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¬,)]).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![10.4. Let Xt be an Itô diffusion on R" and g: R" → R+ a continuous reward
function. Define
9° (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,... and
consider
E" [g(X;) · X,c«] < E"[ lim_g(X-,)]).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f84e8f7-b971-450c-a485-c719bf1153d8%2F62b21907-6478-400c-92d0-3a50e1709344%2Fsd0zu8i_processed.png&w=3840&q=75)
Transcribed Image Text:10.4. Let Xt be an Itô diffusion on R" and g: R" → R+ a continuous reward
function. Define
9° (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,... and
consider
E" [g(X;) · X,c«] < E"[ lim_g(X-,)]).
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