Numerics and Convergence

Example 1

Consider 1D SDE, $X(t)\in \mathbb{R}$ with $t\in [0,T]$ and

$$dX=u(X)dt+\sqrt{2D(X)}dBX(t)=x_0$$

We can use the Euler-Maruyama method to simulate this.

1. Take $n\in {\mathbb{Z}}_{>0}$ and set $\delta t=T/n$. Let $t_i=i\delta t$.
2. For $i=0,\dots ,n$ so $t_n=T$,
   1. Generate $Z_i\sim \mathcal{N}(0,1)$, the standard normal, iid with other $i$‘s
   2. Define $Y_0^{\delta t},Y_1^{\delta t},\dots ,Y_n^{\delta t}$ by:
      1. $Y_0^{\delta t}=x_0$
      2. $Y_{i+1}^{\delta t}=Y_i^{\delta t}+\delta tu\left(Y_i^{\delta t}\right)+\sqrt{\delta t}\sqrt{2D(Y_i^{\delta t})}\cdot Z_i$
3. Return $Y^{\delta t}$.

Here, $Y_i^{\delta t}$ approximates $X(t_i)=X(i\delta t)$, the continuous solution. In fact, $Y_i^{\delta t}$ converges to the true solution. We want to focus on time $T$, where $Y_n^{\delta t}\to X(T)$.

Definition (Discretized Strong Convergence)

In the above example, we say $Y_n^{\delta t}$ converges strongly to $X$ at time $T$ with order $\beta$ if

$$\mathbb{E}\left[|Y_n^{\delta t}-X(T)|\right]=O\left(\delta t^{\beta }\right)$$

as $\delta t\to 0,n\to \infty$, and $n\delta t=T$ fixed (for all function in this function class)

Remarks:

• Strong convergence: convergence of sample paths for fixed realization of $B(t)$.
• Weak: convergence of distributions.
• Typically, we care about weak convergence.
• Euler-Maruyama Method has strong order $1/2$ and weak order $1$.