2432 CHAPTER 71. STOCHASTIC O.D.E. ONE SPACE
arguments about to be given in order to show that this determines a mapping which has asufficiently high power a contraction map. They are also the same arguments to be usedin the following theorem to establish estimates which imply a stopping time is eventuallyinfinity.
Let v1,v2 be two given functions of this sort and let the corresponding u be denoted byu1,u2 respectively. Then
u1 (t)−u2 (t)+∫ t
0N (s,v1(s),v1 (s−h) ,w1 (s))−N (s,v2(s),v2 (s−h) ,w2 (s))ds
=∫ t
0σ (s,v1,ω)−σ (s,v2,ω)dW
Use the Ito formula and the Lipschitz condition on N to obtain an expression of the form
12|u1 (t)−u2 (t)|2−C
∫ t
0|v1− v2|2 ds−C
∫ t
0|u1−u2|2 ds
−12
∫ t
0∥σ (s,v1,ω)−σ (s,v2,ω)∥2 ds≤ |M (t)|
where M (t) is a martingale whose quadratic variation is dominated by
C∫ t
0∥σ (s,v1,ω)−σ (s,v2,ω)∥2 |u1−u2|2 ds
Therefore, using the Lipschitz condition on σ and the Burkholder-Davis-Gundy inequality,the above implies
E
(sup
s∈[0,t]|u1 (s)−u2 (s)|2
)≤ CE
∫ t
0sup
r∈[0,s]|u1 (r)−u2 (r)|2 ds
+CE∫ t
0sup
r∈[0,s]|v1 (r)− v2 (r)|2 ds
+CE
((∫ t
0∥σ (s,v1,ω)−σ (s,v2,ω)∥2 |u1−u2|2 ds
)1/2)
Then a use of Gronwall’s inequality allows this to be simplified to an expression of theform
E
(sup
s∈[0,t]|u1 (s)−u2 (s)|2
)≤CE
∫ t
0sup
r∈[0,s]|v1 (r)− v2 (r)|2 ds
+CE
((∫ t
0∥σ (s,v1,ω)−σ (s,v2,ω)∥2 |u1−u2|2 ds
)1/2)
≤C∫ t
0E
(sup
r∈[0,s]|v1 (r)− v2 (r)|2
)ds+
12
E
(sup
s∈[0,t]|u1 (s)−u2 (s)|2
)