📂Stochastic Differential Equations

Milstein Method Derivation

Method ¹

$$d X(t) = f \left( X_{t} \right) dt + g \left( X_{t} \right) d W_{t} \qquad , t \in [t_{0}, T]$$ Given that the Ito process is a solution to the autonomous stochastic differential equation described above. For equally spaced intervals of $h$, the calculation expressed for the equally spaced points $\left\{ t_{i} \le T : t_{i+1} = t_{i} + h \right\}_{i=0}^{N}$ is a numerical solution to the given differential equation. $$Y_{i+1} = Y_{i} + f \left( Y_{i} \right) h + g \left( Y_{i} \right) \sqrt{h} Z + {{ 1 } \over { 2 }} g \left( Y_{i} \right) {{ d g \left( Y_{i} \right) } \over { d y }} h \left( Z^{2} - 1 \right)$$ Here, $Z$ is a random variable following the standard normal distribution.

Convergence

This solution converges strongly to the $\gamma = 1$th order, and weakly to the $\beta = 1$th order.

Explanation

Milstein 2nd-order Approximation Scheme is a method that improves accuracy by incorporating a 2nd-order correction term into the Euler-Maruyama scheme. The formula might look complex due to the many indices, but it can be neatly written as the following by omitting some parts. $$X_{t + h} = X_{t} + f_{t} h + g_{t} \sqrt{h} Z + {{ 1 } \over { 2 }} g_{t} g_{t}’ h \left( Z^{2} - 1 \right)$$

Derivation

$$\begin{align*} f_{t} &:= f \left( X_{t} \right) \\ g_{t} &:= g \left( X_{t} \right) \end{align*}$$ For convenience, let’s designate it as above. Since $f$ and $g$ are independent of time $t$, it follows that $df / dt = dg / dt = 0$, and if we set it as $f ' (x)$, it represents the derivative of $f$ with respect to $x$.

Ito’s Formula: Assume that the Ito process $\left\{ X_{t} \right\}_{t \ge 0}$ is given. $$d X_{t} = u dt + v d W_{t}$$ If we set a function $V \left( t, X_{t} \right) = V \in C^{2} \left( [0,\infty) \times \mathbb{R} \right)$ such that $Y_{t} := V \left( t, X_{t} \right)$, then $\left\{ Y_{t} \right\}$ is also an Ito process, and the following holds. $$\begin{align*} d Y_{t} =& V_{t} dt + V_{x} d X_{t} + {{ 1 } \over { 2 }} V_{xx} \left( d X_{t} \right)^{2} \\ =& \left( V_{t} + V_{x} u + {{ 1 } \over { 2 }} V_{xx} v^{2} \right) dt + V_{x} v d W_{t} \end{align*}$$

$$d X(t) = f \left( X_{t} \right) dt + g \left( X_{t} \right) d W_{t}$$ Let’s calculate $d f_{t}$ using the Ito formula from the given Ito process. If we set it as $V = f$, according to the Ito formula $$\begin{align*} d f_{t} =& d f \left( X_{t} \right) \\ =& \left( {{ \partial f_{t} } \over { \partial t }} + {{ \partial f_{t} } \over { \partial x }} f_{t} + {{ 1 } \over { 2 }} {{ \partial^{2} f_{t} } \over { \partial x^{2} }} g_{t}^{2} \right) dt + {{ \partial f_{t} } \over { \partial x }} g_{t} d W_{t} \\ =& \left( 0 + f_{t}’ f_{t} + {{ 1 } \over { 2 }} f_{t}’’ g_{t}^{2} \right) dt + f_{t}’ g_{t} d W_{t} \end{align*}$$ it follows. Similarly, if we define $V = g$ and calculate $d g_{t}$, $$\begin{align*} d g_{t} =& g f \left( X_{t} \right) \\ =& \left( {{ \partial g_{t} } \over { \partial t }} + {{ \partial g_{t} } \over { \partial x }} f_{t} + {{ 1 } \over { 2 }} {{ \partial^{2} g_{t} } \over { \partial x^{2} }} g_{t}^{2} \right) dt + {{ \partial g_{t} } \over { \partial x }} g_{t} d W_{t} \\ =& \left( 0 + g_{t}’ f_{t} + {{ 1 } \over { 2 }} g_{t}’’ g_{t}^{2} \right) dt + g_{t}’ g_{t} d W_{t} \end{align*}$$ it follows. If we convert it to the integral form from $t$ to $s$, $$\begin{align*} f_{s} =& f_{t} + \int_{t}^{s} \left( f_{u}’ f_{u} + {{ 1 } \over { 2 }} f_{u}’’ g_{u}^{2} \right) du + \int_{t}^{s} f_{u}’ g_{u} d W_{u} \\ g_{s} =& g_{t} + \int_{t}^{s} \left( g_{u}’ f_{u} + {{ 1 } \over { 2 }} g_{u}’’ g_{u}^{2} \right) du + \int_{t}^{s} g_{u}’ g_{u} d W_{u} \end{align*}$$ it follows. If we substitute this into the integral form of the Ito process $$X_{t+h} = X_{t} + \int_{t}^{t+h} f_{s} ds + \int_{t}^{t+h} g_{s} d W_{s}$$ we obtain the following. $$\begin{align*} X_{t+h} =& X_{t} + \int_{t}^{t + h} \left[ f_{t} + {\color{Red} \int_{t}^{s} \left( f_{u}’ f_{u} + {{ 1 } \over { 2 }} f_{u}’’ g_{u}^{2} \right) du + \int_{t}^{s} f_{u}’ g_{u} d W_{u} } \right] ds \\ & + \int_{t}^{t + h} \left[ g_{t} + {\color{Red} \int_{t}^{s} \left( g_{u}’ f_{u} + {{ 1 } \over { 2 }} g_{u}’’ g_{u}^{2} \right) du} + \int_{t}^{s} g_{u}’ g_{u} d W_{u} \right] d W_{s} \end{align*}$$

Ito’s Multiplication Table: The product of $dt$ and $d W_{t}$ is as follows. $$\begin{align*} \left( dt \right)^{2} =& 0 \\ dt d W_{t} =& 0 \\ d W_{t} dt =& 0 \\ \left( d W_{t} \right)^{2} =& dt \end{align*}$$

The part marked in red according to the Ito’s multiplication table is all $0$. As a result, only the term treating $f_{t}$ and $g_{t}$ as constants and the double integral of the integrand $dW_{u} dW_{s}$ remain and can be written as follows. $$X_{t+h} = X_{t} + f_{t} \int_{t}^{t + h} ds + g_{t} \int_{t}^{t + h} d W_{s} + \int_{t}^{t + h} \int_{t}^{s} g_{u} g_{u}’ d W_{u} d W_{s}$$ The last term $\int_{t}^{t + h} \int_{t}^{s} g_{u} g_{u}’ d W_{u} d W_{s}$ is approximately calculated as follows for a random variable $Z \sim N (0,1)$ following the standard normal distribution, according to the corollary of Ito’s formula $$\begin{equation} \int_{a}^{b} W_{s} d W_{s} = {{ 1 } \over { 2 }} \left[ W_{b}^{2} - W_{a}^{2} \right] - {{ 1 } \over { 2 }} (b-a) \end{equation}$$ and the normality of the increments of the Wiener process, i.e., $$\begin{equation} \left( W_{t+h} - W_{t} \right) \sim \sqrt{h} N \left( 0, 1 \right) \end{equation}$$ Thus, the following is derived. $$\begin{align*} & \int_{t}^{t + h} \int_{t}^{s} g_{u} g_{u}’ d W_{u} d W_{s} \\ \approx& g_{t} g_{t} ' \int_{t}^{t+h} \int_{t}^{s} d W_{u} d W_{s} \\ =& g_{t} g_{t} ' \int_{t}^{t+h} \left( W_{s} - W_{t} \right) d W_{s} \\ =& g_{t} g_{t} ' \left[ \int_{t}^{t+h} W_{s} d W_{s} - W_{t} \left( W_{t+h} - W_{t} \right) \right] \\ =& g_{t} g_{t} ' \left[ \int_{t}^{t+h} W_{s} d W_{s} - W_{t} W_{t+h} + W_{t}^{2} \right] \\ =& g_{t} g_{t} ' \left[ {{ W_{t + h}^{2} } \over { 2 }} - {{ W_{t}^{2} } \over { 2 }} - {{ h } \over { 2 }} - W_{t} W_{t+h} + W_{t}^{2} \right] & \cdots (1) \\ =& {{ 1 } \over { 2 }} g_{t} g_{t} ' \left[ W_{t + h}^{2} + W_{t}^{2} - h - 2 W_{t} W_{t+h} \right] \\ =& {{ 1 } \over { 2 }} g_{t} g_{t} ' \left[ \left( W_{t + h} - W_{t} \right)^{2} - h \right] \\ =& {{ 1 } \over { 2 }} g_{t} g_{t} ' \left[ h Z^{2} - h \right] & \cdots (2) \\ =& {{ 1 } \over { 2 }} g_{t} g_{t}’ h \left( Z^{2} - 1 \right) \end{align*}$$

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