Journal articles: 'Stochastic calculus via regularization'

1

Platen, Eckhard, and Rolando Rebolledo. "Pricing via anticipative stochastic calculus." Advances in Applied Probability 26, no. 4 (December 1994): 1006–21. http://dx.doi.org/10.2307/1427902.

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The paper proposes a general model for pricing of derivative securities. The underlying dynamics follows stochastic equations involving anticipative stochastic integrals. These equations are solved explicitly and structural properties of solutions are studied.

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Platen, Eckhard, and Rolando Rebolledo. "Pricing via anticipative stochastic calculus." Advances in Applied Probability 26, no. 04 (December 1994): 1006–21. http://dx.doi.org/10.1017/s0001867800026732.

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The paper proposes a general model for pricing of derivative securities. The underlying dynamics follows stochastic equations involving anticipative stochastic integrals. These equations are solved explicitly and structural properties of solutions are studied.

3

Atsuji, A. "Nevanlinna Theory via Stochastic Calculus." Journal of Functional Analysis 132, no. 2 (September 1995): 473–510. http://dx.doi.org/10.1006/jfan.1995.1112.

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Cohen, Paula, Robin Hudson, K. Parthasarathy, and Sylvia Pulmannová. "Hall's transformation via quantum stochastic calculus." Banach Center Publications 43, no. 1 (1998): 147–55. http://dx.doi.org/10.4064/-43-1-147-155.

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Cosso, Andrea, and Francesco Russo. "Functional Itô versus Banach space stochastic calculus and strict solutions of semilinear path-dependent equations." Infinite Dimensional Analysis, Quantum Probability and Related Topics 19, no. 04 (December 2016): 1650024. http://dx.doi.org/10.1142/s0219025716500247.

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Functional Itô calculus was introduced in order to expand a functional [Formula: see text] depending on time [Formula: see text], past and present values of the process [Formula: see text]. Another possibility to expand [Formula: see text] consists in considering the path [Formula: see text] as an element of the Banach space of continuous functions on [Formula: see text] and to use Banach space stochastic calculus. The aim of this paper is threefold. (1) To reformulate functional Itô calculus, separating time and past, making use of the regularization procedures which match more naturally the notion of horizontal derivative which is one of the tools of that calculus. (2) To exploit this reformulation in order to discuss the (not obvious) relation between the functional and the Banach space approaches. (3) To study existence and uniqueness of smooth solutions to path-dependent partial differential equations which naturally arise in the study of functional Itô calculus. More precisely, we study a path-dependent equation of Kolmogorov type which is related to the window process of the solution to an Itô stochastic differential equation with path-dependent coefficients. We also study a semilinear version of that equation.

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Barchielli, A., and A. S. Holevo. "Constructing quantum measurement processes via classical stochastic calculus." Stochastic Processes and their Applications 58, no. 2 (August 1995): 293–317. http://dx.doi.org/10.1016/0304-4149(95)00011-u.

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OLIVERA, CHRISTIAN. "STOCHASTIC INTEGRATION WITH RESPECT TO THE CYLINDRICAL WIENER PROCESS VIA REGULARIZATION." Infinite Dimensional Analysis, Quantum Probability and Related Topics 16, no. 03 (September 2013): 1350024. http://dx.doi.org/10.1142/s0219025713500240.

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Following the ideas of F. Russo and P. Vallois, we use the notion of forward integral to introduce a new stochastic integral respect to the cylindrical Wiener process. This integral is an extension of the classical integral. As an application, we prove existence of solution of a parabolic stochastic differential partial equation with anticipating stochastic initial date.

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Meyer-Brandis, Thilo, Bernt Øksendal, and Xun Yu Zhou. "A mean-field stochastic maximum principle via Malliavin calculus." Stochastics 84, no. 5-6 (February 10, 2012): 643–66. http://dx.doi.org/10.1080/17442508.2011.651619.

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Pamen, O. Menoukeu, F. Proske, and H. Binti Salleh. "Stochastic Differential Games in Insider Markets via Malliavin Calculus." Journal of Optimization Theory and Applications 160, no. 1 (April 19, 2013): 302–43. http://dx.doi.org/10.1007/s10957-013-0310-z.

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Flandoli, Franco, and Ciprian A. Tudor. "Brownian and fractional Brownian stochastic currents via Malliavin calculus." Journal of Functional Analysis 258, no. 1 (January 2010): 279–306. http://dx.doi.org/10.1016/j.jfa.2009.05.001.

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McIver, Annabelle, and Carroll Morgan. "A Novel Stochastic Game Via the Quantitative μ-calculus." Electronic Notes in Theoretical Computer Science 153, no. 2 (May 2006): 195–212. http://dx.doi.org/10.1016/j.entcs.2005.10.039.

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Luo, Chao, Li Yu, and Jun Zheng. "Extending Stochastic Network Calculus to Loss Analysis." Scientific World Journal 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/918565.

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Loss is an important parameter of Quality of Service (QoS). Though stochastic network calculus is a very useful tool for performance evaluation of computer networks, existing studies on stochastic service guarantees mainly focused on the delay and backlog. Some efforts have been made to analyse loss by deterministic network calculus, but there are few results to extend stochastic network calculus for loss analysis. In this paper, we introduce a new parameter named loss factor into stochastic network calculus and then derive the loss bound through the existing arrival curve and service curve via this parameter. We then prove that our result is suitable for the networks with multiple input flows. Simulations show the impact of buffer size, arrival traffic, and service on the loss factor.

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HUANG, JUAN, HONG CHEN, and LUOQING LI. "LEAST SQUARE REGRESSION WITH COEFFICIENT REGULARIZATION BY GRADIENT DESCENT." International Journal of Wavelets, Multiresolution and Information Processing 10, no. 01 (January 2012): 1250005. http://dx.doi.org/10.1142/s021969131100447x.

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We propose a stochastic gradient descent algorithm for the least square regression with coefficient regularization. An explicit expression of the solution via sampling operator and empirical integral operator is derived. Learning rates are given in terms of the suitable choices of the step sizes and regularization parameters.

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Zong, Gaofeng. "Nash Equilibrium of Stochastic Partial Differential Game with Partial Information via Malliavin Calculus." Complexity 2023 (October 26, 2023): 1–29. http://dx.doi.org/10.1155/2023/8803764.

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In this article, we consider the Nash equilibrium of stochastic differential game where the state process is governed by a controlled stochastic partial differential equation and the information available to the controllers is possibly less than the general information. All the system coefficients and the objective performance functionals are assumed to be random. We find an explicit strong solution of the linear stochastic partial differential equation with a generalized probabilistic representation for this solution with the benefit of Kunita’s stochastic flow theory. We use Malliavin calculus to derive a stochastic maximum principle for the optimal control and obtain the Nash equilibrium of this type of stochastic differential game problem.

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Feng, Qi, and Wuchen Li. "Entropy Dissipation for Degenerate Stochastic Differential Equations via Sub-Riemannian Density Manifold." Entropy 25, no. 5 (May 11, 2023): 786. http://dx.doi.org/10.3390/e25050786.

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We studied the dynamical behaviors of degenerate stochastic differential equations (SDEs). We selected an auxiliary Fisher information functional as the Lyapunov functional. Using generalized Fisher information, we conducted the Lyapunov exponential convergence analysis of degenerate SDEs. We derived the convergence rate condition by generalized Gamma calculus. Examples of the generalized Bochner’s formula are provided in the Heisenberg group, displacement group, and Martinet sub-Riemannian structure. We show that the generalized Bochner’s formula follows a generalized second-order calculus of Kullback–Leibler divergence in density space embedded with a sub-Riemannian-type optimal transport metric.

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Sivasankar, Sivajiganesan, and Ramalingam Udhayakumar. "Hilfer Fractional Neutral Stochastic Volterra Integro-Differential Inclusions via Almost Sectorial Operators." Mathematics 10, no. 12 (June 15, 2022): 2074. http://dx.doi.org/10.3390/math10122074.

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In our paper, we mainly concentrate on the existence of Hilfer fractional neutral stochastic Volterra integro-differential inclusions with almost sectorial operators. The facts related to fractional calculus, stochastic analysis theory, and the fixed point theorem for multivalued maps are used to prove the result. In addition, an illustration of the principle is provided.

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Wang, Yan, Aimin Song, Lei Wang, and Jie Sun. "Maximum principle via Malliavin calculus for regular-singular stochastic differential games." Optimization Letters 12, no. 6 (February 28, 2017): 1301–14. http://dx.doi.org/10.1007/s11590-017-1120-2.

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Ascione, Giacomo, and Enrica Pirozzi. "Generalized Fractional Calculus for Gompertz-Type Models." Mathematics 9, no. 17 (September 2, 2021): 2140. http://dx.doi.org/10.3390/math9172140.

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This paper focuses on the construction of deterministic and stochastic extensions of the Gompertz curve by means of generalized fractional derivatives induced by complete Bernstein functions. Precisely, we first introduce a class of linear stochastic equations involving a generalized fractional integral and we study the properties of its solutions. This is done by proving the existence and uniqueness of Gaussian solutions of such equations via a fixed point argument and then by showing that, under suitable conditions, the expected value of the solution solves a generalized fractional linear equation. Regularity of the absolute p-moment functions is proved by using generalized Grönwall inequalities. Deterministic generalized fractional Gompertz curves are introduced by means of Caputo-type generalized fractional derivatives, possibly with respect to other functions. Their stochastic counterparts are then constructed by using the previously considered integral equations to define a rate process and a generalization of lognormal distributions to ensure that the median of the newly constructed process coincides with the deterministic curve.

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Alhojilan, Yazid, and Hamdy M. Ahmed. "New Results Concerning Approximate Controllability of Conformable Fractional Noninstantaneous Impulsive Stochastic Evolution Equations via Poisson Jumps." Mathematics 11, no. 5 (February 22, 2023): 1093. http://dx.doi.org/10.3390/math11051093.

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We introduce the conformable fractional (CF) noninstantaneous impulsive stochastic evolution equations with fractional Brownian motion (fBm) and Poisson jumps. The approximate controllability for the considered problem was investigated. Principles and concepts from fractional calculus, stochastic analysis, and the fixed-point theorem were used to support the main results. An example is applied to show the established results.

20

Kendall, David G. "The Mardia–Dryden shape distribution for triangles: a stochastic calculus approach." Journal of Applied Probability 28, no. 1 (March 1991): 225–30. http://dx.doi.org/10.2307/3214753.

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A remarkable distribution found by Mardia and Dryden for the shape of a random triangle in ℝ2 whose vertices are displaced from their initial positions by independent identical symmetrical gaussian perturbations is re-derived via stochastic calculus.

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Kendall, David G. "The Mardia–Dryden shape distribution for triangles: a stochastic calculus approach." Journal of Applied Probability 28, no. 01 (March 1991): 225–30. http://dx.doi.org/10.1017/s0021900200039553.

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A remarkable distribution found by Mardia and Dryden for the shape of a random triangle in ℝ2 whose vertices are displaced from their initial positions by independent identical symmetrical gaussian perturbations is re-derived via stochastic calculus.

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Choi, Young-Seok. "Subband Adaptive Filtering withl1-Norm Constraint for Sparse System Identification." Mathematical Problems in Engineering 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/601623.

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This paper presents a new approach of the normalized subband adaptive filter (NSAF) which directly exploits the sparsity condition of an underlying system for sparse system identification. The proposed NSAF integrates a weightedl1-norm constraint into the cost function of the NSAF algorithm. To get the optimum solution of the weightedl1-norm regularized cost function, a subgradient calculus is employed, resulting in a stochastic gradient based update recursion of the weightedl1-norm regularized NSAF. The choice of distinct weightedl1-norm regularization leads to two versions of thel1-norm regularized NSAF. Numerical results clearly indicate the superior convergence of thel1-norm regularized NSAFs over the classical NSAF especially when identifying a sparse system.

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Yilmaz, Bilgi. "Computation of option greeks under hybrid stochastic volatility models via Malliavin calculus." Modern Stochastics: Theory and Applications 5, no. 2 (2018): 145–65. http://dx.doi.org/10.15559/18-vmsta100.

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Darses, Sebastien, and Emmanuel Lépinette-Denis. "Parabolic Schemes for Quasi-Linear Parabolic and Hyperbolic PDEs via Stochastic Calculus." Stochastic Analysis and Applications 30, no. 1 (January 2012): 67–99. http://dx.doi.org/10.1080/07362994.2012.628914.

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Sivasankar, Sivajiganesan, and Ramalingam Udhayakumar. "New Outcomes Regarding the Existence of Hilfer Fractional Stochastic Differential Systems via Almost Sectorial Operators." Fractal and Fractional 6, no. 9 (September 16, 2022): 522. http://dx.doi.org/10.3390/fractalfract6090522.

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In this paper, we focus on the existence of Hilfer fractional stochastic differential systems via almost sectorial operators. The main results are obtained by using the concepts and ideas from fractional calculus, multivalued maps, semigroup theory, sectorial operators, and the fixed-point technique. We start by confirming the existence of the mild solution by using Dhage’s fixed-point theorem. Finally, an example is provided to demonstrate the considered Hilferr fractional stochastic differential systems theory.

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Kadiev, Ramazan, and Arcady Ponosov. "Input-to-State Stability of Linear Stochastic Functional Differential Equations." Journal of Function Spaces 2016 (2016): 1–12. http://dx.doi.org/10.1155/2016/8901563.

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The purpose of the paper is to show how asymptotic properties, first of all stochastic Lyapunov stability, of linear stochastic functional differential equations can be studied via the property of solvability of the equation in certain pairs of spaces of stochastic processes, the property which we call input-to-state stability with respect to these spaces. Input-to-state stability and hence the desired asymptotic properties can be effectively verified by means of a special regularization, also known as “theW-method” in the literature. How this framework provides verifiable conditions of different kinds of stochastic stability is shown.

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Finardi, E. C., R. D. Lobato, V. L. de Matos, C. Sagastizábal, and A. Tomasgard. "Stochastic hydro-thermal unit commitment via multi-level scenario trees and bundle regularization." Optimization and Engineering 21, no. 2 (July 3, 2019): 393–426. http://dx.doi.org/10.1007/s11081-019-09448-z.

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Wang, Xiaobo, Xuefei Wu, Zhe Nie, and Zengxian Yan. "The pth Moment Exponential Synchronization of Drive-Response Memristor Neural Networks Subject to Stochastic Perturbations." Complexity 2023 (July 18, 2023): 1–10. http://dx.doi.org/10.1155/2023/1335184.

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In this paper, the p t h moment exponential synchronization problems of drive-response stochastic memristor neural networks are studied via a state feedback controller. The dynamics of the memristor neural network are nonidentical, consisting of both asymmetrically nondelayed and delayed coupled, state-dependent, and subject to exogenous stochastic perturbations. The pth moment exponential synchronization of these drive-response stochastic memristor neural networks is guaranteed under some testable and computable sufficient conditions utilizing differential inclusion theory and Filippov regularization. Finally, the correctness and effectiveness of our theoretical results are demonstrated through a numerical example.

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Sivasankar, Sivajiganesan, Ramalingam Udhayakumar, and Venkatesan Muthukumaran. "A new conversation on the existence of Hilfer fractional stochastic Volterra–Fredholm integro-differential inclusions via almost sectorial operators." Nonlinear Analysis: Modelling and Control 28 (February 22, 2023): 1–20. http://dx.doi.org/10.15388/namc.2023.28.31450.

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The existence of Hilfer fractional stochastic Volterra–Fredholm integro-differential inclusions via almost sectorial operators is the topic of our paper. The researchers used fractional calculus, stochastic analysis theory, and Bohnenblust–Karlin’s fixed point theorem for multivalued maps to support their findings. To begin with, we must establish the existence of a mild solution. In addition, to show the principle, an application is presented.

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Ahmed, Hamdy. "Total controllability for noninstantaneous impulsive conformable fractional evolution system with nonlinear noise and nonlocal conditions." Filomat 37, no. 16 (2023): 5287–99. http://dx.doi.org/10.2298/fil2316287a.

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Noninstantaneous impulsive conformable fractional stochastic differential equation with nonlinear noise and nonlocal condition via Rosenblatt process and Poisson jump is studied in this paper. Sufficient conditions for controllability for the considered problem are established. The required results are obtained based on fractional calculus, stochastic analysis, semigroups and Sadovskii?s fixed point theorem. In the end paper, an example is provided to illustrate the applicability of the results.

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Kloeden, Peter E., and Arnulf Jentzen. "Pathwise convergent higher order numerical schemes for random ordinary differential equations." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463, no. 2087 (August 21, 2007): 2929–44. http://dx.doi.org/10.1098/rspa.2007.0055.

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Random ordinary differential equations (RODEs) are ordinary differential equations (ODEs) with a stochastic process in their vector field. They can be analysed pathwise using deterministic calculus, but since the driving stochastic process is usually only Hölder continuous in time, the vector field is not differentiable in the time variable, so traditional numerical schemes for ODEs do not achieve their usual order of convergence when applied to RODEs. Nevertheless deterministic calculus can still be used to derive higher order numerical schemes for RODEs via integral versions of implicit Taylor-like expansions. The theory is developed systematically here and applied to illustrative examples involving Brownian motion and fractional Brownian motion as the driving processes.

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O, Akeju Adeyemi, and Ayoola E. O. "A Malliavin Calculus Computation of the Greeks Theta and Vega of Asian Option and Best of Asset Option." International Journal of Latest Technology in Engineering, Management & Applied Science XII, no. X (2023): 41–54. http://dx.doi.org/10.51583/ijltemas.2023.121006.

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We determine the Theta and the Vega sensitivity of Asian Option (AO) and Best of Asset option (BAO) via the properties of Malliavin calculus. These sensitivities which are represented by the Greeks are obtained with skorohod integral and the integration by part technique for stochastic variation of the Malliavin calculus. The weight functions of the Greeks for Asian Option (AO) and the Best of Asset option (BAO) were derived and this was used to determine expressions for the Greeks.

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Johnson, Murugesan, and Velusamy Vijayakumar. "An Investigation on the Optimal Control for Hilfer Fractional Neutral Stochastic Integrodifferential Systems with Infinite Delay." Fractal and Fractional 6, no. 10 (October 11, 2022): 583. http://dx.doi.org/10.3390/fractalfract6100583.

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The main concern of this manuscript is to study the optimal control problem for Hilfer fractional neutral stochastic integrodifferential systems with infinite delay. Initially, we establish the existence of mild solutions for the Hilfer fractional stochastic integrodifferential system with infinite delay via applying fractional calculus, semigroups, stochastic analysis techniques, and the Banach fixed point theorem. In addition, we establish the existence of mild solutions of the Hilfer fractional neutral stochastic delay integrodifferential system. Further, we investigate the existence of optimal pairs for the Hilfer fractional neutral stochastic delay integrodifferential systems. We provide an illustration to clarify our results.

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MELCHER, TAI. "MALLIAVIN CALCULUS FOR LIE GROUP-VALUED WIENER FUNCTIONS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 12, no. 01 (March 2009): 67–89. http://dx.doi.org/10.1142/s0219025709003537.

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Let G be a Lie group equipped with a set of left invariant vector fields. These vector fields generate a function ξ on Wiener space into G via the stochastic version of Cartan's rolling map. It is shown here that, for any smooth function f with compact support, f(ξ) is Malliavin differentiable to all orders and these derivatives belong to Lp(μ) for all p > 1, where μ is Wiener measure.

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Sivasankar, Sivajiganesan, Ramalingam Udhayakumar, Velmurugan Subramanian, Ghada AlNemer, and Ahmed M. Elshenhab. "Existence of Hilfer Fractional Stochastic Differential Equations with Nonlocal Conditions and Delay via Almost Sectorial Operators." Mathematics 10, no. 22 (November 21, 2022): 4392. http://dx.doi.org/10.3390/math10224392.

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In this article, we examine the existence of Hilfer fractional (HF) stochastic differential systems with nonlocal conditions and delay via almost sectorial operators. The major methods depend on the semigroup of operators method and the Mo¨nch fixed-point technique via the measure of noncompactness, and the fundamental theory of fractional calculus. Finally, to clarify our key points, we provide an application.

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Tsionas, Mike G. "Robust Bayesian Inference in Stochastic Frontier Models." Journal of Risk and Financial Management 12, no. 4 (December 4, 2019): 183. http://dx.doi.org/10.3390/jrfm12040183.

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We use the concept of coarsened posteriors to provide robust Bayesian inference via coarsening in order to robustify posteriors arising from stochastic frontier models. These posteriors arise from tempered versions of the likelihood when at most a pre-specified amount of data is used, and are robust to changes in the model. Specifically, we examine robustness to changes in the distribution of the composed error in the stochastic frontier model (SFM). Moreover, coarsening is a form of regularization, reduces overfitting and makes inferences less sensitive to model choice. The new techniques are illustrated using artificial data as well as in a substantive application to large U.S. banks.

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Han, Yuecai, Zheng Li, and Chunyang Liu. "Option pricing under the fractional stochastic volatility model." ANZIAM Journal 63 (October 2, 2021): 123–42. http://dx.doi.org/10.21914/anziamj.v63.15204.

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We investigate the European call option pricing problem under the fractional stochastic volatility model. The stochastic volatility model is driven by both fractional Brownian motion and standard Brownian motion. We obtain an analytical solution of the European option price via the Itô’s formula for fractional Brownian motion, Malliavin calculus, derivative replication and the fundamental solution method. Some numerical simulations are given to illustrate the impact of parameters on option prices, and the results of comparison with other models are presented. doi:10.1017/S1446181121000225

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Mezerdi, Brahim, and Samia Yakhlef. "A stochastic maximum principle for mixed regular-singular control problems via Malliavin calculus." Afrika Matematika 27, no. 3-4 (May 24, 2015): 409–26. http://dx.doi.org/10.1007/s13370-015-0351-6.

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Alhojilan, Yazid, Hamdy M. Ahmed, and Wafaa B. Rabie. "Stochastic Solitons in Birefringent Fibers for Biswas–Arshed Equation with Multiplicative White Noise via Itô Calculus by Modified Extended Mapping Method." Symmetry 15, no. 1 (January 10, 2023): 207. http://dx.doi.org/10.3390/sym15010207.

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Stochastic partial differential equations have wide applications in various fields of science and engineering. This paper addresses the optical stochastic solitons and other exact stochastic solutions through birefringent fibers for the Biswas–Arshed equation with multiplicative white noise using the modified extended mapping method. This model contains many kinds of soliton solutions, which are always symmetric or anti-symmetric in space. Stochastic bright soliton solutions, stochastic dark soliton solutions, stochastic combo bright–dark soliton solutions, stochastic combo singular-bright soliton solutions, stochastic singular soliton solutions, stochastic periodic solutions, stochastic rational solutions, stochastic Weierstrass elliptic doubly periodic solutions, and stochastic Jacobi elliptic function solutions are extracted. The constraints on the parameters are considered to guarantee the existence of these stochastic solutions. Furthermore, some of the selected solutions are described graphically to demonstrate the physical nature of the obtained solutions.

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Liu, Yongchao, Huifu Xu, and Gui-Hua Lin. "Stability Analysis of Two-Stage Stochastic Mathematical Programs with Complementarity Constraints via NLP Regularization." SIAM Journal on Optimization 21, no. 3 (July 2011): 669–705. http://dx.doi.org/10.1137/100785685.

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Le, Huiling. "A stochastic calculus approach to the shape distribution induced by a complex normal model." Mathematical Proceedings of the Cambridge Philosophical Society 109, no. 1 (January 1991): 221–28. http://dx.doi.org/10.1017/s0305004100069681.

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AbstractAn approach via stochastic calculus was given by Kendall in [7] to the Mardia–Dryden shape distribution of three labelled independent -random points (j = 1, 2, 3). We give here the analogous approach for the general case discussed in [3] in which k labelled random points have a complex normal distribution.

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Scarpa, Luca, and Ulisse Stefanelli. "Doubly nonlinear stochastic evolution equations." Mathematical Models and Methods in Applied Sciences 30, no. 05 (May 2020): 991–1031. http://dx.doi.org/10.1142/s0218202520500219.

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Nonlinear diffusion problems featuring stochastic effects may be described by stochastic partial differential equations of the form [Formula: see text] We present an existence theory for such equations under general monotonicity assumptions on the nonlinearities. In particular, [Formula: see text], [Formula: see text], and [Formula: see text] are allowed to be multivalued, as required by the modelization of solid–liquid phase transitions. In this regard, the equation corresponds to a nonlinear-diffusion version of the classical two-phase Stefan problem with stochastic perturbation. The existence of martingale solutions is proved via regularization and passage-to-the-limit. The identification of the limit is obtained by a lower-semicontinuity argument based on a suitably generalized Itô’s formula. Under some more restrictive assumptions on the nonlinearities, existence and uniqueness of strong solutions follows. Besides the relation above, the theory covers equations with nonlocal terms as well as systems.

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Cruzeiro, Ana Bela. "Stochastic Approaches to Deterministic Fluid Dynamics: A Selective Review." Water 12, no. 3 (March 19, 2020): 864. http://dx.doi.org/10.3390/w12030864.

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We present a stochastic Lagrangian view of fluid dynamics. The velocity solving the deterministic Navier–Stokes equation is regarded as a mean time derivative taken over stochastic Lagrangian paths and the equations of motion are critical points of an associated stochastic action functional involving the kinetic energy computed over random paths. Thus the deterministic Navier–Stokes equation is obtained via a variational principle. The pressure can be regarded as a Lagrange multiplier. The approach is based on Itô’s stochastic calculus. Different related probabilistic methods to study the Navier–Stokes equation are discussed. We also consider Navier–Stokes equations perturbed by random terms, which we derive by means of a variational principle.

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Moumen, Abdelkader, Ammar Alsinai, Ramsha Shafqat, Nafisa A. Albasheir, Mohammed Alhagyan, Ameni Gargouri, and Mohammed M. A. Almazah. "Controllability of fractional stochastic evolution inclusion via Hilfer derivative of fixed point theory." AIMS Mathematics 8, no. 9 (2023): 19892–912. http://dx.doi.org/10.3934/math.20231014.

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<abstract><p>In this study, we use the Hilfer derivative to analyze the approximate controllability of fractional stochastic evolution inclusions (FSEIs) with nonlocal conditions. By assuming that the corresponding linear system is approximately controllable, we obtain a novel set of adequate requirements for the approximate controllability of nonlinear FSEIs in meticulous detail. The fixed-point theorem for multi-valued operators and fractional calculus are used to achieve the results. Finally, we use several instances to demonstrate our findings.</p></abstract>

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Assaad, Obayda, and Ciprian A. Tudor. "Wavelet analysis for the solution to the wave equation with fractional noise in time and white noise in space." ESAIM: Probability and Statistics 25 (2021): 220–57. http://dx.doi.org/10.1051/ps/2021009.

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Via Malliavin calculus, we analyze the limit behavior in distribution of the spatial wavelet variation for the solution to the stochastic linear wave equation with fractional Gaussian noise in time and white noise in space. We propose a wavelet-type estimator for the Hurst parameter of the this solution and we study its asymptotic properties.

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Alpay, Daniel, and Palle Jorgensen. "Finitely additive functions in measure theory and applications." Opuscula Mathematica 44, no. 3 (2024): 323–39. http://dx.doi.org/10.7494/opmath.2024.44.3.323.

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In this paper, we consider, and make precise, a certain extension of the Radon-Nikodym derivative operator, to functions which are additive, but not necessarily sigma-additive, on a subset of a given sigma-algebra. We give applications to probability theory; in particular, to the study of \(\mu\)-Brownian motion, to stochastic calculus via generalized It�-integrals, and their adjoints (in the form of generalized stochastic derivatives), to systems of transition probability operators indexed by families of measures \(\mu\), and to adjoints of composition operators.

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Johnson, Murugesan, and Velusamy Vijayakumar. "An Analysis on the Optimal Control for Fractional Stochastic Delay Integrodifferential Systems of Order 1 < γ < 2." Fractal and Fractional 7, no. 4 (March 25, 2023): 284. http://dx.doi.org/10.3390/fractalfract7040284.

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The purpose of this paper is to investigate the optimal control for fractional stochastic integrodifferential systems of order 1 < γ < 2. To ensure the existence and uniqueness of mild solutions, we first gather a novel list of requirements. Further, the existence of optimal control for the stated issue is given by applying Balder’s theorem. Additionally, we extend our existence outcomes with infinite delay. The outcomes are obtained via fractional calculus, Hölder’s inequality, the cosine family, stochastic analysis techniques, and the fixed point approach. The theory is shown by an illustration, as well.

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Khalil, M., C. A. Tudor, and M. Zili. "Spatial variation for the solution to the stochastic linear wave equation driven by additive space-time white noise." Stochastics and Dynamics 18, no. 05 (September 12, 2018): 1850036. http://dx.doi.org/10.1142/s0219493718500363.

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We study the asymptotic behavior of the spatial quadratic variation for the solution to the stochastic wave equation driven by additive space-time white noise. We prove that the sequence of its renormalized quadratic variations satisfies a central limit theorem (CLT for short). We obtain the rate of convergence for this CLT via the Stein–Malliavin calculus and we also discuss some consequences.

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Liu, Junfeng, and Ciprian A. Tudor. "Central limit theorem for the solution to the heat equation with moving time." Infinite Dimensional Analysis, Quantum Probability and Related Topics 19, no. 01 (March 2016): 1650005. http://dx.doi.org/10.1142/s0219025716500053.

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We consider the solution to the stochastic heat equation driven by the time-space white noise and study the asymptotic behavior of its spatial quadratic variations with “moving time”, meaning that the time variable is not fixed and its values are allowed to be very big or very small. We investigate the limit distribution of these variations via Malliavin calculus.

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Lei, Youming, and Yanyan Wang. "Period-Doubling Bifurcation of Stochastic Fractional-Order Duffing System via Chebyshev Polynomial Approximation." Shock and Vibration 2017 (2017): 1–12. http://dx.doi.org/10.1155/2017/4162363.

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Fractional-order calculus is more competent than integer-order one when modeling systems with properties of nonlocality and memory effect. And many real world problems related to uncertainties can be modeled with stochastic fractional-order systems with random parameters. Therefore, it is necessary to analyze the dynamical behaviors in those systems concerning both memory and uncertainties. The period-doubling bifurcation of stochastic fractional-order Duffing (SFOD for short) system with a bounded random parameter subject to harmonic excitation is studied in this paper. Firstly, Chebyshev polynomial approximation in conjunction with the predictor-corrector approach is used to numerically solve the SFOD system that can be reduced to the equivalent deterministic system. Then, the global and local analysis of period-doubling bifurcation are presented, respectively. It is shown that both the fractional-order and the intensity of the random parameter can be taken as bifurcation parameters, which are peculiar to the stochastic fractional-order system, comparing with the stochastic integer-order system or the deterministic fractional-order system. Moreover, the Chebyshev polynomial approximation is proved to be an effective approach for studying the period-doubling bifurcation of the SFOD system.

Journal articles on the topic 'Stochastic calculus via regularization'

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