Frontiers of Mathematics in China >
Reflected stochastic differential equations driven by G-Brownian motion with nonlinear resistance
Received date: 09 Sep 2014
Accepted date: 22 Sep 2014
Published date: 02 Dec 2015
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We study the uniqueness and existence of solutions of reflected G-stochastic differential equations (RGSDEs) with nonlinear resistance under an integral-Lipschitz condition of coefficients. Moreover, we obtain the comparison theorem for RGSDEs with nonlinear resistance.
Peng LUO . Reflected stochastic differential equations driven by G-Brownian motion with nonlinear resistance[J]. Frontiers of Mathematics in China, 2016 , 11(1) : 123 -140 . DOI: 10.1007/s11464-015-0433-7
1 |
Bihari I.A generalization of a lemma of bellman and its application to uniqueness problems of differential equations. Acta Math Hungar, 1956, 7(1): 81–94
|
2 |
Denis L,Hu M,Peng S.Function spaces and capacity related to a sublinear expectation: application to G-Brownian motion paths. Potential Anal, 2011, 34(2):139–161
|
3 |
El Karoui N.Processus de réflexion dans ℝn. In: Meyer P A, ed. Séminaire deprobabilités IX, Université de Strasbourg. Lecture Notes in Math, Vol 465, Berlin:Springer, 1975, 534–554
|
4 |
El Karoui N,Chaleyat-Maurel M.Un probl`eme de réflexion et ses applications au temps local et aux équations différentielles stochastiques sur ℝ, cas continu. In: Exposés duSéminaire J. Azéma-M. Yor. Held at the Université Pierre et Marie Curie (Paris, 1976-1977). Astérisque, 52, 53, Société Math′ematique de France, Paris, 1978, 117–144
|
5 |
El Karoui N,Kapoudjian C,Pardoux E,Peng S,Quenez M C.Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s. Ann Probab, 1997, 25(2):702–737
|
6 |
Gao F.Pathwise properties and homomorphic flows for stochastic differential equations driven by G-Brownian motion. Stochastic Process Appl, 2009, 119: 3356–3382
|
7 |
Gegout-Petit A,Pardoux E.Equations différentielles stochastiques rétrogrades refl echiesdans un convexe. Stoch Stoch Rep, 1996, 57: 111–128
|
8 |
Hu Y,Tang S.Multi-dimensional BSDE with oblique reflection and optimal switching.Probab Theory Related Fields, 2010, 147: 89–121
|
9 |
Li X,Peng S.Stopping times and related Itô calculus with G-Brownian motion.Stochastic Process Appl, 2011, 121: 1492–1508
|
10 |
Lin Q.Local time and Tanaka formula for the G-Brownian motion. J Math Anal Appl,2013, 398: 315–334
|
11 |
Lin Q.Some properties of stochastic differential equations driven by G-Brownian motion. Acta Math Sin (Engl Ser), 2013, 29: 923–942
|
12 |
Lin Q.Differentiability of stochastic differential equations driven by G-Brownian motion. Sci China Math, 2013, 56: 1087–1107
|
13 |
Lin Y.Stochastic differential equations driven by G-Brownian motion with reflecting boundary. Electron J Probab, 2013, 18(9): 1–23
|
14 |
Lin Y.Équations différentielles stochastiques sous les espérances mathématiquesnonlinéaire et applications. Ph D Thesis. Université de Rennes 1, 2013,tel.archivesouvertes.fr/tel-00955814
|
15 |
Lin Y,Bai X.On the existence and uniqueness of solutions to stochastic differential equations driven by G-Brownian motion with integral-Lipschitz coefficients. Acta Math Appl Sin Engl Ser, 2014, 30(3): 589–610
|
16 |
Lions P L,Sznitman A L.Stochastic differential equations with reflecting boundary conditions. Comm Pure Appl Math, 1984, 37: 511–537
|
17 |
Luo P,Jia G.On monotonicity and order-preservation for multi-dimensional G-diffusion processes. Stoch Anal Appl, 2015, 33(1): 67–90
|
18 |
Luo P,Wang F.Stochastic differential equations driven by G-Brownian motion and ordinary differential equations. Stochastic Process Appl, 2014, 124: 3869–3885
|
19 |
Peng S.Filtration consistent nonlinear expectations and evaluations of contingent claims. Acta Math Appl Sin Engl Ser, 2004, 20(2): 1–24
|
20 |
Peng S.Nonlinear expectations and nonlinear Markov chains. Chin Ann Math Ser B, 2005, 26(2): 159–184
|
21 |
Peng S.G-expectation, G-Brownian motion and related stochastic calculus of Itô type.In: Stochastic Analysis and Applications, Abel Symp, 2. Berlin: Springer, 2007, 541–567
|
22 |
Peng S.Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations. Sci China Ser A, 2009,52(7): 1391–1411
|
23 |
Peng S.Nolinear expectations and stochastic calculus under uncertainty. arXiv:1002.4546v1
|
24 |
Peng S.Backward stochastic differential equation, nonlinear expectation and their applications. In: Proceedings of the International Congress of Mathematicians,Hyderabad, India, 2010
|
25 |
Qian Z,Xu M.Skorohod equation and reflected bacward stochastic differential equations. arXiv: 1103.2078v1
|
26 |
Ramasubramanian S.Reflected backward stochastic differential equations in an orthant. Proc Indian Acad Sci Math Sci, 2002, 112: 347–360
|
27 |
Skorohod A V.Stochastic equations for diffusion processes in a bounded region. Theo Veroyatnost i Primenen, 1961, 6: 287–298
|
28 |
Skorohod A V.Stochastic equations for diffusion processes in a bounded region II. Theo Veroyatnost i Primenen, 1962, 7: 5–25
|
29 |
Stroock D W,Varadhan S R S. Diffusion processes with boundary conditions. CommPure Appl Math, 1971, 24: 147–225
|
30 |
Tanaka H.Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math J, 1979, 9: 163–177
|
31 |
Yamada T.On the uniqueness of solutions of stochastic differential equations with reflecting barrier conditions. In: Meyer P A, ed. S′eminaire de Probabilité X, Université de Strasbourg. Lecture Notes in Math, Vol 511. Berlin: Springer, 1976,240–244
|
32 |
Zhang B,Xu J,Kannan D.Extension and application of Itô’s formula under G-framework. Stoch Anal Appl, 2010, 28: 322–349
|
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