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Consider the optimal stopping problem

Consider the optimal stopping problem T 'jr-{mj = sup E.\" [139. 'P'Xtdt + (\"Xr] , o Where ng : [IX-it + XtdBt ; Kg 2 I 23' U is geometric Brownian motion (Ct,,l3,3 constants, 6' > D). We put tit l D dYg [dXt] _[o:X] dt+[xt]dBt, i'(s,a:) and dY 1 B dZ= [JWE] = GK: fit-F 3X: ng; Zu=(5,ar:,w). t {3th U Then with y) = ew) = 639552, gm) = 3"\";- and 6(315'3119] = 9(3v3) +10 = Cnpl' + u} we have _BG 30'122320 _m3G_ _m AEG as +Q$B$ . 2.5 I 332 +33 IE (p+os+tl)e 3:. Hence 3 0 U = {(s,z,w);AzG-:s,a:,w) > 0} = {5' $2213. Ftom this we conclude {see Exercise 10.6]: przo+3thenr*={} and G'[s,:r:,w:l = G(s,z,w} : 3"\"..1 + w. [10.3.8] If o: