zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Using a geometric Brownian motion to control a Brownian motion and vice versa. (English) Zbl 0911.60040

Consider a one-dimensional controlled process x(t) governed by the equation

dx(t)=a(ξ(t))dt+b(ξ(t))u(ξ(t))dt+[N(ξ(t))] 1/2 dW(t),

where ξ(t):=(x(t),t). The aim of the homing control problem is to minimize the expectation of a functional of the form

J(x)= 0 T(x) [1 2q(ξ(t))u 2 (ξ(t))+λ]dt,

where q0,λ is real and T(x) denotes the exit time from an interval (A,B) for a solution starting from x=x(0)(A,B). In the particular case a=0,b=N=1 and q(ξ(t))=x 2 (t) the optimal control is found by means of the mathematical expectation of a geometric Brownian motion while the optimal process is shown to be a Bessel process. Conversely, if the uncontrolled process is a geometric Brownian motion, then the optimal control is found by means of an expectation of a Brownian motion.

MSC:
60H10Stochastic ordinary differential equations