Optimal Pair-Trading Decision Rules for a Class of Non-Linear Boundary Crossings by Ornstein-Uhlenbeck Processes Page: 26
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We will consider two cases of this threshold (boundary), as listed in the subsequent subsec-
tions.
Let us first consider the standardized OU process Zt satisfying the SDE
dZ= -Z dt + 2dWt, Zo = zo.
The above boundary for the OU process is equivalent to the boundary
(16) g(t) = /e-t - -e',
of the standardized OU process.
THEOREM 4.1. Let Tg(t),zo := inf{t > 0 : Zt g(K) Zo = zo} be the first passage time of Z
to the boundary (K) = /e-. Then the density of Tr) is given by
(-zO + f)(2e2t) _-(-z0+)2
(17) , 2(e2t _ 13/2 2(e2t -1)
PROOF. By equation 9,
P(Zt < g(t), Vt E [0, T]) = P(WS < -zo + ( 1 + s)(t(s)), Vs c [0, S])
= P(WS < -zo + ( 1 + s)/e t(s),Vs E [0, S])
= P(WS < -z0 + ( 1 + s)/e lh(1+s), Vs E [0, S])
=P(W8 < -zo+ ( 1+ s)/(1+ s)-IVs E [0, S])
= P(W, < -zo+f,Vs E [0, S])
Thus the boundary crossing probability of the standardized OU process to the boundary
g(t) = /e-t is equal to the boundary crossing probability of the standard Brownian motion
Ws to the constant boundary g(s) = -zo + 0.
Let us define the first passage time of the standard Brownian motion to the boundary
g(s) = -zo + 0 by Tg(s),zo := inf{s > 0 : s -z0 + /3 W0 = 0}, and let fg(s),o denote its
density function. It is well known that fg(s),o follows the Levy distribution and has density
(-zo + ) (-(-zo0+ /)2
fg(s),o(s) 2=w2s3/2 exp 2s 726
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Tamakloe, Emmanuel Edem Kwaku. Optimal Pair-Trading Decision Rules for a Class of Non-Linear Boundary Crossings by Ornstein-Uhlenbeck Processes, dissertation, December 2021; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc1873709/m1/45/: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .