# 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 $$left | f & # 39; & # 39; (X_t) right |$$,

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