By Kohn R.V.

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**Additional resources for Partial Differential Equations for Finance**

**Sample text**

One might worry that when the spatial dimension is greater than 1 this scheme is utterly impractical, since the number of grid points x to be considered at each time t is of order (∆x) −n in dimension n. This worry is well-founded: our scheme is impractical in higher dimensions. However 6 there are good numerical schemes for multidimensional problems. ) At the end of this step we have computed u(·, T − ∆t) as a function of space. Next Consider the problem with initial time t = T −2∆t. For any initial state x = y(t), the possible controls are now represented by a pair of vectors α(t), α(t + ∆t).

T h(y(s), α(s)) ds + g(y(T )). max (2) 0 The problem is determined by specifying the dynamics f , the initial state x, the final time T , the “running utility” h and the “final utility” g. The problem is solved by finding the optimal control α(s) for 0 < s < T and the value of the maximum. The mathematical and engineering literature often focuses on minimizing some sort of cost; the economic literature on maximizing utility. The two problems are mathematically equivalent. One needs some hypotheses on f to be sure the solution of the ODE defining y(s) exists and is unique.

This viewpoint can be used for all the optimal control problems we’ve discussed (finite-horizon, infinite-horizon, least-time, with or without discounting) but to fix ideas we concentrate on the usual finite-horizon example T h(y(s), α(s)) ds + g(y(T )) u(x, t) = max α t where the controls are restricted by α(s) ∈ A, and the state equation is dy/ds = f (y(s), α(s)) for t < s < T and y(t) = x. ) The Hamilton-Jacobi-Bellman equation in this case is ut + H(∇u, x) = 0 for t < T (8) with u(x, T ) = g(x) at t = T , where H (the “Hamiltonian”) is defined by H(p, x) = max{f (x, a) · p + h(x, a)}.