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Second, the inventory of the investor’s stock account at time t ≥ 0, Nt ∈ N, does not change between the trading times and can be expressed as follows: Q(t) Nt = Nτ (i) = n(k)1τ (k) k=−∞ if τ (i) ≤ t < τ (i + 1) , i = 0, 1, . . 46) where Q(t) = sup{k ≥ 0 | τ (k) ≤ t}. 47) Nτ (i) = Nτ (i)− ⊕ ζ(i), where Nτ (i)− ⊕ ζ(i) : (−∞, 0] → N is defined, for θ ∈ (−∞, 0], by (Nτ (i)− ⊕ ζ(i))(θ) = ∞ k=0 n ˆ (i − k)1{τ (i−k)} (τ (i) + θ) = m(i)1{τ (i)} (τ (i) + θ) + ∞ k=1 n(i − k) +m(i − k)(1{n(i−k)<0,0≤m(i−k)≤−n(i−k)} +1{n(i−k)>0,−n(i−k)≤m(i−k)≤0} ) 1{τ (i−k)} (τ (i) + θ).

T (N ) } is said to be a discrete admissible control if u(N) (kh(N) ) is F(kh(N))-measurable for each k = 0, 1, 2, . . , T (N ) , and ⎤ ⎡ (N ) T E⎣ k=0 2 u(N ) (kh(N ) ) ⎦ < ∞. 1, we let U (N ) [0, T ] be the class of continuous-time admissible control process u ¯(·) = {¯ u(s), s ∈ [0, T ]}, where for each s ∈ [0, T ], u ¯(s) = u ¯(⌊s⌋N ) is F(⌊s⌋N )-measurable and takes only finite different values in U . Given an one-step Markov transition functions p(N ) : (S(N ) )N +1 × U × (N ) N +1 (S ) → [0, 1], where p(N ) (x, u; y) shall be interpreted as the probability that the ζ(k+1)h = y ∈ (S(N) )N +1 given that ζkh = x and u(kh) = u, where h = h(N ) .

4 for the value function under the condition that it is sufficiently smooth. However, it is known in most of optimal control problems, deterministic or stochastic, that the value functions do not meet these smoothness conditions and therefore cannot be a solution to the HBJE in the strong sense. 5. 6, it is shown that the value function is the unique viscosity solution to the HJBE. 7. 8. Chapter 4. This chapter formulates and provides a characterization of the value function for the optimal stopping problem briefly described in B2.

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