By Kohn R.V.
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Extra info for Partial Differential Equations for Finance
Sample text
The HJB equation is |∇u| = 1 in Ω=unit square, with u = 0 at ∂Ω. The value function is defined as u(x)=minimum time of arrival to ∂Ω (among all paths with speed ≤ 1). Simple geometry tells us the solution is the distance function dist (x, ∂Ω), whose graph is a pyramid. We wish to give an entirely PDE proof of this fact. One inequality is always easy. In this case it is the relation u(x) ≤ dist (x, ∂Ω). This is clear, because the right hand side is associated with a specific control law (namely: travel straight toward the nearest point of the boundary, with unit speed).
Consider, for example, the minimum time problem). In such a case our proof of the verification theorem remains OK for paths that avoid the locus of nonsmoothness – or cross it transversely. But there is a problem if the state should happen to hug the locus of nonsmoothness. Said more plainly: if w(x, t) has discontinuous derivatives along some set Γ in space-time, and if a control makes (y(s), s) move along Γ, then the first step in our verification argument d w(y(s), s) = ws (y(s), s) + ∇w(y(s), s) · y(s) ˙ ds doesn’t really make sense (for example, the right hand side is not well-defined).
The only difference is the new term 21 2 ∆u(x, t)∆t on the right. It doesn’t depend on a, so the optimal a is unchanged – it still maximizes h(x, a) + f (x, a) · ∇u – and we conclude, as asserted, that u solves (2). ” We noted in Section 4 that the solution of the deterministic HJB equation can be nonunique. (For example, our geometric Example 2 has the HJB equation |∇u| = 1 with boundary condition u = 0 at the target; it clearly has many almost-everywhere solutions, none of them smooth). ” One characterization of the viscosity solution is this: it is the solution obtained by including a little noise in the problem formulation (with variance , as above), then taking the limit → 0.