Статті в журналах: "Skorohod equations"

1

Buckdahn, Rainer. "Linear skorohod stochastic differential equations." Probability Theory and Related Fields 90, no. 2 (June 1991): 223–40. http://dx.doi.org/10.1007/bf01192163.

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2

Buckdahn, Rainer, and David Nualart. "Skorohod stochastic differential equations with boundary conditions." Stochastics and Stochastic Reports 45, no. 3-4 (December 1993): 211–35. http://dx.doi.org/10.1080/17442509308833862.

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3

Nualart, David, and Michèle Thieullen. "Skorohod stochastic differential equations on random intervals." Stochastics and Stochastic Reports 49, no. 3-4 (August 1994): 149–67. http://dx.doi.org/10.1080/17442509408833917.

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4

Buckdahn, Rainer. "Skorohod stochastic differential equations of diffusion type." Probability Theory and Related Fields 93, no. 3 (September 1992): 297–323. http://dx.doi.org/10.1007/bf01193054.

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5

El-Borai, Mahmoud M., Khairia El-Said El-Nadi, Osama L. Mostafa, and Hamdy M. Ahmed. "Volterra equations with fractional stochastic integrals." Mathematical Problems in Engineering 2004, no. 5 (2004): 453–68. http://dx.doi.org/10.1155/s1024123x04312020.

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Анотація:

Some fractional stochastic systems of integral equations are studied. The fractional stochastic Skorohod integrals are also studied. The existence and uniquness of the considered stochastic fractional systems are established. An application of the fractional Black-Scholes is considered.

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6

Tudor, Ciprian A. "Itô-Skorohod stochastic equations and applications to finance." Journal of Applied Mathematics and Stochastic Analysis 2004, no. 4 (January 1, 2004): 359–69. http://dx.doi.org/10.1155/s1048953304311044.

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We prove an existence and uniqueness theorem for a class of Itô-Skorohod stochastic equations. As an application, we introduce a Black-Scholes market model where the price of the risky asset follows a nonadapted equation.

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7

Bishwal, Jaya P. N. "Maximum likelihood estimation in Skorohod stochastic differential equations." Proceedings of the American Mathematical Society 138, no. 04 (April 1, 2010): 1471. http://dx.doi.org/10.1090/s0002-9939-09-10113-2.

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8

Buckdahn, Rainer. "Anticipative Girsanov transformations and Skorohod stochastic differential equations." Memoirs of the American Mathematical Society 111, no. 533 (1994): 0. http://dx.doi.org/10.1090/memo/0533.

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9

DONEY, R., and T. ZHANG. "Perturbed Skorohod equations and perturbed reflected diffusion processes." Annales de l'Institut Henri Poincare (B) Probability and Statistics 41, no. 1 (January 2005): 107–21. http://dx.doi.org/10.1016/j.anihpb.2004.03.005.

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10

Buckdahn, R., P. Malliavin, and D. Nualart. "Multidimensional linear stochastic differential equations in the skorohod sense." Stochastics and Stochastic Reports 62, no. 1-2 (November 1997): 117–45. http://dx.doi.org/10.1080/17442509708834130.

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11

Zhang, Xicheng. "Skorohod problem and multivalued stochastic evolution equations in Banach spaces." Bulletin des Sciences Mathématiques 131, no. 2 (March 2007): 175–217. http://dx.doi.org/10.1016/j.bulsci.2006.05.009.

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12

Chen, Z. Q., and Z. Zhao. "Switched diffusion processes and systems of elliptic equations: a Dirichlet space approach." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 124, no. 4 (1994): 673–701. http://dx.doi.org/10.1017/s0308210500028596.

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The switched diffusion process associated with a weakly coupled system of elliptic equations is studied via a Dirichlet space approach and is applied to prove the existence theorem of the Cauchy initial problem for the system. A representation theorem for the solution of the Dirichlet boundary value problem and a generalised Skorohod decomposition for the reflecting switched diffusion process are obtained.

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13

Egorov, A. D. "Approximate formulas for the evaluation of the mathematical expectation of functionals from the solution to the linear Skorohod equation." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 57, no. 2 (July 16, 2021): 198–205. http://dx.doi.org/10.29235/1561-2430-2021-57-2-198-205.

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This paper is devoted to the construction of approximate formulas for calculating the mathematical expectation of nonlinear functionals from the solution to the linear Skorohod stochastic differential equation with a random initial condition. To calculate the mathematical expectations of nonlinear functionals from random processes, functional analogs of quadrature formulas have been developed, based on the requirement of their accuracy for functional polynomials of a given degree. Most often, formulas are constructed that are exact for polynomials of the third degree [1–9], which are used to obtain an initial approximation and in combination with approximations of the original random process. In the latter case, they are usually also exact for polynomials of a given degree and are called compound formulas. However, in the case of processes specified in the form of compound functions from other random processes the constructed functional quadrature formulas, as a rule, have great computational complexity and cannot be used for computer implementation. This is exactly what happens in the case of functionals from the solutions of stochastic equations. In [1, 2], the approaches to solving this problem were considered for some types of Ito equations in martingales. The solution of the problem is simplified in the cases when the solution of the stochastic equation is found in explicit form: the corresponding approximations were obtained in the cases of the linear equations of Ito, Ito – Levy and Skorohod in [3–11]. In [7, 8, 11], functional quadrature formulas were constructed that are exact for the approximations of the expansions of the solutions in terms of orthonormal functional polynomials and in terms of multiple stochastic integrals. This work is devoted to the approximate calculation of the mathematical expectations of nonlinear functionals from the solution of the linear Skorokhod equation with a leading Wiener process and a random initial condition. A new approach to the construction of quadrature formulas, exact for functional polynomials of the third degree, based on the use of multiple Stieltjes integrals over functions of bounded variation in the sense of Hardy – Krause, is proposed. A composite approximate formula is also constructed, which is exact for second-order functional polynomials, converging to the exact expectation value, based on a combination of the obtained quadrature formula and an approximation of the leading Wiener process. The test examples illustrating the application of the obtained formulas are considered.

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14

Ma, Jin, and Yusun Wang. "On Variant Reflected Backward SDEs, with Applications." Journal of Applied Mathematics and Stochastic Analysis 2009 (June 18, 2009): 1–26. http://dx.doi.org/10.1155/2009/854768.

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We study a new type of reflected backward stochastic differential equations (RBSDEs), where the reflecting process enters the drift in a nonlinear manner. This type of the reflected BSDEs is based on a variance of the Skorohod problem studied recently by Bank and El Karoui (2004), and is hence named the “Variant Reflected BSDEs” (VRBSDE) in this paper. The special nature of the Variant Skorohod problem leads to a hidden forward-backward feature of the BSDE, and as a consequence this type of BSDE cannot be treated in a usual way. We shall prove that in a small-time duration most of the well-posedness, comparison, and stability results are still valid, although some extra conditions on the boundary process are needed. We will also provide some possible applications where the VRBSDE can be potentially useful. These applications show that the VRBSDE could become a novel tool for some problems in finance and optimal stopping problems where no existing methods can be easily applicable.

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15

Castaing, Charles, Christiane Godet-Thobie, Manuel D. P. Monteiro Marques, and Anna Salvadori. "Evolution Problems with m-Accretive Operators and Perturbations." Mathematics 10, no. 3 (January 20, 2022): 317. http://dx.doi.org/10.3390/math10030317.

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This paper is devoted to the study of perturbation evolution problems involving time-dependent m-accretive operators. We present for a specific class of m-accretive operators with convex weakly compact-valued perturbation, some results about the existence of absolutely continuous solutions and BRVC solutions. We finish by giving several applications to various domains such as relaxation results, second-order evolution inclusions, fractional-order equations coupled with m-accretive operators and Skorohod differential inclusions.

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16

Graham, Carl. "McKean-Vlasov Ito-Skorohod equations, and nonlinear diffusions with discrete jump sets." Stochastic Processes and their Applications 40, no. 1 (February 1992): 69–82. http://dx.doi.org/10.1016/0304-4149(92)90138-g.

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17

Yang, Zhaoqiang. "Optimal Exercise Boundary of American Fractional Lookback Option in a Mixed Jump-Diffusion Fractional Brownian Motion Environment." Mathematical Problems in Engineering 2017 (2017): 1–17. http://dx.doi.org/10.1155/2017/5904125.

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A new framework for pricing the American fractional lookback option is developed in the case where the stock price follows a mixed jump-diffusion fraction Brownian motion. By using Itô formula and Wick-Itô-Skorohod integral a new market pricing model is built. The fundamental solutions of stochastic parabolic partial differential equations are estimated under the condition of Merton assumptions. The explicit integral representation of early exercise premium and the critical exercise price are also given. Numerical simulation illustrates some notable features of American fractional lookback options.

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18

Yang, Zhaoqiang. "Efficient valuation and exercise boundary of American fractional lookback option in a mixed jump-diffusion model." International Journal of Financial Engineering 04, no. 02n03 (June 2017): 1750033. http://dx.doi.org/10.1142/s2424786317500335.

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This study presents an efficient method for pricing the American fractional lookback option in the case where the stock price follows a mixed jump diffusion fraction Brownian motion. By using It ô formula and Wick–It ô–Skorohod integral, a new market pricing model is built. The fundamental solutions of stochastic parabolic partial differential equations are estimated under the condition of Merton assumptions. The explicit integral representation of early exercise premium and the critical exercise price are also given. Numerical simulation illustrates some notable features of American fractional lookback options.

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19

Yang, Zhaoqiang. "A NEW STOPPING PROBLEM AND THE CRITICAL EXERCISE PRICE FOR AMERICAN FRACTIONAL LOOKBACK OPTION IN A SPECIAL MIXED JUMP-DIFFUSION MODEL." Probability in the Engineering and Informational Sciences 34, no. 1 (September 21, 2018): 27–52. http://dx.doi.org/10.1017/s0269964818000311.

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Анотація:

A new stopping problem and the critical exercise price of American fractional lookback option are developed in the case where the stock price follows a special mixed jump diffusion fractional Brownian motion. By using Itô formula and Wick-Itô-Skorohod integral a new market pricing model is built, and the fundamental solutions of stochastic parabolic partial differential equations are deduced under the condition of Merton assumptions. With an optimal stopping problem and the exercise boundary, the explicit integral representation of early exercise premium and the critical exercise price are also derived. Numerical simulation illustrates the asymptotic behavior of this critical boundary.

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20

Costantini, C. "The Skorohod oblique reflection problem in domains with corners and application to stochastic differential equations." Probability Theory and Related Fields 91, no. 1 (March 1992): 43–70. http://dx.doi.org/10.1007/bf01194489.

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21

KACHANOVSKY, N. A. "AN EXTENDED STOCHASTIC INTEGRAL AND A WICK CALCULUS ON PARAMETRIZED KONDRATIEV-TYPE SPACES OF MEIXNER WHITE NOISE." Infinite Dimensional Analysis, Quantum Probability and Related Topics 11, no. 04 (December 2008): 541–64. http://dx.doi.org/10.1142/s0219025708003270.

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Using a general approach that covers the cases of Gaussian, Poissonian, Gamma, Pascal and Meixner measures, we consider an extended stochastic integral and construct elements of a Wick calculus on parametrized Kondratiev-type spaces of generalized functions; consider the interconnection between the extended stochastic integration and the Wick calculus; and give an example of a stochastic equation with a Wick-type nonlinearity. The main results consist of studying the properties of the extended (Skorohod) stichastic integral subject to the particular spaces under consideration; and of studying the properties of a Wick product and Wick versions of holomorphic functions on the parametrized Kondratiev-type spaces. These results are necessary, in particular, in order to describe properties of solutions of normally ordered white noise equations in the "Meixner analysis".

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22

Dyriv, M. M., and N. A. Kachanovsky. "On operators of stochastic differentiation on spaces of regular test and generalized functions of Lévy white noise analysis." Carpathian Mathematical Publications 6, no. 2 (December 25, 2014): 212–29. http://dx.doi.org/10.15330/cmp.6.2.212-229.

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The operators of stochastic differentiation, which are closely related with the extended Skorohod stochastic integral and with the Hida stochastic derivative, play an important role in the classical (Gaussian) white noise analysis. In particular, these operators can be used in order to study properties of the extended stochastic integral and of solutions of stochastic equations with Wick-type nonlinearities. In this paper we introduce and study bounded and unbounded operators of stochastic differentiation in the Levy white noise analysis. More exactly, we consider these operators on spaces from parametrized regular rigging of the space of square integrable with respect to the measure of a Levy white noise functions, using the Lytvynov's generalization of the chaotic representation property. This gives a possibility to extend to the Levy white noise analysis and to deepen the corresponding results of the classical white noise analysis.

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23

Moon, Jun. "State and Control Path-Dependent Stochastic Zero-Sum Differential Games: Viscosity Solutions of Path-Dependent Hamilton–Jacobi–Isaacs Equations." Mathematics 10, no. 10 (May 22, 2022): 1766. http://dx.doi.org/10.3390/math10101766.

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In this paper, we consider the two-player state and control path-dependent stochastic zero-sum differential game. In our problem setup, the state process, which is controlled by the players, is dependent on (current and past) paths of state and control processes of the players. Furthermore, the running cost of the objective functional depends on both state and control paths of the players. We use the notion of non-anticipative strategies to define lower and upper value functionals of the game, where unlike the existing literature, these value functions are dependent on the initial states and control paths of the players. In the first main result of this paper, we prove that the (lower and upper) value functionals satisfy the dynamic programming principle (DPP), for which unlike the existing literature, the Skorohod metric is necessary to maintain the separability of càdlàg (state and control) spaces. We introduce the lower and upper Hamilton–Jacobi–Isaacs (HJI) equations from the DPP, which correspond to the state and control path-dependent nonlinear second-order partial differential equations. In the second main result of this paper, we show that by using the functional Itô calculus, the lower and upper value functionals are viscosity solutions of (lower and upper) state and control path-dependent HJI equations, where the notion of viscosity solutions is defined on a compact κ-Hölder space to use several important estimates and to guarantee the existence of minimum and maximum points between the (lower and upper) value functionals and the test functions. Based on these two main results, we also show that the Isaacs condition and the uniqueness of viscosity solutions imply the existence of the game value. Finally, we prove the uniqueness of classical solutions for the (state path-dependent) HJI equations in the state path-dependent case, where its proof requires establishing an equivalent classical solution structure as well as an appropriate contradiction argument.

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24

Kachanovsky, N. A. "Operators of stochastic differentiation on spaces of nonregular generalized functions of Levy white noise analysis." Carpathian Mathematical Publications 8, no. 1 (June 30, 2016): 83–106. http://dx.doi.org/10.15330/cmp.8.1.83-106.

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Анотація:

The operators of stochastic differentiation, which are closely related with the extended Skorohod stochastic integral and with the Hida stochastic derivative, play an important role in the classical (Gaussian) white noise analysis. In particular, these operators can be used in order to study some properties of the extended stochastic integral and of solutions of stochastic equations with Wick-type nonlinearities. During recent years the operators of stochastic differentiation were introduced and studied, in particular, in the framework of the Meixner white noise analysis, in the same way as on spaces of regular test and generalized functions and on spaces of nonregular test functions of the Levy white noise analysis. In the present paper we make the next natural step: introduce and study operators of stochastic differentiation on spaces of nonregular generalized functions of the Levy white noise analysis (i.e., on spaces of generalized functions that belong to the so-called nonregular rigging of the space of square integrable with respect to the measure of a Levy white noise functions). In so doing, we use Lytvynov's generalization of the chaotic representation property. The researches of the present paper can be considered as a contribution in a further development of the Levy white noise analysis.

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25

Liu, Kefan, Jingyao Chen, Jichao Zhang, and Yueting Yang. "Application of fuzzy Malliavin calculus in hedging fixed strike lookback option." AIMS Mathematics 8, no. 4 (2023): 9187–211. http://dx.doi.org/10.3934/math.2023461.

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<abstract><p>In this paper, we develop a Malliavin calculus approach for hedging a fixed strike lookback option in fuzzy space. Due to the uncertainty in financial markets, it is not accurate to describe the problems of option pricing and hedging in terms of randomness alone. We consider a fuzzy pricing model by introducing a fuzzy stochastic differential equation with Skorohod sense. In this way, our model simultaneously involves randomness and fuzziness. A well-known hedging strategy for vanilla options is so-called $ \Delta $-hedging, which is usually derived from the Itô formula and some properties of partial differentiable equations. However, when dealing with some complex path-dependent options (such as lookback options), the major challenge is that the payoff function of these options may not be smooth, resulting in the estimates are computationally expensive. With the help of the Malliavin derivative and the Clark-Ocone formula, the difficulty will be readily solved, and it is also possible to apply this hedging strategy to fuzzy space. To obtain the explicit expression of the fuzzy hedging portfolio for lookback options, we adopt the Esscher transform and reflection principle techniques, which are beneficial to the calculation of the conditional expectation of fuzzy random variables and the payoff function with extremum, respectively. Some numerical examples are performed to analyze the sensitivity of the fuzzy hedging portfolio concerning model parameters and give the permissible range of the expected hedging portfolio of lookback options with uncertainty by a financial investor's subjective judgment.</p></abstract>

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26

Wagner, Wolfgang. "Skorohod, A. V.: Stochastic Equations for Complex Systems (Mathematics and its applications. East European Series). D. Reidel Publiahing Company, Dordrecht 1988,196 S., US $69.00; UKE 43.50; Dfl. 140.00." Biometrical Journal 31, no. 2 (1989): 212. http://dx.doi.org/10.1002/bimj.4710310210.

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27

Mandrekar, V., and U. V. Naik-Nimbalkar. "Weak uniqueness of martingale solutions to stochastic partial differential equations in Hilbert spaces." Theory of Stochastic Processes 25(41), no. 1 (December 21, 2020): 78–89. http://dx.doi.org/10.37863/tsp-5986263728-06.

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We prove the uniqueness of martingale solutions for stochastic partial differential equations generalizing the work in Mandrekar and Skorokhod (1998). The main idea used is to reduce this problem to the case in Mandrekar and Skorokhod using the techniques introduced in Filipović et al. (2010).

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28

Döring, Leif, Lukas Gonon, David J. Prömel, and Oleg Reichmann. "On Skorokhod embeddings and Poisson equations." Annals of Applied Probability 29, no. 4 (August 2019): 2302–37. http://dx.doi.org/10.1214/18-aap1454.

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29

Sun, Xichao, and Ming Li. "Stochastic Fractional Heat Equations Driven by Fractional Noises." Mathematical Problems in Engineering 2015 (2015): 1–16. http://dx.doi.org/10.1155/2015/421705.

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This paper is concerned with the following stochastically fractional heat equation on(t,x)∈[0,T]×Rddriven by fractional noise:∂u(t,x)/∂t=Dδαu(t,x)+WH(t,x)⋄u(t,x), where the Hurst parameterH=(h0,h1,…,hd)and⋄denotes the Skorokhod integral. A unique solution of that equation in an appropriate Hilbert space is constructed. Moreover, the Lyapunov exponent of the solution is estimated, and the Hölder continuity of the solution on both space and time parameters is discussed. On the other hand, the absolute continuity of the solution is also obtained.

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30

Levajkovic, Tijana, and Dora Selesi. "Chaos expansion methods for stochastic differential equations involving the Malliavin derivative, Part I." Publications de l'Institut Math?matique (Belgrade) 90, no. 104 (2011): 65–84. http://dx.doi.org/10.2298/pim1104065l.

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We consider Gaussian, Poissonian, fractional Gaussian and fractional Poissonian white noise spaces, all represented through the corresponding orthogonal basis of the Hilbert space of random variables with finite second moments, given by the Hermite and the Charlier polynomials. There exist unitary mappings between the Gaussian and Poissonian white noise spaces. We investigate the relationship of the Malliavin derivative, the Skorokhod integral, the Ornstein-Uhlenbeck operator and their fractional counterparts on a general white noise space.

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31

Лотоцкий, С. В., S. V. Lototskii, Борис Л. Розовский, and Boris L. Rozovskii. "A unified approach to stochastic evolution equations using the Skorokhod integral." Teoriya Veroyatnostei i ee Primeneniya 54, no. 2 (2009): 288–303. http://dx.doi.org/10.4213/tvp2703.

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32

Lototsky, S. V., and B. L. Rozovskii. "A Unified Approach to Stochastic Evolution Equations Using the Skorokhod Integral." Theory of Probability & Its Applications 54, no. 2 (January 2010): 189–202. http://dx.doi.org/10.1137/s0040585x97984152.

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33

Lin, Yiqing, and Abdoulaye Soumana Hima. "Reflected stochastic differential equations driven by G-Brownian motion in non-convex domains." Stochastics and Dynamics 19, no. 03 (May 30, 2019): 1950025. http://dx.doi.org/10.1142/s0219493719500254.

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Анотація:

In this paper, we first review the penalization method for solving deterministic Skorokhod problems in non-convex domains and establish estimates for problems with [Formula: see text]-Hölder continuous functions. With the help of these results obtained previously for deterministic problems, we pathwisely define the reflected [Formula: see text]-Brownian motion and prove its existence and uniqueness in a Banach space. Finally, multi-dimensional reflected stochastic differential equations driven by [Formula: see text]-Brownian motion are investigated via a fixed-point argument.

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34

de Raynal, Paul-Éric Chaudru, Gilles Pagès, and Clément Rey. "Numerical methods for Stochastic differential equations: two examples." ESAIM: Proceedings and Surveys 64 (2018): 65–77. http://dx.doi.org/10.1051/proc/201864065.

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Анотація:

The goal of this paper is to present a series of recent contributions arising in numerical probability. First we present a contribution to a recently introduced problem: stochastic differential equations with constraints in law, investigated through various theoretical and numerical viewpoints. Such a problem may appear as an extension of the famous Skorokhod problem. Then a generic method to approximate in a weak way the invariant distribution of an ergodic Feller process by a Langevin Monte Carlo simulation. It is an extension of a method originally developed for diffusions and based on the weighted empirical measure of an Euler scheme with decreasing step. Finally, we mention without details a recent development of a multilevel Langevin Monte Carlo simulation method for this type of problem.

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35

Mohammed, Mogtaba. "Homogenization of nonlinear hyperbolic stochastic equation via Tartar’s method." Journal of Hyperbolic Differential Equations 14, no. 02 (May 16, 2017): 323–40. http://dx.doi.org/10.1142/s0219891617500096.

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In this paper, We establish new homogenization results for stochastic nonlinear hyperbolic equations with periodically oscillating coefficients. We use a delicate blending of Tartar’s method of oscillating test functions and deep probabilistic compactness results due to Prokhorov and Skorokhod. We prove that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized stochastic hyperbolic problem with constant coefficients. We also prove the convergence of the associated energies.

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36

Nikitin, A., and O. Baliasnikova. "Optimization of functionals under uncertainties for Ito-Skorokhod stochastic differential equations in Hilbert spaces." Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics, no. 3 (2018): 65–70. http://dx.doi.org/10.17721/1812-5409.2018/3.9.

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Анотація:

In the article for the stochastic differential equations of Ito-Skorokhod, problems of optimization of functionals under conditions of uncertainty in Hilbert spaces are investigated. Purpose of the article is to investigate some properties of stochastic differential equations in Hilbert spaces. These objects arise in diverse areas of applied mathematics as models for various natural phenomena, in particular, the evolution of complex systems with infinitely many degrees of freedom. For instance, one may think of the liquid fuel motion in the tank of a spacecraft. Spacecraft constructors should take into account this motion, for it influences heavily the path of a spacecraft. Also, optimization of the motion is an issue of principal importance. It is not trivial to carry over the results concerning stochastic differential equations in finite-dimensional spaces to the infinite dimensional case. We give some statements, in which the existence, uniqueness is proved and the explicit form μ-optimal controls for such equations is constructed, in particular, μ-optimal control is found as a linear inverse relationship.

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37

Manna, Utpal, and Debopriya Mukherjee. "Optimal relaxed control of stochastic hereditary evolution equations with Lévy noise." ESAIM: Control, Optimisation and Calculus of Variations 25 (2019): 61. http://dx.doi.org/10.1051/cocv/2018066.

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Existence theory of optimal relaxed control problem for a class of stochastic hereditary evolution equations driven by Lévy noise has been studied. We formulate the problem in the martingale sense of Stroock and Varadhan to establish existence of optimal controls. The construction of the solution is based on the classical Faedo–Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod theorem for nonmetric spaces, and certain compactness properties of the class of Young measures on Suslin metrizable control sets. As application of the abstract theory, Oldroyd and Jeffreys fluids have been studied and existence of optimal relaxed control is established. Existence and uniqueness of a strong solution and uniqueness in law for the two-dimensional Oldroyd and Jeffreys fluids are also shown.

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38

Geiss, Christel, Céline Labart, and Antti Luoto. "Mean square rate of convergence for random walk approximation of forward-backward SDEs." Advances in Applied Probability 52, no. 3 (September 2020): 735–71. http://dx.doi.org/10.1017/apr.2020.17.

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AbstractLet (Y, Z) denote the solution to a forward-backward stochastic differential equation (FBSDE). If one constructs a random walk $B^n$ from the underlying Brownian motion B by Skorokhod embedding, one can show $L_2$-convergence of the corresponding solutions $(Y^n,Z^n)$ to $(Y, Z).$ We estimate the rate of convergence based on smoothness properties, especially for a terminal condition function in $C^{2,\alpha}$. The proof relies on an approximative representation of $Z^n$ and uses the concept of discretized Malliavin calculus. Moreover, we use growth and smoothness properties of the partial differential equation associated to the FBSDE, as well as of the finite difference equations associated to the approximating stochastic equations. We derive these properties by probabilistic methods.

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39

ANKIRCHNER, STEFAN, GREGOR HEYNE, and PETER IMKELLER. "A BSDE APPROACH TO THE SKOROKHOD EMBEDDING PROBLEM FOR THE BROWNIAN MOTION WITH DRIFT." Stochastics and Dynamics 08, no. 01 (March 2008): 35–46. http://dx.doi.org/10.1142/s0219493708002160.

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We solve Skorokhod's embedding problem for Brownian motion with linear drift (Wt + κ t)t≥0 by means of techniques of stochastic control theory. The search for a stopping time T such that the law of WT + κ T coincides with a prescribed law μ possessing the first moment is based on solutions of backward stochastic differential equations of quadratic type. This new approach generalizes an approach by Bass [3] of the classical version of Skorokhod's embedding problem using martingale representation techniques.

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40

Ninouh, Abdelhakim, Boulakhras Gherbal, and Nassima Berrouis. "Existence of optimal controls for systems of controlled forward-backward doubly SDEs." Random Operators and Stochastic Equations 28, no. 2 (June 1, 2020): 93–112. http://dx.doi.org/10.1515/rose-2020-2031.

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AbstractWe wish to study a class of optimal controls for problems governed by forward-backward doubly stochastic differential equations (FBDSDEs). Firstly, we prove existence of optimal relaxed controls, which are measure-valued processes for nonlinear FBDSDEs, by using some tightness properties and weak convergence techniques on the space of Skorokhod {\mathbb{D}} equipped with the S-topology of Jakubowski. Moreover, when the Roxin-type convexity condition is fulfilled, we prove that the optimal relaxed control is in fact strict. Secondly, we prove the existence of a strong optimal controls for a linear forward-backward doubly SDEs. Furthermore, we establish necessary as well as sufficient optimality conditions for a control problem of this kind of systems. This is the first theorem of existence of optimal controls that covers the forward-backward doubly systems.

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41

Pilipenko, A. Yu. "On the Skorokhod mapping for equations with reflection and possible jump-like exit from a boundary." Ukrainian Mathematical Journal 63, no. 9 (February 2012): 1415–32. http://dx.doi.org/10.1007/s11253-012-0588-2.

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42

Cacciafesta, Federico, and Anne-Sophie de Suzzoni. "Invariance of Gibbs measures under the flows of Hamiltonian equations on the real line." Communications in Contemporary Mathematics 22, no. 02 (February 15, 2019): 1950012. http://dx.doi.org/10.1142/s0219199719500123.

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We prove that the Gibbs measures [Formula: see text] for a class of Hamiltonian equations written as [Formula: see text] on the real line are invariant under the flow of [Formula: see text] in the sense that there exist random variables [Formula: see text] whose laws are [Formula: see text] (thus independent from [Formula: see text]) and such that [Formula: see text] is a solution to [Formula: see text]. Besides, for all [Formula: see text], [Formula: see text] is almost surely not in [Formula: see text] which provides as a direct consequence the existence of global weak solutions for initial data not in [Formula: see text]. The proof uses Prokhorov’s theorem, Skorohod’s theorem, as in the strategy in [N. Burq, L. Thomann and N. Tzvetkov, Remarks on the Gibbs measures for nonlinear dispersive equations, preprint (2014); arXiv:1412.7499v1 [math.AP]] and Feynman–Kac’s integrals.

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43

Yurchenko, I. V., and V. K. Yasynskyy. "Existence of Lyapunov–Krasovskii Functionals for Stochastic Functional Differential Ito–Skorokhod Equations Under the Condition of Solutions’ Stability on Probability with Finite Aftereffect." Cybernetics and Systems Analysis 54, no. 6 (November 2018): 957–70. http://dx.doi.org/10.1007/s10559-018-0099-8.

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44

Li, Hanwu, and Yongsheng Song. "Backward Stochastic Differential Equations Driven by G-Brownian Motion with Double Reflections." Journal of Theoretical Probability, September 13, 2020. http://dx.doi.org/10.1007/s10959-020-01038-5.

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Abstract In this paper, we study the reflected backward stochastic differential equations driven by G-Brownian motion with two reflecting obstacles, which means that the solution lies between two prescribed processes. A new kind of approximate Skorohod condition is proposed to derive the uniqueness and existence of the solutions. The uniqueness can be proved by a priori estimates and the existence is obtained via a penalization method.

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45

Cass, Thomas, and Nengli Lim. "Skorohod and rough integration for stochastic differential equations driven by Volterra processes." Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 57, no. 1 (February 1, 2021). http://dx.doi.org/10.1214/20-aihp1074.

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46

Droniou, Jérôme, Beniamin Goldys, and Kim-Ngan Le. "Design and convergence analysis of numerical methods for stochastic evolution equations with Leray–Lions operator." IMA Journal of Numerical Analysis, March 4, 2021. http://dx.doi.org/10.1093/imanum/draa105.

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Abstract The gradient discretization method (GDM) is a generic framework, covering many classical methods (finite elements, finite volumes, discontinuous Galerkin, etc.), for designing and analysing numerical schemes for diffusion models. In this paper we study the GDM for a general stochastic evolution problem based on a Leray–Lions type operator. The problem contains the stochastic $p$-Laplace equation as a particular case. The convergence of the gradient scheme (GS) solutions is proved by using discrete functional analysis techniques, Skorohod theorem and the Kolmogorov test. In particular, we provide an independent proof of the existence of weak martingale solutions for the problem. In this way we lay foundations and provide techniques for proving convergence of the GS approximating stochastic partial differential equations.

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47

Sun, Chengfeng, Hongjun Gao, Hui Liu, and Jie Zhang. "Martingale solutions of the stochastic 2D primitive equations with anisotropic viscosity." ESAIM: Probability and Statistics, April 20, 2022. http://dx.doi.org/10.1051/ps/2022006.

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The stochastic 2D primitive equations with anisotropic viscosity are studied in this paper. The existence of the martingale solutions and pathwise uniqueness of the solutions are obtained. The proof is based on anisotropic estimates, the compactness method, tightness criteria and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces.

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48

Fichtner, Karl-Heinz, Steffen Klaere, and Volkmar Liebscher. "Solving a class of linear Skorokhod stochastic differential equations." Communications on Stochastic Analysis 9, no. 4 (December 1, 2015). http://dx.doi.org/10.31390/cosa.9.4.02.

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49

"Skorohod's stochastic differential equations with reflecting boundary condition." Stochastic Processes and their Applications 21, no. 1 (December 1985): 43–44. http://dx.doi.org/10.1016/0304-4149(85)90307-2.

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50

Grün, Günther, and Lorenz Klein. "Zero-contact angle solutions to stochastic thin-film equations." Journal of Evolution Equations 22, no. 3 (July 16, 2022). http://dx.doi.org/10.1007/s00028-022-00818-2.

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AbstractWe establish existence of nonnegative martingale solutions to stochastic thin-film equations with quadratic mobility for compactly supported initial data under Stratonovich noise. Based on so-called $$\alpha $$ α -entropy estimates, we show that almost surely these solutions are classically differentiable in space almost everywhere in time and that their derivative attains the value zero at the boundary of the solution’s support. From a physics perspective, this means that they exhibit a zero-contact angle at the three-phase contact line between liquid, solid, and ambient fluid. These $$\alpha $$ α -entropy estimates are first derived for almost surely strictly positive solutions to a family of stochastic thin-film equations augmented by second-order linear diffusion terms. Using Itô’s formula together with stopping time arguments, Jakubowski’s modification of the Skorokhod theorem, and martingale identification techniques, the passage to the limit of vanishing regularization terms gives the desired existence result.

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