Probability Theory – Integrability in Dynkins formula

To let

  • $ ( Omega, mathcal A, operatorname P) $ to be a complete probability space
  • $ W $ a Brownian movement $ ( Omega, mathcal A, operatorname P) $
  • $ b, sigma: mathbb R to mathbb R $ Borel be measurable with $$ | b (t, x) | ^ 2 + | sigma (t, x) | ^ 2 le C_1 (1+ | x | ^ 2) ; ; ; text {for all} t ge0 text {and} x in mathbb R tag1 $$ for some $ C_1 ge0 $ and $$ | b (t, x) -b (t, y) | ^ 2 + | sigma (t, x) – sigma (t, y) | ^ 2 le C_2 | xy | ^ 2 ; ; ; text {for all} t ge0 text {and} x, y in mathbb R tag2 $$ for some $ C_2 ge0 $
  • $ (X_t) _ {t ge0} $ be a strong solution of $$ { rm d} X_t = b (t, X_t) { rm d} t + sigma (t, X_t) { rm d} W_t tag3 $$

In addition, let $$ (L_tf) (x): = b (t, x) f ((x) + frac12 sigma ^ 2 (t, x) f & # 39; # (X) ; ; ; ; text {for} x in mathbb R text {and} f in C ^ 2 ( mathbb R) $$ to the $ t ge0 $, Now let it go $ f in C ^ 2 ( mathbb R) $, After the Ito formula $$ f (t, X_t) = f (0, X_0) + int_0 ^ t (L_sf) (X_s) : { rm d} s + underscore { int_0 ^ t sigma f & # 39; X_s) : { rm d} W_s} _ {=: : M_t} ; ; ; text {for all} t ge0 tag4. $$ Accepted, $ f & # 39; $ is limited, we get that $ M $ is a martingale.

Can we show that? $$ operator-name E left[int_0^tleft|(L_sf)(X_s)right|:{rm d}sright]< infty tag5 $$ for all $ t ge0 $?

I read here (in Corollary 6.5) that this would be the case. However, I do not see how we can bind ourselves $ left | f & # 39; & # 39; (X_t) right | $,