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Conditional local defect bounds for the Milstein Taylor increment

Statement

Let $X$ solve the finite-dimensional autonomous SDE in Milstein updates, iterated Brownian integrals, and commutative noise with $b, σ_{1}, \dots, σ_{m} \in C_{b}^{3} (R^{d})$ , using the $C_{b}^{3}$ convention in that Definition. Let $0 < h \leq 1$ and let $[t, t + h]$ be a deterministic interval. Define $L_{0} f : = D f b + \frac{1}{2} \sum_{k = 1}^{m} D^{2} f [σ_{k}, σ_{k}]$ and $L_{i} f : = D f σ_{i}$ for $i \geq 1$ . Then there is a remainder $R_{t, h} \in L^{2} (R^{d})$ such that $X_{t + h} = X_{t} + b (X_{t}) h + \sum_{j = 1}^{m} σ_{j} (X_{t}) Δ W^{j} + \sum_{i, j = 1}^{m} L_{i} σ_{j} (X_{t}) I_{i j} + R_{t, h},$ where $Δ W^{j} = W_{t + h}^{j} - W_{t}^{j}$ and $I_{i j} = \int_{t}^{t + h} (W_{s}^{i} - W_{t}^{i}) d W_{s}^{j}$ . Moreover, for the conditioning sigma algebra $F_{t}$ there is a conditional-expectation version $r_{t, h} = E [R_{t, h} ∣ F_{t}]$ with $∣ r_{t, h} ∣ \leq C h^{2}, E [∣ R_{t, h} ∣^{2} ∣ F_{t}] \leq C h^{3} almost surely,$ where $C$ depends only on $d, m$ and the coefficient bounds, not on $t, h$ or $X_{t}$ .

Facts & Assumptions

Given: AC, a Brownian It^o solution on a usual filtered probability space, the coefficient bounds in the Statement, and $0 < h \leq 1$ .

[L1]

Milstein updates, iterated Brownian integrals, and commutative noise: $I_{i j} = \int_{t}^{t + h} (W_{s}^{i} - W_{t}^{i}) d W_{s}^{j}$ , $L_{i} f = D f σ_{i}$ , and $C_{b}^{3}$ means all derivatives through order three, including the function itself, are bounded and continuous.

[L2]

Second and fourth moments of bounded-coefficient SDE increments: for $s \in [t, t + h]$ and every $A \in F_{t}$ , $E [1_{A} ∣ X_{s} - X_{t} ∣^{2}] \leq C (s - t) P (A)$ .

[F1]

Continuous Brownian Ito processes: $X$ has the drift-plus-It^o-integral decomposition, Brownian increments after $t$ are independent of $F_{t}$ , and finite-energy It^o integrals are martingales.

[F2]

Multidimensional Ito formula for Brownian-driven processes: for $f \in C^{1, 2}$ , $d f (X) = L_{0} f (X) d t + \sum_{i} L_{i} f (X) d W^{i}$ .

[F3]

Event-weighted Ito isometry after a conditioning time: for $A \in F_{t}$ , each scalar It^o integral over $(t, t + h]$ satisfies $E [1_{A} ∣ \int H, d W^{j} ∣^{2}] = E [1_{A} \int ∣ H ∣^{2} d s]$ .

[F4]

The Ito integral process has a continuous martingale version: a finite-energy It^o integral increment has conditional mean zero at its start time.

[F5]

Conditional expectation given a sigma algebra: a conditional expectation version has the same integrals as the original variable on every event in the conditioning sigma algebra.

[F6]

Conditional expectation exists by radon nikodym and Basic algebra and order properties of conditional expectation: under AC the conditional expectation versions used below exist, are linear, and preserve expectation and order.

[A1]

The Axiom of Choice: AC is used only through the declared Brownian, It^o, and conditional-expectation interfaces [F1]--[F6].

Proof

Given: The assumptions above.

1.1L1F1F2A1

Apply [F2] to $f = σ_{j}$ on $[t, s]$ . With the operators in [L1], $σ_{j} (X_{s}) = σ_{j} (X_{t}) + \int_{t}^{s} (L_{0} σ_{j}) (X_{r}) d r + \sum_{i} \int_{t}^{s} (L_{i} σ_{j}) (X_{r}) d W_{r}^{i} .$ Integrating this identity against $d W_{s}^{j}$ , summing over $j$ , and substituting the SDE decomposition of $X_{t + h} - X_{t}$ isolates the constant coefficient terms. Thus the remainder is exactly $R_{t, h} = R_{b} + R_{0} + R_{1},$ where $R_{b} = \int_{t}^{t + h} (b (X_{s}) - b (X_{t})) d s$ , $R_{0} = \sum_{j} \int_{t}^{t + h} (\int_{t}^{s} (L_{0} σ_{j}) (X_{r}) d r) d W_{s}^{j}$ , and $R_{1} = \sum_{i, j} \int_{t}^{t + h} (\int_{t}^{s} ((L_{i} σ_{j}) (X_{r}) - (L_{i} σ_{j}) (X_{t})) d W_{r}^{i}) d W_{s}^{j}$ . The subtraction in $R_{1}$ leaves precisely $\sum_{i, j} (L_{i} σ_{j}) (X_{t}) I_{i j}$ .

1.2L2F2F4F5F6A1

Since $L_{0} b$ is bounded, apply [F2] to $b (X_{s})$ and take event integrals over $A \in F_{t}$ . The stochastic increments have zero event integral by [F4]; componentwise this gives $∣ E [1_{A} R_{b}] ∣ \leq ∥ L_{0} b ∥_{\infty} h^{2} P (A) / 2$ . By [F5]--[F6], the vector conditional expectation $r_{b} = E [R_{b} ∣ F_{t}]$ , formed coordinatewise, exists. If any coordinate exceeded its bound on an $F_{t}$ event, the defining event-integral identity [F5] would contradict the bound just proved, so $∣ r_{b} ∣ \leq C h^{2}$ after increasing $C$ by a dimension factor. Also, the pathwise inequality $∣ \int_{t}^{t + h} g (s) d s ∣^{2} \leq h \int_{t}^{t + h} ∣ g (s) ∣^{2} d s$ follows by applying $0 \leq \int_{t}^{t + h} (∣ g ∣ - a)^{2} d s$ with $a = h^{- 1} \int_{t}^{t + h} ∣ g ∣ d s$ ; [L2] therefore gives $E [1_{A} ∣ R_{b} ∣^{2}] \leq h ∥ D b ∥_{\infty}^{2} \int_{t}^{t + h} C (s - t) P (A) d s \leq C h^{3} P (A)$ .

1.3F3F4A1

Each $L_{0} σ_{j}$ is bounded. Therefore the inner integral in $R_{0}$ has magnitude at most $C (s - t)$ . Its outer It^o integrals have zero conditional mean by [F4]. Apply the scalar event-weighted identity [F3] to each component of each outer vector integral and use $∣ \sum_{j} v_{j} ∣^{2} \leq m \sum_{j} ∣ v_{j} ∣^{2}$ ; this gives $E [1_{A} ∣ R_{0} ∣^{2}] \leq C \int_{t}^{t + h} (s - t)^{2} d s P (A) \leq C h^{3} P (A)$ .

1.4L1L2F3F4A1

Each $L_{i} σ_{j}$ is globally Lipschitz: its derivative is a sum of products of first and second derivatives of the $σ_{k}$ , bounded by [L1]. The inner integral $J_{i j} (s) = \int_{t}^{s} ((L_{i} σ_{j}) (X_{r}) - (L_{i} σ_{j}) (X_{t})) d W_{r}^{i}$ has zero conditional mean and, by [F3] coordinatewise and [L2], satisfies $E [1_{A} ∣ J_{i j} (s) ∣^{2}] \leq C \int_{t}^{s} E [1_{A} ∣ X_{r} - X_{t} ∣^{2}] d r \leq C (s - t)^{2} P (A)$ . Each outer integral has zero conditional mean by [F4]; applying the scalar identity [F3] componentwise and $∣ \sum_{i j} v_{i j} ∣^{2} \leq m^{2} \sum_{i j} ∣ v_{i j} ∣^{2}$ gives $E [1_{A} ∣ R_{1} ∣^{2}] \leq C h^{3} P (A)$ .

2.1F5F6step 1.2step 1.3step 1.4

The sum $R_{0} + R_{1}$ has conditional mean zero, while $R_{b}$ has conditional mean $r_{b}$ from step 1.2; set $r_{t, h} = r_{b}$ . The inequality $∣ x + y + z ∣^{2} \leq 3 (∣ x ∣^{2} + ∣ y ∣^{2} + ∣ z ∣^{2})$ and steps 1.2--1.4 show $E [1_{A} ∣ R_{t, h} ∣^{2}] \leq C h^{3} P (A)$ for every $A \in F_{t}$ . By the defining event-integral property [F5] and order property [F6], this is exactly $E [∣ R_{t, h} ∣^{2} ∣ F_{t}] \leq C h^{3}$ almost surely.

3.1A1step 1.1step 1.2step 2.1∎

Steps 1.2 and 2.1 give the conditional mean and second-moment bounds in the Statement, and step 1.1 gives the exact Taylor expansion. Every constant is uniform in the starting state because the coefficients and derivatives are bounded. AC enters only through [A1].

Depends on

Used by

Strong order one for scalar and commutative-noise Milstein schemes Theorem

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Sources

Kloeden and Platen, Numerical Solution of Stochastic Differential Equations, Sections 5.5, 5.7, 10.3, and 10.6, printed pp. 180-190, 345-351, and 356-365 (standard reference, not scraped)
Desmond J. Higham, An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations, SIAM Review 43 (2001), Section 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)