Stochastic calculus5/31/2023 ![]() Moreover, i = − 1, σ is noise strength which is constant in the space and W t is standard Wiener process and it has been defined as in the following form ,, : W t = ∫ 0 t A η d W ( η ), η < t ,where η is stochastic variable. In this article, we introduce stochastic dispersive Schrödinger–Hirota equation with parabolic law nonlinearity with multiplicative white noise via Ito calculus in the following form: i ∂ u ∂ t + α ∂ 2 u ∂ x 2 + β 1 | u | 2 + β 2 | u | 4 u + i λ ∂ 3 u ∂ x 3 + β 3 | u | 2 + β 4 | u | 4 ∂ u ∂ x + σ ∂ W ∂ t u = 0 ,in which u = u x, t, λ and α are solution function, third-order dispersion (3OD), group velocity dispersion (GVD) terms coefficients and β 1, β 2, β 3, β 4 are the coefficients of the nonlinear terms, respectively. Similarly, in ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, , researchers obtained optical soliton solutions of different forms of models under different conditions. ![]() In , Aydin, suggested the stochastic nonlinear Schrödinger equation and examined noise effect on optical soliton solutions of this equation. presented soliton solutions of the Biswas-Arshed equation with multiplicative noise using the Itô calculus method. used the exp-function method to obtain optical solitons in birefringent fibers with Kerr nonlinearity. proposed exact solutions to some stochastic equation systems for ion sound and Langmuir waves. examined the stochastic nonlinear Shrödinger equation using two different approach. modeled optical soliton solutions of the higher order nonlinear Schrödinger equation with variable coefficients. Some of these studies are as follows: In , Belic et al. ![]() Researchers have developed stochastic models of nonlinear evolution equations in many studies. This causes the use of stochastic terms in order to obtain more effective results in the mathematical models established for the examination of adverse environmental effects. Examples of these natural situations are sound effects, magnetic and electric field effects, material quality and ambient temperature. However, some situations in nature cause stochastic losses in data transmission. These methods, with dispersion and optic solitons in the focus, are the modified Kudryashov’s algorithm , Laplace–Adomian decomposition , improved Adomian decomposition ,, the ansatz method , Riccati equation and F-expansion , exp-function method , G ′ / G expansion scheme, the extended auxiliary equation approach, sine–Gordon ,, stationary solitons , Lie group analysis , Hamiltonian perturbations , traveling wave hypothesis ,, semi-inverse variational principle , unified Riccati equation, new mapping scheme, addendum to the Kudryashov’s method , Riccati–Bernoulli approach , the extended Kudryashov method , the generalized Jacobi elliptic expansion and many others. Depending on the importance of optical solitons, many optical models and solution techniques related to these models have been developed and used. Today, optical solitons are used to transmit huge amounts of data over great distances at the speed of light. Soliton waves form the basis of optical communication because they are stable and can travel great distances. Including full mathematical statements and rigorous proofs, this book is completely self-contained and suitable for lecture courses as well as self-study.One of the most important parts of communication is the concept of optical soliton. Stochastic Calculus will be particularly useful to advanced undergraduate and graduate students wishing to acquire a solid understanding of the subject through the theory and exercises. The final chapter provides detailed solutions to all exercises, in some cases presenting various solution techniques together with a discussion of advantages and drawbacks of the methods used. The core of the book covers stochastic calculus, including stochastic differential equations, the relationship to partial differential equations, numerical methods and simulation, as well as applications of stochastic processes to finance. It is the only textbook on the subject to include more than two hundred exercises with complete solutions.Īfter explaining the basic elements of probability, the author introduces more advanced topics such as Brownian motion, martingales and Markov processes. This book provides a comprehensive introduction to the theory of stochastic calculus and some of its applications.
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