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Dynkin's formula

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Theorem in stochastic analysis

In mathematics — specifically, in stochastic analysis — Dynkin's formula is a theorem giving the expected value of any suitably smooth function applied to a Feller process 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 {\displaystyle X} be a Feller process with infinitesimal generator A {\displaystyle A} . For a point x {\displaystyle x} in the state-space of X {\displaystyle X} , let P x {\displaystyle \mathbf {P} ^{x}} denote the law of X {\displaystyle X} given initial datum X 0 = x {\displaystyle X_{0}=x} , and let E x {\displaystyle \mathbf {E} ^{x}} denote expectation with respect to P x {\displaystyle \mathbf {P} ^{x}} . Then for any function f {\displaystyle f} in the domain of A {\displaystyle A} , and any stopping time τ {\displaystyle \tau } with E [ τ ] < + ∞ {\displaystyle \mathbf {E} <+\infty } , Dynkin's formula holds:

E x [ f ( X τ ) ] = f ( x ) + E x [ ∫ 0 τ A f ( X s ) d s ] . {\displaystyle \mathbf {E} ^{x}=f(x)+\mathbf {E} ^{x}\left.}

Example: Itô diffusions

Let X {\displaystyle X} be the R n {\displaystyle \mathbf {R} ^{n}} -valued Itô diffusion solving the stochastic differential equation

d X t = b ( X t ) d t + σ ( X t ) d B t . {\displaystyle \mathrm {d} X_{t}=b(X_{t})\,\mathrm {d} t+\sigma (X_{t})\,\mathrm {d} B_{t}.}

The infinitesimal generator A {\displaystyle A} of X {\displaystyle X} is defined by its action on compactly-supported C 2 {\displaystyle C^{2}} (twice differentiable with continuous second derivative) functions f : R n → R {\displaystyle f:\mathbf {R} ^{n}\to \mathbf {R} } as

A f ( x ) = lim t ↓ 0 E x [ f ( X t ) ] − f ( x ) t {\displaystyle Af(x)=\lim _{t\downarrow 0}{\frac {\mathbf {E} ^{x}-f(x)}{t}}}

or, equivalently,

A f ( x ) = ∑ i b i ( x ) ∂ f ∂ x i ( x ) + 1 2 ∑ i , j ( σ σ ⊤ ) i , j ( x ) ∂ 2 f ∂ x i ∂ x j ( x ) . {\displaystyle Af(x)=\sum _{i}b_{i}(x){\frac {\partial f}{\partial x_{i}}}(x)+{\frac {1}{2}}\sum _{i,j}{\big (}\sigma \sigma ^{\top }{\big )}_{i,j}(x){\frac {\partial ^{2}f}{\partial x_{i}\,\partial x_{j}}}(x).}

Since this X {\displaystyle X} is a Feller process, Dynkin's formula holds. In fact, if τ {\displaystyle \tau } is the first exit time of a bounded set B ⊂ R n {\displaystyle B\subset \mathbf {R} ^{n}} with E [ τ ] < + ∞ {\displaystyle \mathbf {E} <+\infty } , then Dynkin's formula holds for all C 2 {\displaystyle C^{2}} functions f {\displaystyle f} , without the assumption of compact support.

Application: Brownian motion exiting the ball

Dynkin's formula can be used to find the expected first exit time τ K {\displaystyle \tau _{K}} of a Brownian motion B {\displaystyle B} from the closed ball K = { x ∈ R n : | x | ≤ R } , {\displaystyle K=\{x\in \mathbf {R} ^{n}:\,|x|\leq R\},} which, when B {\displaystyle B} starts at a point a {\displaystyle a} in the interior of K {\displaystyle K} , is given by

E a [ τ K ] = 1 n ( R 2 − | a | 2 ) . {\displaystyle \mathbf {E} ^{a}={\frac {1}{n}}{\big (}R^{2}-|a|^{2}{\big )}.}

This is shown as follows. Fix an integer j. The strategy is to apply Dynkin's formula with X = B {\displaystyle X=B} , τ = σ j = min { j , τ K } {\displaystyle \tau =\sigma _{j}=\min\{j,\tau _{K}\}} , and a compactly-supported f ∈ C 2 {\displaystyle f\in C^{2}} with f ( x ) = | x | 2 {\displaystyle f(x)=|x|^{2}} on K {\displaystyle K} . The generator of Brownian motion is Δ / 2 {\displaystyle \Delta /2} , where Δ {\displaystyle \Delta } denotes the Laplacian operator. Therefore, by Dynkin's formula,

E a [ f ( B σ j ) ] = f ( a ) + E a [ ∫ 0 σ j 1 2 Δ f ( B s ) d s ] = | a | 2 + E a [ ∫ 0 σ j n d s ] = | a | 2 + n E a [ σ j ] . {\displaystyle {\begin{aligned}\mathbf {E} ^{a}\left&=f(a)+\mathbf {E} ^{a}\left\\&=|a|^{2}+\mathbf {E} ^{a}\left=|a|^{2}+n\mathbf {E} ^{a}.\end{aligned}}}

Hence, for any j {\displaystyle j} ,

E a [ σ j ] ≤ 1 n ( R 2 − | a | 2 ) . {\displaystyle \mathbf {E} ^{a}\leq {\frac {1}{n}}{\big (}R^{2}-|a|^{2}{\big )}.}

Now let j → + ∞ {\displaystyle j\to +\infty } to conclude that τ K = lim j → + ∞ σ j < + ∞ {\displaystyle \tau _{K}=\lim _{j\to +\infty }\sigma _{j}<+\infty } almost surely, and so E a [ τ K ] = ( R 2 − | a | 2 ) / n {\displaystyle \mathbf {E} ^{a}=(R^{2}-|a|^{2})/n} as claimed.

References

  1. Kallenberg (2021), Lemma 17.21, p383.
  2. Øksendal (2003), Definition 7.3.1, p124.
  3. Øksendal (2003), Theorem 7.3.3, p126.
  4. ^ Øksendal (2003), Theorem 7.4.1, p127.
  5. Øksendal (2003), Example 7.4.2, p127.

Sources

  • Dynkin, Eugene B.; trans. J. Fabius; V. Greenberg; A. Maitra; G. Majone (1965). Markov processes. Vols. I, II. Die Grundlehren der Mathematischen Wissenschaften, Bände 121. New York: Academic Press Inc. (See Vol. I, p. 133)
  • Kallenberg, Olav (2021). Foundations of Modern Probability (third ed.). Springer. ISBN 978-3-030-61870-4.
  • Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer. ISBN 3-540-04758-1. (See Section 7.4)
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