30K (Stochastic Financial Models)

Let

V (n, x)

be the minimum expected remaining cost after time

n

, given

X_{n} = x

. The terminal condition is

V (N, x) = x^{2}

V (n - 1, x) = u \in R in f {u^{2} + E [V (n, x + u + ξ_{n})]} .

b

V (n, x) = A_{n} + B_{n} x + C_{n} x^{2}

. The terminal values are

(A_{N}, B_{N}, C_{N}) = (0, 0, 1)

. Since the noise is centred, backward induction keeps

B_{n} = 0

u^{*} = - \frac{C _{n}}{1 + C _{n}} x, qq u a d C_{n - 1} = \frac{C _{n}}{1 + C _{n}}, qq u a d A_{n - 1} = A_{n} + C_{n} σ^{2} .

Hence

C_{n} = \frac{1}{N - n + 1}, B_{n} = 0, A_{n} = σ^{2} j = n + 1 \sum N \frac{1}{N - j + 1}

(with the empty sum equal to zero). This proves the asserted quadratic form of the value function.

n

, the minimizing feedback from part (b) is

u_{n}^{*} = - \frac{X _{n - 1}}{N - n + 2} .

Starting with

u_{1}^{*} = - X_{0} / (N + 1)

and repeatedly substituting

X_{j} = X_{j - 1} + u_{j}^{*} + ξ_{j}

gives by induction

X_{n - 1} = \frac{N - n + 2}{N + 1} X_{0} + \sum_{j = 1}^{n - 1} \frac{N - n + 2}{N - j + 1} ξ_{j} .

Therefore

u_{n}^{*} = - \frac{X _{0}}{N + 1} - \frac{ξ _{1}}{N} - \frac{ξ _{2}}{N - 1} - \dots - \frac{ξ _{n - 1}}{N - n + 2},

as required.

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