71.3. STOCHASTIC DIFFERENTIAL EQUATIONS 2435
Then stopping the equation with this stopping time, we can write
uτnn (t,ω)−u0(ω)+
∫ t
0X[0,τn]N (s,uτn
n (s,ω),uτnn (s−h,ω) ,wτn
n (s,ω) ,ω)ds
=∫ t
0X[0,τn] f (s,ω)ds+
∫ t
0X[0,τn]σ (s,uτn
n ,ω)dW. (71.3.23)
Then using the growth condition 71.3.20 and the Ito formula,
12|uτn
n (t)|2H ≤C (u0,w0, f )+C∫ t
0|uτn
n |2H ds+ sup
s∈[0,t]|M (t)|
where M (t) is a martingale whose quadratic variation is dominated by∫ t
0∥σ (s,uτn
n )∥2 |uτnn |
2 ds
Then it follows by the Burkholder-Davis-Gundy inequality
E
(sup
s∈[0,t]|uτn
n (s)|2H
)≤ E (C (u0,w0, f ))+C
∫ t
0E
(sup
r∈[0,s]|uτn
n (r)|2 dr
)ds
+CE
((∫ t
0∥σ (s,uτn
n )∥2 |uτnn |
2 ds)1/2
)Now apply Gronwall’s inequality and modify the constants so that
E
(sup
s∈[0,t]|uτn
n (s)|2H
)≤ E (C (u0,w0, f ))+CE
((∫ t
0∥σ (s,uτn
n )∥2 |uτnn |
2 ds)1/2
)
≤ E (C (u0,w0, f ))+12
E
(sup
s∈[0,t]|uτn
n (s)|2H
)+CE
(∫ t
0∥σ (s,uτn
n )∥2 ds)
Then, using the linear growth condition on σ , it follows on modification of the constantsagain that
E
(sup
s∈[0,t]|uτn
n (s)|2H
)≤ E (C (u0,w0, f ))+CE
(∫ t
0|uτn
n |2 ds)
≤ E (C (u0,w0, f ))+CE
(∫ t
0sup
r∈[0,s]|uτn
n |2 dr
)
and so, another application of Gronwall’s inequality implies that
E
(sup
s∈[0,T ]|uτn
n (s)|2H
)≤ E (C (u0,w0, f ))< ∞