Milstein updates, iterated Brownian integrals, and commutative noise

Assume AC and the Brownian filtered-space conventions in Continuous Brownian Ito processes. Let $d,m\ge 1$ be finite integers, let $W=(W^1,\dots ,W^m)$ be the driving standard Brownian motion, and consider the autonomous It^o SDE $d{X}_t=b(X_t)dt+{\sum }_{j=1}^m{\sigma }_j(X_t)d{W}_t^j,$ where $b:{\mathbb{R}}^d\to {\mathbb{R}}^d$ and each ${\sigma }_j:{\mathbb{R}}^d\to {\mathbb{R}}^d$ is differentiable. For a differentiable vector-valued function $f$, $Df(x)$ is its Jacobian and $L_{i}f(x):=Df(x){\sigma }_i(x)$; this definition is applied componentwise when $f$ is matrix-valued.

On a deterministic grid $t_n=nh$, define $\mathrm{\Delta }{W}_n^j=W_{t_{n+1}}^j-W_{t_n}^j$ and the iterated It^o integral $I_{ij,n}:={\int }_{t_n}^{t_{n+1}}(W_s^i-W_{t_n}^i)d{W}_s^j,1\le i,j\le m\mathrm{.}$ The integrand is predictable and square-integrable, so this is the $L^2$ It^o integral of Ito integral for square-integrable predictable processes; its finite energy is $\mathbb{E}{\int }_{t_n}^{t_{n+1}}(W_s^i-W_{t_n}^i{)}^{2}ds=h^2\mathrm{/}2$ by Ito isometry and linearity in predictable L2.

The general Milstein Taylor update is $Y_{n+1}=Y_n+b(Y_n)h+{\sum }_{j=1}^m{\sigma }_j(Y_n)\mathrm{\Delta }{W}_n^j+{\sum }_{i,j=1}^m{L}_i{\sigma }_j(Y_n)I_{ij,n}\mathrm{.}$ The diffusion fields have commutative noise when $L_i{\sigma }_j(x)=L_j{\sigma }_i(x)$ for every $x$ and every $i,j$. Under that condition, the equivalent update using only Brownian increments is $Y_{n+1}=Y_n+b(Y_n)h+{\sum }_{j=1}^m{\sigma }_j(Y_n)\mathrm{\Delta }{W}_n^j+\frac{1}{2}{\sum }_{i,j=1}^m{L}_i{\sigma }_j(Y_n)(\mathrm{\Delta }{W}_n^i\mathrm{\Delta }{W}_n^j-{\delta }_{ij}h)\mathrm{.}$ For scalar noise ($d=m=1$), this reads $Y_{n+1}=Y_n+b(Y_n)h+\sigma (Y_n)\mathrm{\Delta }{W}_n+\frac{1}{2}\sigma (Y_n){\sigma }^{\mathrm{\prime }}(Y_n)((\mathrm{\Delta }{W}_n{)}^2-h)\mathrm{.}$

For every $1\le i,j\le m$, define the Brownian area component $A_{ij,n}:=(I_{ij,n}-I_{ji,n})\mathrm{/}2$ and the symmetric part $S_{ij,n}:=(I_{ij,n}+I_{ji,n})\mathrm{/}2$. In particular, $A_{ii,n}=0$ and $S_{ii,n}=I_{ii,n}$. Then $A_{ji,n}=-A_{ij,n}$, $S_{ji,n}=S_{ij,n}$, and reindexing the sums gives ${\sum }_{i,j}{L}_i{\sigma }_j{I}_{ij,n}=\frac{1}{2}{\sum }_{i,j}(L_i{\sigma }_j+L_j{\sigma }_i)S_{ij,n}+\frac{1}{2}{\sum }_{i,j}(L_i{\sigma }_j-L_j{\sigma }_i)A_{ij,n}\mathrm{.}$ The second sum vanishes under commutative noise; otherwise the general update includes the area components. A noncommuting Milstein implementation must sample or approximate the needed areas and account for their approximation error and computational cost; this definition makes no claim that an exact area sample is free or that a particular area sampler has a specified error rate.

For this pair, $C_b^k({\mathbb{R}}^d;{\mathbb{R}}^q)$ means that every component and every partial derivative of orders at most $k$ exists, is continuous, and is bounded. This notation is componentwise for vector and matrix valued functions.