Statistics Homeworks
H.W. 7 - Optional theory part
Compare also with other possible simulation schemes which have been proposed
(eg. Milstein, Runge-Kutta, Heun's, ...), pointing out possible differences.
The Euler-Murayama (E-M) method formula is the following:
\[ X_{n+1} - X_{n} = \mu X_n\Delta t_n + \sigma X_n \Delta B_n\]
In addition to it, which was used in the homeworks, there are different numerical methods for solving the stochastic differential equations. Let's see them in the next sections.
Milstein method
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The Milstein method is an extension of the Euler method which is able to increase the accuracy adding a second-order "correction" term (in green) by applying Ito's
lemma to the \(a()\) and \(b()\) functions:
\[X_{n+1} - X_n = a(X_n)\Delta t + b(X_n)\Delta B_n \color{green}{+ 0.5\sigma^2 X_n\left((\Delta B_n)^2-\Delta t\right)}\]
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It has both strong and weak convergence of order 1 whereas the E-M method has weak convergence of order 1, but strong convergence only of order 0.5.
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In general, it is more stable than Euler for SDEs with a high degree of noise.
Runge-Kutta methods
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R-K method can applies higher order method to SDEs like the fourth-order Runge-Kutta "RK4", which is defined using the
following recursion formula:
\[ y_{n+1} = y_n + \frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4) \]
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It achieves the same convergence properties as the Milstein method, but it doesn't need to compute derivates of the \(b(\cdot)\).
Heun's method
It is known as an improved or modified E-M method, or a similar two-stage R-K method.
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The update formula is the following:
\[ y_{n+1} = y_n + \frac{h}{2} \left( f(t_n, y_n) + f(t_{n+1}, \widetilde{y}_{n+1}) \right) \]
where \(\widetilde{y}_{n+1}\) is an intermediate estimate obtained using Euler's method:
\[ \widetilde{y}_{n+1} = y_n + h \cdot f(t_n, y_n) \]
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It converges more rapidly compared to the Euler method beacuse, being a second-order method, the error decreases
with the square of the step size \(h\). As \(h\) approaches zero, Heun's method provides more accurate results.
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Heun's method offers improved accuracy compared to the Euler method. The error is proportional to \(h^2\).