📂Stochastic Differential Equations

Strong and Weak Convergence of Numerical Solutions to SDEs

Buildup

$$ d X_{t} = f \left( t, X_{t} \right) dt + g \left( t , X_{t} \right) d W_{t} \qquad , t \in \left[ t_{0} , T \right] $$ Given a Stochastic Differential Equation as above, let’s assume that the time is discretized as $t_{0} < t_{1} < \cdots < t_{N}$. Choosing a sufficiently large $N \in \mathbb{N}$ and setting $\Delta = \left( T - t_{0} \right) / N \in (0,1)$ turns it into equal spacing. If the solution of the SDE is $X(t)$ and its numerical approximation is $Y(T)$, then the average difference between them, $$ E \left| X(T) - Y(T) \right| $$ would be a reasonable measure of how accurate the numerical approximation is.

Definition ¹

If there exist constants $C$ and $\gamma$ that satisfy the following without depending on $\Delta$, then $Y_{\Delta}$ is said to converge strongly with order $\gamma$ towards solution $X$. $$ E \left| X(T) - Y_{\Delta} (T) \right| \le C \Delta^{\gamma} $$ If there exist a polynomial function $h$ and constants $C_{h}$, $\beta$ that satisfy the following without depending on $\Delta$, then $Y_{\Delta}$ is said to converge weakly with order $\gamma$ towards solution $X$. $$ \left| E \left( h \left( X (T) \right) \right) - E \left( h \left( Y_{\Delta} (T) \right) \right) \right| \le C_{h} \Delta^{\beta} $$

Explanation

$X(T)$ and $Y_{\Delta}(T)$ are probabilistic variables at the ending point $T$. The meaning of the expression is that, on average, when the divergence at the last point is $\Delta \to 0$, it gets closer to $0$. Thus, both adequately represent the concept of ‘convergence’.

Weak convergence is aptly named because there is room to modify the equation with the polynomial function $h$, not the solution itself. In contrast, strong convergence is the opposite concept. Generally, given some smoothing conditions on $f$ and $g$, the order of weak convergence is higher than that of strong convergence.