# Bifurcation of Extremals in Optimal Control by Jacob Kogan By Jacob Kogan

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Optimal stopping: General facts Sn = ξ1 + · · · + ξn for n ≥ 1 ; X0 = x , Xn = x + Sn for n ≥ 1 , and M = supn≥0 Sn . Let Px be the probability distribution of the sequence (Xn )n≥0 with X0 = x from R . It is clear that the sequence (Xn )n≥0 is a Markov chain started at x . 75) τ ∈M where the supremum is taken over the class M of all Markov (stopping) times τ satisfying Px (τ < ∞) = 1 for all x ∈ R . 76) ¯ of all Markov times. where the supremum is taken over the class M The problem of ﬁnding the value functions Vn (x) and V¯n (x) is of interest for the theory of American options because these functions represent arbitrage-free (fair, rational) prices of “Power options” under the assumption that any exercise ¯ respectively.

18) Section 1. Discrete time 15 for all 0 ≤ n ≤ N . 9). 2 above. The proof is complete. 4. 9) can be written in a more compact form as follows. 19) for x ∈ E where F : E → R is a measurable function for which F (X1 ) ∈ L1 (Px ) for x ∈ E . 20) for 1 ≤ n ≤ N where Q denotes the n -th power of Q . 20) form a constructive method for ﬁnding V N when Law(X1 | Px ) is known for x ∈ E . n 5. Let us now discuss the case when X is a time-inhomogeneous Markov chain. Setting Zn = (n, Xn ) for n ≥ 0 one knows that Z = (Zn )n≥0 is a timehomogeneous Markov chain.

S. s. as well as Vn = ∞ for all n ≥ 0 . 52) are strict. 6. 3) holds. 52) hold for all n ≥ 0 . Proof. 53) 12 Chapter I. Optimal stopping: General facts for all n ≥ 0 . In particular, it follows that (Sn∞ )n≥0 is a supermartingale. s. s. 3) we see that ((Sn∞ )− )n≥0 is uniformly integrable. 54) for all τ ∈ Mn . s. s. 55) for all τ ∈ Mn . s. for all n ≥ 0 . s. for all n ≥ 0 . s. for all n ≥ 0 . Finally, the third identity Vn∞ = Vn follows by the monotone convergence theorem. The proof of the theorem is complete.