11.13. Consider the stochastic control problem Þ(s,x) = T --[] <-mode] = sup E¹8, U where the (1-dimensional) system X, is given by dXt=dx = (1-ut)dt + dBt. The control ut=ut(w) can assume any value in U = [0, 1] and T= inf{t> 8; X ≤0} (the time of bankruptcy). Show that if p≥ 2 then the optimal control is u = 1 for all t and the corresponding value function is Þ(s,x) = e¯Pª ² (1 − e¯√²²) ; x ≥ 0.
11.13. Consider the stochastic control problem Þ(s,x) = T --[] <-mode] = sup E¹8, U where the (1-dimensional) system X, is given by dXt=dx = (1-ut)dt + dBt. The control ut=ut(w) can assume any value in U = [0, 1] and T= inf{t> 8; X ≤0} (the time of bankruptcy). Show that if p≥ 2 then the optimal control is u = 1 for all t and the corresponding value function is Þ(s,x) = e¯Pª ² (1 − e¯√²²) ; x ≥ 0.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Stochastic control problem
![11.13. Consider the stochastic control problem
T
(s,x) = sup ES, I
= sup 5^= [] -² ude]
U
where the (1-dimensional) system X, is given by
dXt = dx = (1 - ut)dt + dBt.
The control ut = ut(w) can assume any value in U = [0, 1] and
T = inf{t> s; X ≤0} (the time of bankruptcy).
Show that if p > 2 then the optimal control is
u = 1
for all t
and the corresponding value function is
Þ(s,x) = e
-P² ²/1 (1 - e-√²P = )
x ≥ 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f84e8f7-b971-450c-a485-c719bf1153d8%2F1a1a86a3-c0c9-46b5-82cc-ac39dce13a55%2Fss5gy55_processed.png&w=3840&q=75)
Transcribed Image Text:11.13. Consider the stochastic control problem
T
(s,x) = sup ES, I
= sup 5^= [] -² ude]
U
where the (1-dimensional) system X, is given by
dXt = dx = (1 - ut)dt + dBt.
The control ut = ut(w) can assume any value in U = [0, 1] and
T = inf{t> s; X ≤0} (the time of bankruptcy).
Show that if p > 2 then the optimal control is
u = 1
for all t
and the corresponding value function is
Þ(s,x) = e
-P² ²/1 (1 - e-√²P = )
x ≥ 0.
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