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Brownian finite-dimensional density

Example

Assume the Axiom of Choice. Let $B$ be a standard Brownian motion and let $0 < t_{1} < \dots < t_{n}$ , where $n \geq 1$ . Put $t_{0} = 0$ and $x_{0} = 0$ . Then the law of $(B_{t_{1}}, \dots, B_{t_{n}})$ has, with respect to Lebesgue measure on $R^{n}$ , the density

$p (x_{1}, \dots, x_{n}) = \prod_{j = 1}^{n} \frac{1}{\sqrt{2 π (t_{j} - t_{j - 1})}} \exp ⁣ (- \frac{(x_{j} - x_{j - 1})^{2}}{2 (t_{j} - t_{j - 1})}) .$

Facts & Assumptions

Given: AC, a standard Brownian motion $B$ , and $0 < t_{1} < \dots < t_{n}$ with $n \geq 1$ ; write $h_{j} = t_{j} - t_{j - 1} > 0$ .

[F1]

Brownian increments $Δ_{j} = B_{t_{j}} - B_{t_{j - 1}}$ are mutually independent and have laws $N (0, h_{j})$ . Brownian motion, Brownian covariance is equivalent to independent stationary normal increments.

[F2]

Under AC, $N (0, 1)$ has density $ϕ (z) = e^{- z^{2} / 2} / \sqrt{2 π}$ , and $N (0, h)$ is the pushforward under $z \mapsto \sqrt{h} z$ . Standard normal and normal laws.

[F3]

Independent random elements have product joint law. Independent random elements have product joint law.

[F4]

A nonnegative measurable function defines a measure by indefinite integration; sigma-finite product measures exist, have the rectangle formula, and are unique. The indefinite integral of a nonnegative measurable function is a measure, For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique.

[F5]

Tonelli holds for nonnegative product-measurable functions, and finite products of one-dimensional Lebesgue measure agree with Euclidean Lebesgue measure on Borel sets. Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, On Borel subsets of $R^{m + n}$ , the product $λ_{m} \times λ_{n}$ agrees with $λ_{m + n}$ .

[F6]

The declared supplier lem-c-one-change-of-variables-for-nonnegative-borel-functions-via-radon-uniqueness gives: assuming countable choice, a $C^{1}$ diffeomorphism $T : U \to V$ satisfies $\int_{V} f = \int_{U} (f \circ T) ∣ \det D T ∣$ for every nonnegative Borel $f$ .

[F7]

Continuous partial derivatives give the total derivative, whose matrix is the Jacobian, and a triangular matrix has determinant equal to the product of its diagonal entries. If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative, The determinant of a triangular matrix is the product of its diagonal entries.

[F8]

The declared supplier thm-choice-implies-dependent-implies-countable-choice gives that AC implies countable choice, and The Axiom of Choice fixes the ambient assumption.

Verification

technique · direct

1.1F2F6F8

For each $j$ , let $σ_{j} = \sqrt{h_{j}} > 0$ and $g_{j} (y) = \frac{1}{σ_{j}} ϕ (y / σ_{j}) = \frac{1}{\sqrt{2 π h_{j}}} e^{- y^{2} / (2 h_{j})} .$ For a Borel set $A \subseteq R$ , apply [F6] to the dilation $T_{j} (z) = σ_{j} z$ and the nonnegative Borel function $1_{A} g_{j}$ . Since $T_{j}^{'} = σ_{j}$ , this gives $\int_{A} g_{j} (y) d y = \int_{R} 1_{A} (σ_{j} z) g_{j} (σ_{j} z) σ_{j} d z = \int_{R} 1_{A} (σ_{j} z) ϕ (z) d z .$ By the pushforward definition in [F2], $g_{j}$ is therefore a density of $N (0, h_{j})$ .

1.2F1F7algebra

Define $T : R^{n} \to R^{n}$ and $S : R^{n} \to R^{n}$ by $(T y)_{j} = \sum_{k = 1}^{j} y_{k}, (S x)_{j} = x_{j} - x_{j - 1}, x_{0} = 0.$ Direct telescoping gives $S T = T S = id$ . Their coordinate partial derivatives are constant, so [F7] makes them $C^{1}$ with their displayed matrices. The matrix of $S$ is lower triangular with every diagonal entry one; hence $∣ \det D S ∣ = 1$ . Thus $S$ is a $C^{1}$ diffeomorphism with inverse $T$ . Moreover, $T Δ = (B_{t_{1}}, \dots, B_{t_{n}})$ almost surely by telescoping and $B_{0} = 0$ almost surely.

2.1step 1.1F1F3F4F5

Put $q (y_{1}, \dots, y_{n}) = \prod_{j = 1}^{n} g_{j} (y_{j})$ . Induction on $n$ , using Tonelli and the Borel equality $λ_{n - 1} \times λ_{1} = λ_{n}$ , shows that the measure $Q (E) = \int_{E} q d λ_{n}$ has on every Borel rectangle the value $\prod_{j} \int_{A_{j}} g_{j} d λ_{1}$ . The base $n = 1$ is step 1.1; the induction also gives $Q (R^{n}) = 1$ . Thus [F4] and uniqueness of the product measure identify $Q$ with the product of the $N (0, h_{j})$ laws. By [F1] and [F3], $Q$ is exactly the joint law of $Δ = (Δ_{1}, \dots, Δ_{n})$ .

3.1step 1.1step 2.1step 1.2F6F8

For a Borel set $A \subseteq R^{n}$ , step 2.1 and the almost-sure identity in step 1.2 give $P ((B_{t_{1}}, \dots, B_{t_{n}}) \in A) = \int_{T^{- 1} A} q (y) d y .$ Apply [F6] to $S$ and $f (y) = 1_{T^{- 1} A} (y) q (y)$ . Because $T (S x) = x$ and $∣ \det D S ∣ = 1$ , the right side becomes $\int_{R^{n}} 1_{A} (x) q (S x) d x = \int_{A} \prod_{j = 1}^{n} g_{j} (x_{j} - x_{j - 1}) d x .$ Substituting the formula from step 1.1 is exactly the stated density.

4.1givenstep 1.1step 1.2step 3.1F1F2F5F6F8∎

The strict inequalities make every $h_{j}$ positive, so no division by zero or singular normal density occurs. For $n = 1$ , the formula is the $N (0, t_{1})$ density with $x_{0} = t_{0} = 0$ ; the empty case $n = 0$ is excluded. The triangular determinant is one even when $n = 1$ . AC is used through the Brownian and normal-law suppliers, and it supplies the countable choice required by [F5] and [F6]; the finite triangular transformation makes no additional choice.

Remarks

The suppliers of [F6] and [F8] are homed on euclidean-surface-measure-divergence-and-green-identities (order 458.0021) and weak-choice-principles-and-sierpinskis-theorem (order 665), while this examples page has order 288.132. Step-5b resolution moved those citations from item-level forward_refs to deps, since both suppliers are published and load bearing, and [F6] and [F8] name them by ID rather than linking because their A pages sit the other way along the reading order. The batch-2 manifest whitelists both pages under this page's forwardRefs, so the page-level dependency is declared as well; rehoming this example to either of those subjects would be an owner-only reading-order change.

Source notes

Sousi, Section 6.1 (printed p. 51), supplies the Brownian independent-increment structure. The density and the triangular change-of-variables calculation are derived explicitly above from the library's normal-density, product-measure, and Borel change-of-variables results.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

78 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Perla Sousi, Advanced Probability, Section 6.1 (standard reference, not scraped)