Let the functions b, satisfy condition (5.2.1) of Theorem 5.2.1, with a constant C independent of...

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Let the functions b, satisfy condition (5.2.1) of Theorem 5.2.1, with a constant C independent of t, i.e. |b(t, x)|+|(t, x)| C(1 + |x|) for all R and all t0. Let Xt be a solution of dxt=b(t, Xt)dt + (t, Xt)dBt. Show that E[|X|] (1+E[|Xo|])ekt 1 - for some constant K independent of t. (Hint: Use Dynkin's formula with f(x) = |x|2 and 7 = t ^TR, where R = inf {t > 0; |Xt| R}, and let R to achieve the inequality t E[|X|] E[|Xo|] + K (1+E[|X|])ds, Let the functions b, satisfy condition (5.2.1) of Theorem 5.2.1, with a constant C independent of t, i.e. |b(t, x)|+|(t, x)| C(1 + |x|) for all R and all t0. Let Xt be a solution of dxt=b(t, Xt)dt + (t, Xt)dBt. Show that E[|X|] (1+E[|Xo|])ekt 1 - for some constant K independent of t. (Hint: Use Dynkin's formula with f(x) = |x|2 and 7 = t ^TR, where R = inf {t > 0; |Xt| R}, and let R to achieve the inequality t E[|X|] E[|Xo|] + K (1+E[|X|])ds,