Volterra equations with fractional stochastic integrals

Mathematical Problems in Engineering

Volumn 2004, Issue 5, 2004, Pages 453-468

  El Borai, Mahmoud M ^a   El Nadi, Khairia El Said ^a   Mostafa, Osama L ^a   Ahmed, Hamdy M ^a  

^a Alexandria University Faculty Of Science   (Egypt)

Author keywords

[No Author keywords available]

Indexed keywords

BOUNDARY CONDITIONS; DIFFERENTIAL EQUATIONS; FUNCTIONS; MATHEMATICAL MODELS; NUMERICAL ANALYSIS; NUMERICAL METHODS; PROBABILITY DENSITY FUNCTION; SET THEORY; STOCHASTIC PROGRAMMING;

SOLUTION PROCESS; STOCHASTIC INTEGRAL EQUATIONS; STOCHASTIC VOLTERRA EQUATIONS; VOLTERRA EQUATIONS;

INTEGRAL EQUATIONS;

EID: 15744394643     PISSN: 1024123X     EISSN: None     Source Type: Journal    
DOI: 10.1155/S1024123X04312020     Document Type: Article

References (22)

1. C. Bender and R. J. Elliott, Binary market models and discrete wick products, preprint, 2002.
2. M. A. Berger and V. J. Mizel, Volterra equations with Itô integrals. I, J. Integral Equations 2 (1980), no. 3, 187-245.
3. _, Volterra equations with ltô integrals. II, J. Integral Equations 2 (1980), no. 4, 319-337.
4. _, An extension of the stochastic integral, Ann. Probab. 10 (1982), no. 2, 435-450.
5. M. M. El-Borai, Some probability densities and fundamental solutions of fractional evolution equations, Chaos Solitons Fractals 14 (2002), no. 3, 433-440.
6. A. M. Kolodh, On the existence solutions of stochastic Volterra equations, Theory of Random Processes 11 (1983), 51-57 (Russian).
7. B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev. 10 (1968), 422-437.
8. P.-A. Meyer, Flot d'une équation différentielle stochastique (d'après Malliavin, Bismut, Kunita) [Flow of a stochastic differential equation (following Malliavin, Bismut, Kunita)], Seminar on Probability, XV (University of Strasbourg, Strasbourg, 1979/1980), Lecture Notes in Math., vol. 850, Springer-Verlag, Berlin, 1981, pp. 103-117 (French).
9. D. Nualart, Noncausal stochastic integrals and calculus, Stochastic Analysis and Related Topics (Silivri, 1986), Lecture Notes in Math., vol. 1316, Springer-Verlag, Berlin, 1988, pp. 80-129.
10. D. Nualart and É. Pardoux, Stochastic calculus with anticipating integrands, Probab. Theory Related Fields 78 (1988), no. 4, 535-581.
11. D. Nualart and M. Zakai, Generalized stochastic integrals and the Malliavin calculus, Probab. Theory Related Fields 73 (1986), no. 2, 255-280.
12. D. Ocone and É. Pardoux, A generalized Itô-Ventzell formula. Application to a class of anticipating stochastic differential equations, Ann. Inst. H. Poincaré Probab. Statist. 25 (1989), no. 1, 39-71.
13. _, Linear stochastic differential equations with boundary conditions, Probab. Theory Related Fields 82 (1989), no. 4, 489-526.
14. S. Ogawa, Sur la question d'existence de solutions d'une équation différentielle stochastique du type noncausal [On the existence of solutions of a stochastic differential equation of noncausal type], J. Math. Kyoto Univ. 24 (1984), no. 4, 699-704 (French).
15. É. Pardoux and P. Protter, A two-sided stochastic integral and its calculus, Probab. Theory Related Fields 76 (1987), no. 1, 15-49.
16. _, Stochastic Volterra equations with anticipating coefficients, Ann. Probab. 18 (1990), no. 4, 1635-1655.
17. P. Protter, Volterra equations driven by semimartingales, Ann. Probab. 13 (1985), no. 2, 519-530.
18. A. N. V. Rao and C. P. Tsokos, On the existence, uniqueness, and stability behavior of a random solution to a nonlinear perturbed stochastic integro-differential equation, Information and Control 27 (1975), 61-74.
19. Y. Shiota, A linear stochastic integral equation containing the extended ltô integral, Math. Rep. Toyama Univ. 9 (1986), 43-65.
20. A. V. Skorohod, On a generalization of the stochastic integral, Teor. Verojatnost. i Primenen. 20 (1975), no. 2, 223-238 (Russian).
21. A.-S. Sznitman, Martingales dépendant d'un paramètre: une formule d'Itô [Martingales depending on a parameter: an Ito formula], Z. Wahrsch. Verw. Gebiete 60 (1982), no. 1, 41-70 (French).
22. W. Wyss, The fractional diffusion equation, J. Math. Phys. 27 (1986), no. 11, 2782-2785.