10.4. Let Xt be an Itô diffusion on R" and g: R" → R+ a continuous rewardfunction. Define9° (x) = sup{E*[g(X,)]; t stopping time, E*[r] < }.Show that g° = g*.(Hint: If 7 is a stopping time put T = TAk for k = 1,2,... andconsiderE" [g(X;) · X,c«] < E"[ lim_g(X-,)]).

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Answered: Let Xt be an Itô diffusion on R" and g:…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Let Xt be an Itô diffusion on R" and g: R" → R+ a continuous reward function. Define g°(x) = sup{E*[g(X,)]; † stopping time, E*[7] < o}. Show that g° = g*. (Hint: If 7 is a stopping time put Tk = TAk for k = 1,2,... and consider E" [g(X-) · X-c»] < E°[ lim_g(X¬,)]).

Let Xt be an Itô diffusion on R" and g: R" → R+ a continuous reward function. Define g°(x) = sup{E[g(X,)]; † stopping time, E[7] < o}. Show that g° = g*. (Hint: If 7 is a stopping time put Tk = TAk for k = 1,2,... and consider E" [g(X-) · X-c»] < E°[ lim_g(X¬,)]).