In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
— specifically, in
stochastic analysis — Dynkin's formula is a theorem giving the
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
of any suitably smooth statistic of an
Itō diffusion at a
stopping time. It may be seen as a stochastic generalization of the (second)
fundamental theorem of calculus. It is named after the
Russian
mathematician Eugene Dynkin.
Statement of the theorem
Let ''X'' be the R
''n''-valued Itō diffusion solving the
stochastic differential equation
:
For a point ''x'' ∈ R
''n'', let P
''x'' denote the law of ''X'' given initial datum ''X''
0 = ''x'', and let E
''x'' denote expectation with respect to P
''x''.
Let ''A'' be the
infinitesimal generator of ''X'', defined by its action on
compactly-supported ''C''
2 (twice differentiable with continuous second derivative) functions ''f'' : R
''n'' → R as
:
or, equivalently,
:
Let ''τ'' be a
stopping time with E
''x'' 'τ''nbsp;< +∞, and let ''f'' be ''C''
2 with compact support. Then Dynkin's formula holds:
:
In fact, if ''τ'' is the first exit time for a
bounded set
:''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.
In mathematical analysis and related areas of mat ...
''B'' ⊂ R
''n'' with E
''x'' 'τ''nbsp;< +∞, then Dynkin's formula holds for all ''C''
2 functions ''f'', without the assumption of compact support.
Example
Dynkin's formula can be used to find the expected first exit time ''τ''
''K'' of
Brownian motion ''B'' from the
closed ball
which, when ''B'' starts at a point ''a'' in the
interior
Interior may refer to:
Arts and media
* ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas
* ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck
* ''The Interior'' (novel), by Lisa See
* Interior de ...
of ''K'', is given by
Choose an
integer ''j''. The strategy is to apply Dynkin's formula with ''X'' = ''B'', ''τ'' = ''σ''
''j'' = min(''j'', ''τ''
''K''), and a compactly-supported ''C''
2 ''f'' with ''f''(''x'') = , ''x'',
2 on ''K''. The generator of Brownian motion is Δ/2, where Δ denotes the
Laplacian operator. Therefore, by Dynkin's formula,
Hence, for any ''j'',
Now let ''j'' → +∞ to conclude that ''τ''
''K'' = lim
''j''→+∞''σ''
''j'' < +∞
almost surely and
as claimed.
References
* (See Vol. I, p. 133)
* {{cite book
, last = Øksendal
, first = Bernt K.
, authorlink = Bernt Øksendal
, title = Stochastic Differential Equations: An Introduction with Applications
, edition = Sixth
, publisher=Springer
, location = Berlin
, year = 2003
, isbn = 3-540-04758-1
(See Section 7.4)
Stochastic differential equations
Probability theorems