Kolmogorov equation and large-time behaviour for fractional Brownian motion driven linear SDE's

Michal Vyoral

Applications of Mathematics (2005)

  • Volume: 50, Issue: 1, page 63-81
  • ISSN: 0862-7940

Abstract

top
We consider a stochastic process X t x which solves an equation d X t x = A X t x d t + Φ d B t H , X 0 x = x where A and Φ are real matrices and B H is a fractional Brownian motion with Hurst parameter H ∈ ( 1 / 2 , 1 ) . The Kolmogorov backward equation for the function u ( t , x ) = 𝔼 f ( X t x ) is derived and exponential convergence of probability distributions of solutions to the limit measure is established.

How to cite

top

Vyoral, Michal. "Kolmogorov equation and large-time behaviour for fractional Brownian motion driven linear SDE's." Applications of Mathematics 50.1 (2005): 63-81. <http://eudml.org/doc/33205>.

@article{Vyoral2005,
abstract = {We consider a stochastic process $X_t^x$ which solves an equation \[ \{\mathrm \{d\}\}X\_t^x = AX\_t^x\mathrm \{d\}t + \Phi \{\mathrm \{d\}\}B^H\_t,\quad X\_0^x = x \] where $A$ and $\Phi $ are real matrices and $B^H$ is a fractional Brownian motion with Hurst parameter $H \in (1/2,1)$. The Kolmogorov backward equation for the function $u(t,x) = \mathbb \{E\} f(X^x_t)$ is derived and exponential convergence of probability distributions of solutions to the limit measure is established.},
author = {Vyoral, Michal},
journal = {Applications of Mathematics},
keywords = {fractional Brownian motion; Kolmogorov backwards equation; linear stochastic equation; Kolmogorov backwards equation; linear stochastic equation},
language = {eng},
number = {1},
pages = {63-81},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Kolmogorov equation and large-time behaviour for fractional Brownian motion driven linear SDE's},
url = {http://eudml.org/doc/33205},
volume = {50},
year = {2005},
}

TY - JOUR
AU - Vyoral, Michal
TI - Kolmogorov equation and large-time behaviour for fractional Brownian motion driven linear SDE's
JO - Applications of Mathematics
PY - 2005
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 50
IS - 1
SP - 63
EP - 81
AB - We consider a stochastic process $X_t^x$ which solves an equation \[ {\mathrm {d}}X_t^x = AX_t^x\mathrm {d}t + \Phi {\mathrm {d}}B^H_t,\quad X_0^x = x \] where $A$ and $\Phi $ are real matrices and $B^H$ is a fractional Brownian motion with Hurst parameter $H \in (1/2,1)$. The Kolmogorov backward equation for the function $u(t,x) = \mathbb {E} f(X^x_t)$ is derived and exponential convergence of probability distributions of solutions to the limit measure is established.
LA - eng
KW - fractional Brownian motion; Kolmogorov backwards equation; linear stochastic equation; Kolmogorov backwards equation; linear stochastic equation
UR - http://eudml.org/doc/33205
ER -

References

top
  1. Stochastic calculus with respect to fractional Brownian motion with Hurst parameter lesser than 1 / 2 , Stochastic Processes Appl. 86 (2000), 121–139. (2000) MR1741199
  2. 10.1016/S0764-4442(00)01594-9, C.R. Acad. Sci. Paris 331 (2000), 75–80. (2000) MR1780221DOI10.1016/S0764-4442(00)01594-9
  3. 10.1155/S104895339600038X, J.  Appl. Math. Stochastic Anal. 9 (1996), 439–448. (1996) MR1429266DOI10.1155/S104895339600038X
  4. 10.1023/A:1008634027843, Potential Anal. 10 (1999), 177–214. (1999) MR1677455DOI10.1023/A:1008634027843
  5. An Introduction to p -Variation and Young Integrals Concentrated Advanced Course, Maphysto, Centre for Mathematical Physics and Stochastics, University of Aarhus, Aarhus, 1999. (1999) 
  6. 10.1137/S036301299834171X, SIAM J. Control Optimization 38 (2000), 582–612. (2000) MR1741154DOI10.1137/S036301299834171X
  7. 10.1142/S0219493702000340, Stoch. Dyn. 2 (2002), 225–250. (2002) MR1912142DOI10.1142/S0219493702000340
  8. 10.1016/S0167-7152(98)00147-3, Statist. Probab. Lett. 41 (1999), 337–346. (1999) MR1666072DOI10.1016/S0167-7152(98)00147-3
  9. Matrix Analysis, Cambridge University Press, Cambridge, 1985. (1985) MR0832183
  10. Long-term storage capacity in reservoirs, Trans. Amer. Soc. Civil Eng. 116 (1951), 400–410. (1951) 
  11. Methods of using long-term storage in reservoirs, Proc. Inst. Civil Engineers, Part  I (1956), 519–590. (1956) 
  12. Fractional Ornstein-Uhlenbeck processes, http://www.math.washington.edu/ ejpecp/EjpVol8/paper3.abs.html. 
  13. Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum, C. R. (Dokl.) Acad. Sci. URSS (N.S.) 26 (1940), 115–118. (1940) MR0003441
  14. 10.1016/S0304-4149(01)00145-4, Stochastic Process. Appl. 98 (2002), 289–315. (2002) Zbl1059.60068MR1887537DOI10.1016/S0304-4149(01)00145-4
  15. 10.1137/1010093, SIAM Rev. 10 (1968), 422–437. (1968) MR0242239DOI10.1137/1010093
  16. Evolution equations driven by a fractional Brownian motion, J. Funct. Anal (to appear). (to appear) MR1994773
  17. Differential equations driven by fractional Brownian motion, Collect. Math. 53 (2002), 55–81. (2002) MR1893308
  18. Real Analysis, Macmillan, New York, 1963. (1963) Zbl0121.05501MR0151555
  19. Fractional Langevin equation, Phys. Rev.  E 64, 051106 (2001), . (2001) 
  20. 10.1007/s004400050171, Probab. Theory Relat. Fields 111 (1998), 333–374. (1998) Zbl0918.60037MR1640795DOI10.1007/s004400050171
  21. 10.1142/S0219493702000406, Stoch. Dyn. 2 (2002), 265–280. (2002) Zbl1016.91053MR1912144DOI10.1142/S0219493702000406

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.