By Kohn R.V.
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G. John). But let’s be less ambitious: let’s just show that the numerical solution converges to the solution of the PDE as ∆x and ∆t tend to 0 while obeying the stability restriction (1). e. g(t + ∆t, x) − g(t, x) → gt ∆t and as ∆t → 0 g(t, x + ∆x) − 2g(t, x) + g(t, x − ∆x) → gxx (∆x)2 as ∆x → 0 if g is smooth enough. Let f be the numerical solution, g the PDE solution, and consider h = f − g evaluated at gridpoints. Consistency gives h((j + 1)∆t, k∆x) = ∆t ∆t (∆x)2 h(j∆t, (k + 1)∆x) + (∆x)2 h(j∆t, (k − 1)∆x) ∆t +(1 − 2 (∆x) 2 )h(j∆t, k∆x) + ∆te(j∆t, k∆x) with |e| uniformly small as ∆x and ∆t tend to zero.
T Its value function satisfies the HJB equation ut + H(∇u, x) = 0 for t < T, u(x, T ) = g(x), with Hamiltonian H(p, x) = max{f (x, a) · p + h(x, a)}. a∈A (1) Let us show (heuristically) that when the state is perturbed by a little noise, the value function of resulting stochastic control problem solves the perturbed HJB equation ut + H(∇u, x) + where H is still given by (1), and ∆u = 1 2 ∆u = 0 2 (2) ∂2u i ∂x2 . i Our phrase “perturbing the state by a little noise” means this: we replace the ODE governing the state by the stochastic differential equation (SDE) dy = f (y, α)ds + dw, 1 keeping the initial condition y(t) = x.
We did all the work already in Section 2: the argument given there shows that the solution is t u(x, t) = 0 ∂G (x, 0, t − s)φ0 (s) ds − ∂y 2 t 0 ∂G (x, 1, t − s)φ1 (s) ds ∂y where G is given by (6). In particular, ∂G ∂y (x, 0, t) is the probability that the random walker, starting from x, hits the boundary first at 0 and arrives there at time t. Similarly, − ∂G ∂y (x, 1, t) is the probability that the random walker, starting from x, hits the boundary first at 1 and arrives there at time t. The “separation of variables” solution method just presented is not limited to the constantcoefficient heat equation.