In this note we present some results on stochastic dynamics with respect to a fractional Brownian motion from a L2-approximation approach. A new and simple definition of fractional stochastic integral is introduced and a theorem of existence and uniqueness for fractional stochastic differential equations is established.

References

References
1.
Kubilius
,
K.
, and
Melichov
,
D.
,
2010
, “
Quadratic Variations and Estimation of the Hurst Index of the Solution of Stochastic Differential Equations Driven by a Fractional Brownian Motion
,”
Liathuanian Mathematical Journal
,
26
(
4
), pp.
401
417
.10.1007/s10986-010-9095-z
2.
Elliott
,
R. J.
, and
Van der Hoek
,
J.
,
2007
, “
Ito Formulas for Fractional Brownian Motion
,” Advances in Mathematical Finance, XXXVIII, Applied and Numerical Harmonic Analysis, Birkhauser, Boston, Boston MA, pp.
59
81
.
3.
Alòs
,
E.
,
Mazet
,
O.
, and
Nualart
D.
,
2000
, “
Stochastic Calculus With Respect to Fractional Brownian Motion With Hurst Parameter Less Than 1/2
,”
Stoch. Process. Appl.
,
86
(
1
), pp.
121
130
.10.1016/S0304-4149(99)00089-7
4.
Berzino
,
C.
, and
Léon
,
J. R.
,
2008
, “
Estimation in Models Driven by Fractional Brownian Motion
,”
Annales de l'Inst. Henri Poincaré de probabilités
, Springer-Verlag, Berlin-Heidelberg,
44
(
2
), pp.
191
139
.10.1214/07-AIHP105
5.
Rostek
,
S.
,
2009
,
Option Pricing in Fractional Brownian Motion
,
Springer
.
6.
Marinucci
,
D.
, and
Robinson
P.M.
,
1999
, “
Weak Convergence to Fractional Brownian Motion
,”
Stoch. Process. Appl.
,
80
, pp.
103
120
.10.1016/S0304-4149(98)00073-8
7.
Dung
,
N. T.
,
2011
, “
Fractional Geometric Mean-Reversion Processes
,”
J. Math. Anal. Appl.
,
38
(
1
), pp.
396
402
.10.1016/j.jmaa.2011.03.016
8.
Dung
,
N. T.
,
2011
, “
Semimartingale Approximation of Fractional Brownian Motion and Its Applications
,”
Comput. Math. Appl.
,
61
(
7
), pp.
1844
1854
.10.1016/j.camwa.2011.02.013
9.
Dung
,
N. T.
,
2012
, “
On Delayed Logistic Equation Driven by Fractional Brownian Motions
,”
ASME J. Comput. Nonlinear Dyn.
,
7
, p.
031005
.10.1115/1.4005932
10.
Dung
,
N. T.
,
2012
, “
Mackey-Glass Equation Driven by Fractional Brownian Motion
,”
Physica A
,
391
, pp.
5465
5492
.10.1016/j.physa.2012.06.013
11.
Dung
,
N. T.
,
2013
, “
Fractional Stochastic Differential Equation With Applications to Finance
,”
J. Math. Anal. Appl.
,
397
, pp.
334
348
.10.1016/j.jmaa.2012.07.062
12.
Thao
,
T. H.
, and
Christine
T. A.
,
2003
, “
Évolution Des Cours Gouvernée Par un Processus de Type ARIMA Fractionaire
,”
Studia Babes-Bolyai, Mathematica
,
38
(
2
), pp.
107
115
.
13.
Thao
,
T. H.
,
2006
, “
An Approximate Approach to Fractional Analysis for Finance
,”
Nonlinear Anal.
,
7
(
1
), pp.
124
132
.10.1016/j.nonrwa.2004.08.012
14.
Oksendal
,
B.
,
2008
,
Stochastic Calculus for Fractional Brownian Motion and Applications
,
Springer
-Verlag, Berlin-Heidelberg.
15.
Taqqu
,
M.
,
2003
, “
Fractional Brownian Motion and Long-Range Dependence
,”
Theory and Applications of Long-Range Dependence
,
Birkhäuser
, Boston-Basel-Berlin, pp.
5
38
.
16.
Feyel
,
D.
, and
De la Pradelle
,
A.
,
1999
, “
On Fractional Brownian Motion,
Potential Analysis
,
10
, pp.
273
288
.10.1023/A:1008630211913
You do not currently have access to this content.