On Random Discretization for Ito Diffusion Processes

Proof. Note that(n t−1)∧N∑k=0∫ τk+1τ kb(·, ·)dW s +∫ tτ ntb(·, ·)dW sis a continuous time martingale. Hence, by the Burkholder-Davis-Gundy’sinequality,⎛[ EB1 2 ∑ N ∫ ] p⎞τk+1≤ KE ⎝ (b(s, X s ) − b(τ k , X s )) 2 ds ⎠≤ KE≤ KE≤ KEk=0τ k⎛[ ∑ N ∫ τk+1⎝k=0τ k[(1 + |X s |) 2 ∆τ k dssup (1 + |X s |) 2p · max0≤t≤1(max0≤k≤N) 1/22p∆τk.0≤k≤N ∆τ p k] p⎞⎠]By using the B-D-G’s inequality again, we have⎛[ EB2 2 ∑ N ∫ ] p⎞τk+1≤ KE ⎝ (b(τ k , X s ) − b(τ k , X τk )) 2 ds ⎠≤ KE+ KEk=0τ k⎛[ ∑ N ∫ τk+1(∫ s⎝k=0τ k⎛[ ∑ N ∫ τk+1⎝k=0τ k) 2ds] p⎞|a(t, X t )|dt ⎠τ k∣∫ ∣∣∣ s∣ ] p⎞∣∣∣ 2sup b(t, X t )dW t ds ⎠ .τ k ≤s≤τ k+1 τ kThe first term is easily estimated by KE ( max 0≤k≤N ∆τ 4p ) 1/2k and by the B-D-G’s inequality, the second term is estimated as⎛[ ∑ N ∫ τk+1∣∫ ∣∣∣ s∣ ] p⎞∣∣∣ 2E ⎝sup b(t, X t )dW t ds ⎠k=0τ k τ k ≤s≤τ k+1 τ k(∣∫ ∣∣∣ s∣ ∣∣∣ 2p )≤ E sup sup b(t, X t )dW t0≤k≤N τ k ≤s≤τ k+1 τ k(∫ s∣ ∣∣∣ 2p )= E sup∣ b(t, X t )I τns ≤t≤sdW t0≤s≤1 0( [∫ 1] p)≤ KE b 2 (t, X t )I τN ≤t≤1dt≤ KE(0sup |b(t, X t )| 2p · max ∆τ p k0≤t≤10≤k≤NSimilarly, we can estimate ||A 1 || 2 and ||A 2 || 2 .) (≤ KEmax0≤k≤N) 1/22p∆τk.4