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A simple, one-dimensional example indicating the lack of uniqueness is 0 = − −xWx + x2 + 18 Wx2 . There are two C∞ solutions with W (0) = 0: √ √ W 1 (x) = (2 − 2)x2 and W 2 (x) = (2 + 2)x2 , and an infinite number of viscosity solutions such as √ if x ≤ 1 (2 − √2)x2 √ W (x) = (2 + 2)x2 − 2 2 if x > 1. 22). As with the finite time-horizon case, there will be two major parts to the proof of the above statement. Here, we will start with what is referred to as a verification theorem — which states that a solution of the PDE must be the value function.

34) This is an upper bound on the size of ε-optimal u which is independent of T (using e−cf T ≤ 1). 2 Viscosity Solutions In the previous section, we concentrated on the relationship between the DPP and the control problem value function for the example problem classes we will concentrate on. As noted earlier, the DPE is obtained by an infinitesimal limit in the DPP, and takes the form of a nonlinear, first-order Hamilton–Jacobi– Bellman PDE (HJB PDE) in these problem classes. In the finite time-horizon problem, it is a time-dependent PDE over (s, T ) × Rn with terminal-time boundary data.

2 below and [88]) 0 ≤ W (x) ≤ cf γ2 − δ 2 |x| 2m2σ ∀ x ∈ Rn . 23) We now indicate the more specific DPP that one can obtain in this context. 10. 13). Let δ > 0 be sufficiently small . 23) holds, and such that with γ 2 = γ 2 − δ one still has the inequality (γ 2 c2f )/(2m2σ αl ) > 1. Then for any ε > 0, for all x ∈ Rn T W (x) = sup U,ε,|x| u∈U0,T where U,ε,|x| U0,T . 25) . δ δ 2m2σ cf Proof. The following proof is adapted from [88]. 21), T W (x) = sup U u∈U0,T 0 l(ξr ) − γ2 |ur |2 dr + W (ξT ) 2 .