Convergence of the Euler Scheme for a Class of Stochastic ...

for s ∈ [0, τ ∧ T 1 ]. Substituting (3.2), (3.3) and (3.4) into (3.1) then reveals
that
[
]
E sup | x ∆t (t) − x(t) | 2
0≤t≤τ∧T 1
≤
2K 1 (D)(T + 4) E
= 2K 1 (D)(T + 4) E
≤
≤
4K 1 (D)(T + 4) E
4K 1 (D)(T + 4) E
∫ τ∧T1
0
∫ τ∧T1
0
∫ τ∧T1
0
∫ τ∧T1
∫ T1
+4K 1 (D)(T + 4)
0
0
E
| ˆx ∆t (s) − x(s) | 2 ds
| ˆx ∆t (s) − x ∆t (s) + x ∆t (s) − x(s) | 2 ds
(
| ˆx∆t (s) − x ∆t (s) | 2 + | x ∆t (s) − x(s) | 2) ds
| ˆx ∆t (s) − x ∆t (s) | 2 ds
[
]
sup | x ∆t (s ′ ) − x(s ′ ) | 2 ds. (3.5)
0≤s ′ ≤τ∧s
Bounding the first term on the right-hand side of (3.5) and then applying
the Gronwall inequality leads to a bound on E [ sup 0≤t≤τ∧T | x ∆t (t) − x(t) | 2] .
Inspection of (2.4) reveals that ˆx ∆t (s) = x ∆t ([s/∆t]∆t) where [s/∆t] is the
integer part of s/∆t. We can now use (2.3) to show that
| ˆx ∆t (s)−x ∆t (s)| 2 =|x ∆t ([s/∆t]∆t)−x ∆t (s)| 2
= |
∫ s
[s/∆t]∆t
f(x ∆t ([s/∆t]∆t))du +
∫ s
[s/∆t]∆t
g(x ∆t ([s/∆t]∆t))dB(u)| 2
≤ 2 |f(x ∆t ([s/∆t]∆t))| 2 ∆t 2 + 2|g(x ∆t ([s/∆t]∆t))| 2 |B(s) − B([s/∆t]∆t)| 2
≤ 2K 2 (D)∆t 2 + 2K 2 (D) |B(s)−B([s/∆t]∆t)| 2 .
Note that the last line follows provided s ∈ [0, τ ∧T 1 ] and condition (2.2) holds.
If T ∆t < 1 this inequality leads to
∫ τ∧T1
E
0
| ˆx ∆t (s)−x ∆t (s)| 2 ds ≤ 2K 2 (D)T 1 ∆t 2
≤
∫ T1
+2K 2 (D) E |B(s)−B([s/∆t]∆t)| 2 ds
0
2K 2 (D)T ∆t 2 + 2K 2 (D)T 1 m∆t
≤ 2K 2 (D)(mT + 1)∆t (3.6)
(recall that our Brownian motion has dimension m). Using this result in (3.5)
shows that
[
]
E sup |x ∆t (t)−x(t)| 2 ≤ C 1 (D)∆t
0≤t≤τ∧T 1
∫ [
T1
+ C 2 (D) E
0
]
sup |x ∆t (r)−x(r)| 2 ds
0≤r≤τ∧s