parciales. La estructura lineal de los modelos nos permite buscar ciertas soluciones probabilísticas e incluso aproximar su función de densidad de probabilidad, lo cual es un objetivo complicado en general. Una parte muy importante de la disertación se dedica a las ecuaciones diferenciales lineales de segundo orden aleatorias, donde los coeficientes de la ecuación son procesos estocásticos y las condiciones iniciales son variables aleatorias. El estudio de esta clase de ecuaciones diferenciales en el contexto aleatorio está motivado principalmente por su importante papel en la Física Matemática. Empezamos resolviendo la ecuación diferencial de Legendre aleatorizada en el sentido de media cuadrática, lo que permite la aproximación de la esperanza y la varianza de la solución estocástica. La metodología se extiende al caso general de ecuaciones diferenciales lineales de segundo orden aleatorias con coeficientes analíticos (expresables como series de potencias), mediante el conocido método de Fröbenius. Se lleva a cabo un estudio comparativo con métodos espectrales basados en expansiones de caos polinomial. Por otro lado, el método de Fröbenius junto con la simulación de Monte Carlo se utilizan para aproximar la función de densidad de probabilidad de la solución. Para acelerar el procedimiento de Monte Carlo, se proponen varios métodos de reducción de la varianza basados en reglas de cuadratura y estrategias multinivel. La última parte sobre ecuaciones diferenciales lineales de segundo orden aleatorias estudia un problema aleatorio de tipo Poisson de difusión-reacción, en el que la función de densidad de probabilidad es aproximada mediante un esquema numérico de diferencias finitas. En la tesis también se tratan ecuaciones diferenciales ordinarias aleatorias con retardo discreto y constante. Estudiamos el caso lineal y autónomo, cuando el coeficiente de la componente no retardada i el parámetro del término retardado son ambos variables aleatorias mientras que la condición inicial es un proceso estocástico. Se demuestra que la solución determinista construida con el método de los pasos y que involucra la función exponencial retardada es una solución probabilística en el sentido de Lebesgue. Finalmente, el último capítulo lo dedicamos a la ecuación en derivadas parciales lineal de advección, sujeta a velocidad y condición inicial estocásticas. Resolvemos la ecuación en el sentido de media cuadrática y damos nuevas expresiones para la función de densidad de probabilidad de la solución, incluso en el caso de velocidad no Gaussiana.[CA] Aquesta tesi tracta l'anàlisi d'equacions diferencials amb paràmetres d'entrada aleatoris, en la forma de variables aleatòries o processos estocàstics amb qualsevol mena de distribució de probabilitat. En modelització, els coeficients d'entrada són fixats a partir de dades experimentals, les quals solen comportar incertesa pels errors de mesurament. A més a més, el comportament del fenomen físic sota estudi no segueix patrons estrictament deterministes. És per tant més realista treballar amb models matemàtics amb aleatorietat en la seua formulació. La solució, considerada en el sentit de camins aleatoris o en el sentit de mitjana quadràtica, és un procés estocàstic suau, la incertesa del qual s'ha de quantificar. La quantificació de la incertesa és sovint duta a terme calculant els principals estadístics (esperança i variància) i, si es pot, la funció de densitat de probabilitat. En aquest treball, estudiem models aleatoris lineals, basats en equacions diferencials ordinàries amb retard i sense, i en equacions en derivades parcials. L'estructura lineal dels models ens fa possible cercar certes solucions probabilístiques i inclús aproximar la seua funció de densitat de probabilitat, el qual és un objectiu complicat en general. Una part molt important de la dissertació es dedica a les equacions diferencials lineals de segon ordre aleatòries, on els coeficients de l'equació són processos estocàstics i les condicions inicials són variables aleatòries. L'estudi d'aquesta classe d'equacions diferencials en el context aleatori està motivat principalment pel seu important paper en Física Matemàtica. Comencem resolent l'equació diferencial de Legendre aleatoritzada en el sentit de mitjana quadràtica, el que permet l'aproximació de l'esperança i la variància de la solució estocàstica. La metodologia s'estén al cas general d'equacions diferencials lineals de segon ordre aleatòries amb coeficients analítics (expressables com a sèries de potències), per mitjà del conegut mètode de Fröbenius. Es duu a terme un estudi comparatiu amb mètodes espectrals basats en expansions de caos polinomial. Per altra banda, el mètode de Fröbenius juntament amb la simulació de Monte Carlo són emprats per a aproximar la funció de densitat de probabilitat de la solució. Per a accelerar el procediment de Monte Carlo, es proposen diversos mètodes de reducció de la variància basats en regles de quadratura i estratègies multinivell. L'última part sobre equacions diferencials lineals de segon ordre aleatòries estudia un problema aleatori de tipus Poisson de difusió-reacció, en què la funció de densitat de probabilitat és aproximada mitjançant un esquema numèric de diferències finites. En la tesi també es tracten equacions diferencials ordinàries aleatòries amb retard discret i constant. Estudiem el cas lineal i autònom, quan el coeficient del component no retardat i el paràmetre del terme retardat són ambdós variables aleatòries mentre que la condició inicial és un procés estocàstic. Es prova que la solució determinista construïda amb el mètode dels passos i que involucra la funció exponencial retardada és una solució probabilística en el sentit de Lebesgue. Finalment, el darrer capítol el dediquem a l'equació en derivades parcials lineal d'advecció, subjecta a velocitat i condició inicial estocàstiques. Resolem l'equació en el sentit de mitjana quadràtica i donem noves expressions per a la funció de densitat de probabilitat de la solució, inclús en el cas de velocitat no Gaussiana.This work has been supported by the Spanish Ministerio de Economía y Competitividad grant MTM2017–89664–P. I acknowledge the doctorate scholarship granted by Programa de Ayudas de Investigación y Desarrollo (PAID), Universitat Politècnica de València.Jornet Sanz, M. (2020). Mean square solutions of random linear models and computation of their probability density function [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/138394TESI

Pointwise convergence of Fourier series. Carleson’s theorem

Treballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2016, Director: María Jesús Carro RossellIn this project we study the pointwise convergence of Fourier series. Our main goal is the proof of Carleson’s theorem, which states, roughly speaking, that the Fourier series of any periodic and square integrable function converges to the function almost everywhere. The proof will be based on that presented in the article Pointwise convergence of Fourier series, by Charles Fefferman (see [4]). The structure and the notations will be similar to those of the article, but the proofs and the concepts will be explained in much more detail. In Chapter 1 we revise the history of Fourier series until the proof of Carleson’s theorem by Fefferman [1] [3]. We also explain the structure of the project in detail. In Chapter 2 we relate the convergence problem of Fourier series to the boundedness of an operator. In the third chapter, using dyadic grids, we decompose the mentioned operator in simpler operators. In the fourth chapter we handle some technicalities concerning the dyadic grids chosen. In Chapter 5 we give the intuition for the proof of Carleson’s theorem and we specify the main goal. In the sixth chapter the main lemmas of the project are proved, which give as a consequence the proof of Carleson’s theorem in the seventh chapter

Improving Kernel Methods for Density Estimation in Random Differential Equations Problems

[EN] Kernel density estimation is a non-parametric method to estimate the probability density function of a random quantity from a finite data sample. The estimator consists of a kernel function and a smoothing parameter called the bandwidth. Despite its undeniable usefulness, the convergence rate may be slow with the number of realizations and the discontinuity and peaked points of the target density may not be correctly captured. In this work, we analyze the applicability of a parametric method based on Monte Carlo simulation for the density estimation of certain random variable transformations. This approach has important applications in the setting of differential equations with input random parameters.This work has been supported by the Spanish Ministerio de Economia, Industria y Competitividad (MINECO), the Agencia Estatal de Investigacion (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-P.Cortés, J.; Jornet Sanz, M. (2020). Improving Kernel Methods for Density Estimation in Random Differential Equations Problems. Mathematical and Computational Applications (Online). 25(2):1-9. https://doi.org/10.3390/mca25020033S19252Calatayud, J., Cortés, J.-C., Díaz, J. A., & Jornet, M. (2020). Constructing reliable approximations of the probability density function to the random heat PDE via a finite difference scheme. Applied Numerical Mathematics, 151, 413-424. doi:10.1016/j.apnum.2020.01.012Calatayud, J., Cortés, J.-C., & Jornet, M. (2018). The damped pendulum random differential equation: A comprehensive stochastic analysis via the computation of the probability density function. Physica A: Statistical Mechanics and its Applications, 512, 261-279. doi:10.1016/j.physa.2018.08.024Calatayud, J., Cortés, J.-C., Díaz, J. A., & Jornet, M. (2019). Density function of random differential equations via finite difference schemes: a theoretical analysis of a random diffusion-reaction Poisson-type problem. Stochastics, 92(4), 627-641. doi:10.1080/17442508.2019.1645849Calatayud, J., Cortés, J.-C., Dorini, F. A., & Jornet, M. (2020). Extending the study on the linear advection equation subject to stochastic velocity field and initial condition. Mathematics and Computers in Simulation, 172, 159-174. doi:10.1016/j.matcom.2019.12.014Jornet, M., Calatayud, J., Le Maître, O. P., & Cortés, J.-C. (2020). Second order linear differential equations with analytic uncertainties: Stochastic analysis via the computation of the probability density function. Journal of Computational and Applied Mathematics, 374, 112770. doi:10.1016/j.cam.2020.112770Tang, K., Wan, X., & Liao, Q. (2020). Deep density estimation via invertible block-triangular mapping. Theoretical and Applied Mechanics Letters, 10(3), 143-148. doi:10.1016/j.taml.2020.01.023Botev, Z., & Ridder, A. (2017). Variance Reduction. Wiley StatsRef: Statistics Reference Online, 1-6. doi:10.1002/9781118445112.stat0797

Random differential equations with discrete delay

[EN] In this article, we study random differential equations with discrete delay with initial condition The uncertainty in the problem is reflected via the outcome omega. The initial condition g(t) is a stochastic process. The term x(t) is a stochastic process that solves the random differential equation with delay in a probabilistic sense. In our case, we use the random calculus approach. We extend the classical Picard theorem for deterministic ordinary differential equations to calculus for random differential equations with delay, via Banach fixed-point theorem. We also relate solutions with sample-path solutions. Finally, we utilize the theoretical ideas to solve the random autonomous linear differential equation with discrete delay.This work has been supported by the Spanish Ministerio de Economía y Competitividad grant MTM2017 89664 PCalatayud-Gregori, J.; Cortés, J.; Jornet-Sanz, M. (2019). Random differential equations with discrete delay. Stochastic Analysis and Applications. 37(5):699-707. https://doi.org/10.1080/07362994.2019.1608833S699707375Fridman, E., & Shaikhet, L. (2017). Stabilization by using artificial delays: An LMI approach. Automatica, 81, 429-437. doi:10.1016/j.automatica.2017.04.015Shaikhet, L., & Korobeinikov, A. (2015). Stability of a stochastic model for HIV-1 dynamics within a host. Applicable Analysis, 95(6), 1228-1238. doi:10.1080/00036811.2015.1058363Caraballo, T., Colucci, R., & Guerrini, L. (2018). On a predator prey model with nonlinear harvesting and distributed delay. Communications on Pure & Applied Analysis, 17(6), 2703-2727. doi:10.3934/cpaa.2018128Caraballo, T., J. Garrido-Atienza, M., Schmalfuss, B., & Valero, J. (2017). Attractors for a random evolution equation with infinite memory: Theoretical results. Discrete & Continuous Dynamical Systems - B, 22(5), 1779-1800. doi:10.3934/dcdsb.2017106Krapivsky, P. L., Luck, J. M., & Mallick, K. (2011). On stochastic differential equations with random delay. Journal of Statistical Mechanics: Theory and Experiment, 2011(10), P10008. doi:10.1088/1742-5468/2011/10/p10008Liu, S., Debbouche, A., & Wang, J. (2017). On the iterative learning control for stochastic impulsive differential equations with randomly varying trial lengths. Journal of Computational and Applied Mathematics, 312, 47-57. doi:10.1016/j.cam.2015.10.028Dorini, F. A., Cecconello, M. S., & Dorini, L. B. (2016). On the logistic equation subject to uncertainties in the environmental carrying capacity and initial population density. Communications in Nonlinear Science and Numerical Simulation, 33, 160-173. doi:10.1016/j.cnsns.2015.09.009Slama, H., El-Bedwhey, N. A., El-Depsy, A., & Selim, M. M. (2017). Solution of the finite Milne problem in stochastic media with RVT Technique. The European Physical Journal Plus, 132(12). doi:10.1140/epjp/i2017-11763-6Nouri, K., Ranjbar, H., & Torkzadeh, L. (2019). Modified stochastic theta methods by ODEs solvers for stochastic differential equations. Communications in Nonlinear Science and Numerical Simulation, 68, 336-346. doi:10.1016/j.cnsns.2018.08.013Lupulescu, V., O’Regan, D., & ur Rahman, G. (2014). Existence results for random fractional differential equations. Opuscula Mathematica, 34(4), 813. doi:10.7494/opmath.2014.34.4.813Strand, J. . (1970). Random ordinary differential equations. Journal of Differential Equations, 7(3), 538-553. doi:10.1016/0022-0396(70)90100-2Villafuerte, L., Braumann, C. A., Cortés, J.-C., & Jódar, L. (2010). Random differential operational calculus: Theory and applications. Computers & Mathematics with Applications, 59(1), 115-125. doi:10.1016/j.camwa.2009.08.061Granas, A., & Dugundji, J. (2003). Fixed Point Theory. Springer Monographs in Mathematics. doi:10.1007/978-0-387-21593-

Lp-calculus approach to the random autonomous linear differential equation with discrete delay

[EN] In this paper, we provide a full probabilistic study of the random autonomous linear differential equation with discrete delay , with initial condition x(t)=g(t), -t0. The coefficients a and b are assumed to be random variables, while the initial condition g(t) is taken as a stochastic process. Using Lp-calculus, we prove that, under certain conditions, the deterministic solution constructed with the method of steps that involves the delayed exponential function is an Lp-solution too. An analysis of Lp-convergence when the delay tends to 0 is also performed in detail.This work has been supported by the Spanish Ministerio de Economia y Competitividad Grant MTM2017-89664-P. The author Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia.Calatayud-Gregori, J.; Cortés, J.; Jornet-Sanz, M. (2019). Lp-calculus approach to the random autonomous linear differential equation with discrete delay. Mediterranean Journal of Mathematics. 16(4):1-17. https://doi.org/10.1007/s00009-019-1370-6S117164Smith, H.: An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics. Springer, New York (2011)Driver, Y.: Ordinary and Delay Differential Equations. Applied Mathematical Science Series. Springer, New York (1977)Kuang, Y.: Delay Differential Equations: with Applications in Population Dynamics. Academic Press, Cambridge (2012)Bocharov, G.A., Rihan, F.A.: Numerical modelling in biosciences using delay differential equations. J. Comput. Appl. Math. 125, 183–199 (2000). https://doi.org/10.1016/S0377-0427(00)00468-4Jackson, M., Chen-Charpentier, B.M.: Modeling plant virus propagation with delays. J. Comput. Appl. 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Springer, New York (1998)Shaikhet, L.: Lyapunov Functionals and Stability of Stochastic Functional Differential Equations. Springer, New York (2013)Hartung, F., Pituk, M.: Recent Advances in Delay Differential and Differences Equations. Springer-Verlag, Berlin Heidelberg (2014)Shaikhet, L.: Stability of equilibrium states of a nonlinear delay differential equation with stochastic perturbations. Int. J. Robust Nonlinear Control 27(6), 915–924 (2016). https://doi.org/10.1002/rnc.3605Shaikhet, L.: About some asymptotic properties of solution of stochastic delay differential equation with a logarithmic nonlinearity. Funct. Differ. Equ. 4(1–2), 57–67 (2017)Fridman, E., Shaikhet, L.: Delay-induced stability of vector second-order systems via simple Lyapunov functionals. Automatica 74, 288–296 (2016). https://doi.org/10.1016/j.automatica.2016.07.034Benhadri, M., Zeghdoudi, H.: Mean square asymptotic stability in nonlinear stochastic neutral Volterra-Levin equations with Poisson jumps and variable delays. Functiones et Approximatio Commentarii Mathematici 58(2), 157–176 (2018). https://doi.org/10.7169/facm/1657Nouri, K., Ranjbar, H.: Improved Euler-Maruyama method for numerical solution of the Itô stochastic differential systems by composite previous-current-step idea. Mediterr. J. Math. 15, 140 (2018). https://doi.org/10.1007/s00009-018-1187-8Santonja, F., Shaikhet, L.: Probabilistic stability analysis of social obesity epidemic by a delayed stochastic model. Nonlinear Anal. Real World Appl. 17, 114–125 (2014). https://doi.org/10.1016/j.nonrwa.2013.10.010Santonja, F., Shaikhet, L.: Analysing social epidemics by delayed stochastic models. Discret. Dyn. Nat. Soc. 2012, 13 (2012). https://doi.org/10.1155/2012/530472 . (Article ID 530472)Liu, L., Caraballo, T.: Analysis of a stochastic 2D-Navier-Stokes model with infinite delay. J. Dyn. Differ. Equ. pp 1–26 (2018). https://doi.org/10.1007/s10884-018-9703-xCaraballo, T., Colucci, R., Guerrini, L.: On a predator prey model with nonlinear harvesting and distributed delay. Commun. Pure Appl. Anal. 17(6), 2703–2727 (2018). https://doi.org/10.3934/cpaa.2018128Smith, R.C.: Uncertainty Quantification. Theory, Implementation and Applications. SIAM, Philadelphia (2014)Soong, T.T.: Random Differential Equations in Science and Engineering. Academic Press, New York (1973)Nouri, K., Ranjbar, H.: Mean square convergence of the numerical solution of random differential equations. Mediterr. J. Math. 12(3), 1123–1140 (2015). https://doi.org/10.1007/s00009-014-0452-8Zhou, T.: A stochastic collocation method for delay differential equations with random input. Adv. Appl. Math. Mech. 6(4), 403–418 (2014). https://doi.org/10.4208/aamm.2012.m38Shi, W., Zhang, C.: Generalized polynomial chaos for nonlinear random delay differential equations. Appl. Numer. Math. 115, 16–31 (2017). https://doi.org/10.1016/j.apnum.2016.12.004Lupulescu, V., Abbas, U.: Fuzzy delay differential equations. Fuzzy Optim. Decis. Mak. 11(1), 91–111 (2012). https://doi.org/10.1007/s10700-011-9112-7Liu, S., Debbouche, A., Wang, J.R.: Fuzzy delay differential equations. On the iterative learning control for stochastic impulsive differential equations with randomly varying trial lengths. J. Comput. Appl. Math. 312, 47–57 (2017). https://doi.org/10.1016/j.cam.2015.10.028Krapivsky, P.L., Luck, J.L., Mallick, K.: On stochastic differential equations with random delay. J. Stat. Mech. Theory Exp. (2011). https://doi.org/10.1088/1742-5468/2011/10/P10008Garrido-Atienza, M.J., Ogrowsky, A., Schmalfuss, B.: Random differential equations with random delays. Stoch. Dyn. 11(2–3), 369–388 (2011). https://doi.org/10.1142/S0219493711003358Khusainov, D.Y., Ivanov, A.F., Kovarzh, I.V.: Solution of one heat equation with delay. Nonlinear Oscil. 12, 260–282 (2009). https://doi.org/10.1007/s11072-009-0075-3Asl, F.M., Ulsoy, A.G.: Analysis of a system of linear delay differential equations. J. Dyn. Syst. Meas. Control 125, 215–223 (2003). https://doi.org/10.1115/1.1568121Kyrychko, Y.N., Hogan, S.J.: On the use of delay equations in engineering applications. J. Vib. Control 16(7–8), 943–960 (2010). https://doi.org/10.1177/1077546309341100Villafuerte, L., Braumann, C.A., Cortés, J.C., Jódar, L.: Random differential operational calculus: theory and applications. Comput. Math. Appl. 59(1), 115–125 (2010). https://doi.org/10.1016/j.camwa.2009.08.061Strand, J.L.: Random ordinary differential equations. J. Diff. Equ. 7(3), 538–553 (1970). https://doi.org/10.1016/0022-0396(70)90100-2Khusainov, D.Y., Pokojovy, M.: Solving the linear 1d thermoelasticity equations with pure delay. Int. J. Math. Math. Sci. 2015, 1–11 (2015). https://doi.org/10.1155/2015/47926

Approximate solutions of randomized non-autonomous complete linear differential equations via probability density functions

[EN] Solving a random differential equation means to obtain an exact or approximate expression for the solution stochastic process, and to compute its statistical properties, mainly the mean and the variance functions. However, a major challenge is the computation of the probability density function of the solution. In this article we construct reliable approximations of the probability density function to the randomized non-autonomous complete linear differential equation by assuming that the diffusion coefficient and the source term are stochastic processes and the initial condition is a random variable. The key tools to construct these approximations are the random variable transformation technique and Karhunen-Loeve expansions. The study is divided into a large number of cases with a double aim: firstly, to extend the available results in the extant literature and, secondly, to embrace as many practical situations as possible. Finally, a wide variety of numerical experiments illustrate the potentiality of our findings.This work has been supported by the Spanish Ministerio de Economía y Competitividad grant MTM2017-89664-P. The author Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigación y Desarrollo (PAID), Universitat Politècnica de València.Calatayud-Gregori, J.; Cortés, J.; Jornet-Sanz, M. (2019). Approximate solutions of randomized non-autonomous complete linear differential equations via probability density functions. Electronic Journal of Differential Equations. 2019:1-40. http://hdl.handle.net/10251/139661S140201

Uncertainty quantification for random parabolic equations with non-homogeneous boundary conditions on a bounded domain via the approximation of the probability density function

[EN] This paper deals with the randomized heat equation defined on a general bounded interval [L-1, L-2] and with nonhomogeneous boundary conditions. The solution is a stochastic process that can be related, via changes of variable, with the solution stochastic process of the random heat equation defined on [0,1] with homogeneous boundary conditions. Results in the extant literature establish conditions under which the probability density function of the solution process to the random heat equation on [0,1] with homogeneous boundary conditions can be approximated. Via the changes of variable and the Random Variable Transformation technique, we set mild conditions under which the probability density function of the solution process to the random heat equation on a general bounded interval [L-1, L-2] and with nonhomogeneous boundary conditions can be approximated uniformly or pointwise. Furthermore, we provide sufficient conditions in order that the expectation and the variance of the solution stochastic process can be computed from the proposed approximations of the probability density function. Numerical examples are performed in the case that the initial condition process has a certain Karhunen-Loeve expansion, being Gaussian and non-Gaussian.This work has been supported by Spanish Ministerio de Economía y Competitividad grant MTM2017 89664 P. The author Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigación y Desarrollo (PAID), Universitat Politècnica de València.Calatayud-Gregori, J.; Cortés, J.; Jornet-Sanz, M. (2019). Uncertainty quantification for random parabolic equations with non-homogeneous boundary conditions on a bounded domain via the approximation of the probability density function. Mathematical Methods in the Applied Sciences. 42(17):5649-5667. https://doi.org/10.1002/mma.5333S564956674217Holden, H., Øksendal, B., Ubøe, J., & Zhang, T. (2010). Stochastic Partial Differential Equations. doi:10.1007/978-0-387-89488-1Casabán, M.-C., Company, R., Cortés, J.-C., & Jódar, L. (2014). Solving the random diffusion model in an infinite medium: A mean square approach. Applied Mathematical Modelling, 38(24), 5922-5933. doi:10.1016/j.apm.2014.04.063Xu, Z., Tipireddy, R., & Lin, G. (2016). Analytical approximation and numerical studies of one-dimensional elliptic equation with random coefficients. Applied Mathematical Modelling, 40(9-10), 5542-5559. doi:10.1016/j.apm.2015.12.041CalatayudJ CortésJC JornetM.On the approximation of the probability density function of the randomized heat equation.https://arxiv.org/pdf/1802.04190.pdfStrand, J. . (1970). Random ordinary differential equations. Journal of Differential Equations, 7(3), 538-553. doi:10.1016/0022-0396(70)90100-2Vaart, A. W. van der. (1998). Asymptotic Statistics. doi:10.1017/cbo9780511802256Villafuerte, L., Braumann, C. A., Cortés, J.-C., & Jódar, L. (2010). Random differential operational calculus: Theory and applications. Computers & Mathematics with Applications, 59(1), 115-125. doi:10.1016/j.camwa.2009.08.061Pitman, J. (1993). Probability. doi:10.1007/978-1-4612-4374-8Williams, D. (1991). Probability with Martingales. doi:10.1017/cbo9780511813658LawlessJF.Truncated Distributions: Wiley StatsRef: Statistics Reference Online;2014

The damped pendulum random differential equation: A comprehensive stochastic analysis via the computation of the probability density function

[EN] This paper deals with the damped pendulum random differential equation: (X) over double dot(t)+2 omega(0)xi(X) over dot(t) + omega X-2(0)(t) = Y(t), t is an element of [0, T], with initial conditions X(0) = X-0 and (X) over dot(0) = X-1. The forcing term Y(t) is a stochastic process and X-0 and X-1 are random variables in a common underlying complete probability space (Omega, F, P). The term X(t) is a stochastic process that solves the random differential equation in both the sample path and in the L-P senses. To understand the probabilistic behavior of X(t), we need its joint finite-dimensional distributions. We establish mild conditions under which X(t) is an absolutely continuous random variable, for each t, and we find its probability density function f(X(t))(x). Thus, we obtain the first finite-dimensional distributions. In practice, we deal with two types of forcing term: Y(t) is a Gaussian process, which occurs with the damped pendulum stochastic differential equation of Ito type; and Y(t) can be approximated by a sequence {Y-N(t)}(N-1)(infinity) in L-2([0, T] x Omega), which occurs with Karhunen-Loeve expansions and some random power series. Finally, we provide numerical examples in which we choose specific random variables X-0 and X-1 and a specific stochastic process Y(t), and then, we find the probability density function of X(t). (C) 2018 Elsevier B.V. All rights reserved.This work has been supported by the Spanish Ministerio de Economia y Competitividad grant MTM2017-89664-P. Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia. The authors are grateful for the valuable comments raised by the reviewers that have improved the final version of the paper.Calatayud-Gregori, J.; Cortés, J.; Jornet-Sanz, M. (2018). The damped pendulum random differential equation: A comprehensive stochastic analysis via the computation of the probability density function. Physica A Statistical Mechanics and its Applications. 512:261-279. https://doi.org/10.1016/j.physa.2018.08.024S26127951

Improving the approximation of the first and second order statistics of the response stochastic process to the random Legendre differential equation

[EN] In this paper, we deal with uncertainty quantification for the random Legendre differential equation, with input coefficient A and initial conditions X-0 and X-1. In a previous study (Calbo et al. in Comput Math Appl 61(9):2782-2792, 2011), a mean square convergent power series solution on (-1/e, 1/e) was constructed, under the assumptions of mean fourth integrability of X-0 and X-1, independence, and at most exponential growth of the absolute moments of A. In this paper, we relax these conditions to construct an L-p solution (1 <= p <= infinity) to the random Legendre differential equation on the whole domain (-1, 1), as in its deterministic counterpart. Our hypotheses assume no independence and less integrability of X-0 and X-1. Moreover, the growth condition on the moments of A is characterized by the boundedness of A, which simplifies the proofs significantly. We also provide approximations of the expectation and variance of the response process. The numerical experiments show the wide applicability of our findings. A comparison with Monte Carlo simulations and gPC expansions is performed.This work has been supported by the Spanish Ministerio de Economia y Competitividad Grant MTM2017-89664-P. Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia.Calatayud-Gregori, J.; Cortés, J.; Jornet-Sanz, M. (2019). Improving the approximation of the first and second order statistics of the response stochastic process to the random Legendre differential equation. Mediterranean Journal of Mathematics. 16(3):1-14. https://doi.org/10.1007/s00009-019-1338-6S114163Soong, T.T.: Random Differential Equations in Science and Engineering. Academic Press, New York (1973)Strand, J.L.: Random ordinary differential equations. J. Differ. Equ. 7(3), 538–553 (1970)Smith, R.C.: Uncertainty quantification. Theory, implementation, and application. 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Fórmula de Paridad para Opciones Financieras Europeas

En este trabajo se establece una relación matemática, denominada Fórmula de Paridad, que relaciona las primas de una opción de compra u opción Call y de una opción de venta u opción Put de tipo europeo que han sido suscritas sobre un mismo activo subyacente, con un mismo período de vencimiento y un mismo precio de ejercicio. Se establecen dos versiones de la Fórmula de Paridad, dependiendo de si el activo subyacente paga o no dividendos durante la vida del contrato de la opción.Cortés López, JC.; Jornet Sanz, M. (2018). Fórmula de Paridad para Opciones Financieras Europeas. http://hdl.handle.net/10251/107087DE