Acta mathematica scientia,Series B ›› 2000, Vol. 20 ›› Issue (3): 341-358.

• Articles • Previous Articles     Next Articles

MULTI-DIMENSIONAL GEOMETRIC BROWNIAN MOTIONS, ONSAGER-MACHLUP FUNCTIONS, AND APPLICATIONS TO MATHEMATICAL FINANCE

 胡耀忠   

  1. Department of Mathematics, University of Kansas, 405 Snow Hall, Lawrence, KS 66045-2142. USA
    Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China
  • Received:1999-04-08 Online:2000-05-20 Published:2000-05-20
  • Supported by:

    Partly supported by the General Research Fund of the University of Kansas.
    e-mail:hu math.ukans.edu

Abstract:

The solutions of the following bilinear stochastic differential equation are stud-
ied
dxt =
Xm
k=1
Ak
t xtdwk(t) + Btxtdt
where Ak
t , Bt are (deterministic) continuous matrix-valued functions of t and w1(t), · · ·,
wm(t) are m independent standard Brownian motions. Conditions are given such that the
solution is positive if the initial condition is positive. The equation the most probable path
must satisfy is also derived and applied to a mathematical finance problem.

Key words: Multi-dimensional geometric Brownian motions, Onsager-Machlup func-
tions,
most probable path, positivity, most likely interest rate

CLC Number: 

  • 60H10
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