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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 :\mathrm X_ = b(X_) \, \mathrm t + \sigma (X_) \, \mathrm B_. 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 :A f (x) = \lim_ \frac or, equivalently, :A f (x) = \sum_ b_ (x) \frac (x) + \frac1 \sum_ \big( \sigma \sigma^ \big)_ (x) \frac (x). 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: :\mathbf^
(X_) X, or x, is the twenty-fourth and third-to-last Letter (alphabet), letter in the Latin alphabet, used in the English alphabet, modern English alphabet, the alphabets of other western European languages and others worldwide. Its English a ...
= f(x) + \mathbf^ \left \int_^ A f (X_) \, \mathrm s \right 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 K = K_ = \, 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 \mathbf^
tau_ Tau (uppercase Τ, lowercase τ, or \boldsymbol\tau; el, ταυ ) is the 19th letter of the Greek alphabet, representing the voiceless alveolar plosive, voiceless dental or alveolar plosive . In the system of Greek numerals, it has a value o ...
= \frac1 \big( R^ - , a , ^ \big). 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, \begin \mathbf^ \left f \big( B_ \big) \right&= f(a) + \mathbf^ \left \int_^ \frac1 \Delta f (B_) \, \mathrm s \right\\ &= , a , ^ + \mathbf^ \left \int_^ n \, \mathrm s \right\\ &= , a , ^ + n \mathbf^
sigma_ Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used a ...
\end Hence, for any ''j'', \mathbf^
sigma_ Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used a ...
\leq \frac1 \big( R^ - , a , ^ \big). Now let ''j'' → +∞ to conclude that ''τ''''K'' = lim''j''→+∞''σ''''j'' < +∞ almost surely and \mathbf^
tau_ Tau (uppercase Τ, lowercase τ, or \boldsymbol\tau; el, ταυ ) is the 19th letter of the Greek alphabet, representing the voiceless alveolar plosive, voiceless dental or alveolar plosive . In the system of Greek numerals, it has a value o ...
= \frac1 \big( R^ - , a , ^ \big), 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