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Product identity and conditional moments for iterated Brownian integrals

Statement

Let the finite-dimensional Brownian It^o setting and the integrals $I_{i j, n}$ be as in Milstein updates, iterated Brownian integrals, and commutative noise. For every grid interval of length $h > 0$ , $I_{i j, n} + I_{j i, n} = Δ W_{n}^{i} Δ W_{n}^{j} - δ_{i j} h .$ In the scalar case, writing $I_{n} = I_{11, n}$ and $Δ W_{n} = Δ W_{n}^{1}$ , this gives $I_{n} = \frac{1}{2} ((Δ W_{n})^{2} - h), E [I_{n} ∣ F_{t_{n}}] = 0, E [I_{n}^{2} ∣ F_{t_{n}}] = \frac{h^{2}}{2}, E [Δ W_{n} I_{n} ∣ F_{t_{n}}] = 0.$ Consequently, under the commutativity condition in the Definition, the double-integral term in the general Milstein Taylor increment equals $\sum_{i, j} L_{i} σ_{j} (Y_{n}) I_{i j, n} = \frac{1}{2} \sum_{i, j} L_{i} σ_{j} (Y_{n}) (Δ W_{n}^{i} Δ W_{n}^{j} - δ_{i j} h) .$

Facts & Assumptions

Given: AC, a filtered probability space with an adapted standard $m$ -dimensional Brownian motion whose increments after $t_{n}$ are independent of $F_{t_{n}}$ , and the grid interval $[t_{n}, t_{n} + h]$ .

[L1]

Milstein updates, iterated Brownian integrals, and commutative noise: $I_{i j, n} = \int_{t_{n}}^{t_{n} + h} (W_{s}^{i} - W_{t_{n}}^{i}) d W_{s}^{j}$ , and the commutative condition is $L_{i} σ_{j} = L_{j} σ_{i}$ .

[F1]

Multidimensional Ito formula for Brownian-driven processes: for a Brownian It^o process, the second-order term in It^o's formula is one half of the Hessian paired with the instantaneous covariance matrix; the theorem assumes AC and the usual-filtration/integrability conventions stated in its body.

[F2]

Continuous Brownian Ito processes: Brownian increments are independent of the current sigma algebra and have their stated centered normal laws.

[F3]

Standard normal and normal laws: a standard normal has density $(2 π)^{- 1 / 2} e^{- x^{2} / 2}$ ; its odd moments vanish and its even moments follow by integration by parts.

[F4]

Conditional expectation given a sigma algebra: conditional expectation is the measurable integrable version characterized by its integrals on events in the conditioning sigma algebra.

[A1]

The Axiom of Choice: AC is assumed by the cited Brownian and It^o interfaces; no further choice is used here.

Proof

Given: The assumptions above.

1.1F1L1A1

Apply [F1] to $f (x) = x_{i} x_{j}$ on $[t_{n}, t_{n} + h]$ . It gives $W_{t_{n} + h}^{i} W_{t_{n} + h}^{j} - W_{t_{n}}^{i} W_{t_{n}}^{j} = \int_{t_{n}}^{t_{n} + h} W_{s}^{i} d W_{s}^{j} + \int_{t_{n}}^{t_{n} + h} W_{s}^{j} d W_{s}^{i} + δ_{i j} h$ . Expand each integrand as its value at $t_{n}$ plus its Brownian increment; the two constant terms cancel from the two sides, leaving $Δ W_{n}^{i} Δ W_{n}^{j} = I_{i j, n} + I_{j i, n} + δ_{i j} h$ .

2.1F2F3step 1.1

Setting $i = j = 1$ in step 1.1 yields $I_{n} = ((Δ W_{n})^{2} - h) / 2$ . By [F2], $Δ W_{n}$ is independent of $F_{t_{n}}$ and has law $N (0, h)$ . Scaling the density in [F3] and integrating by parts give $E [Δ W_{n} ∣ F_{t_{n}}] = 0$ , $E [(Δ W_{n})^{2} ∣ F_{t_{n}}] = h$ , $E [(Δ W_{n})^{3} ∣ F_{t_{n}}] = 0$ , and $E [(Δ W_{n})^{4} ∣ F_{t_{n}}] = 3 h^{2}$ .

3.1F2F3F4step 2.1

Substituting the four conditional moments from step 2.1 into $I_{n} = ((Δ W_{n})^{2} - h) / 2$ gives $E [I_{n} ∣ F_{t_{n}}] = (h - h) / 2 = 0$ , $E [I_{n}^{2} ∣ F_{t_{n}}] = (3 h^{2} - 2 h^{2} + h^{2}) / 4 = h^{2} / 2$ , and $E [Δ W_{n} I_{n} ∣ F_{t_{n}}] = (0 - h \cdot 0) / 2 = 0$ . Each equality is an equality of conditional-expectation versions by the event-integral definition [F4].

4.1L1step 1.1∎

Under commutativity, pair the summand with indices $(i, j)$ with the summand $(j, i)$ . The coefficients agree by [L1], so the sum equals $\frac{1}{2} \sum_{i, j} L_{i} σ_{j} (Y_{n}) (I_{i j, n} + I_{j i, n})$ . Substituting step 1.1 gives the displayed Brownian-increment formula. The argument includes $i = j$ , where the summand is one half of $L_{i} σ_{i} (Y_{n}) ((Δ W_{n}^{i})^{2} - h)$ .

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Sources

Kloeden and Platen, Numerical Solution of Stochastic Differential Equations, Chapter 5.2 and Section 10.3, printed pp. 167-176 and 345-351 (standard reference, not scraped)
Desmond J. Higham, An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations, SIAM Review 43 (2001), Sections 3 and 6 (standard reference, not scraped)
M. Tretyakov, Lecture Notes for a Mini-Course on Numerical Methods for SDEs, Section 6.4 (standard reference, not scraped)