Georgian Mathematical Journal Volume 10 (2003), Number 2, 237–246

DYNAMIC PROGRAMMING AND MEAN-VARIANCE HEDGING IN DISCRETE TIME

S. GUGUSHVILI

Abstract. We consider the mean-variance hedging problem in the discrete time setting. Using the dynamic programming approach we obtain recurrent equations for an optimal strategy. Additionally, some technical restrictions of the previous works are removed.

2000 Mathematics Subject Classification: 91B28, 60H30, 90C39. Key words and phrases: Mean-variance hedging, dynamic programming, optimality principle.

1. Introduction

Let (Ω, F, P) be a probability space endowed with a filtration F = (F_{n}), n = 0, 1, …, N(F_{0} is trivial, F_{N} = F) and let X = (X_{n}) be a sequence of random variables, adapted to this filtration, such that for all n, E((ΔX_{n})^{2}|F_{n−1}) < ∞. Denote by Π the set of sequences of predictable random variables π = (π_{n}), n = 1, 2, …, N, such that ∑_{i=1}^{N} π_{i}ΔX_{i} is square integrable.

For a given real number c and a square integrable random variable H ∈ F_{N} let us consider the optimization problem:

minimize E[(H − c − ∑_{i=1}^{N} π_{i}ΔX_{i})^{2} ] over all π ∈ Π.  (1.1)

The corresponding solution π^{∗} will be called an optimal strategy.

(1.1) is a discrete time analogue of the mean-variance hedging problem originally introduced by Föllmer and Sondermann [2]. Here X_{n} can be interpreted as the price of a risky asset at time n, H as a contigent claim due at time N and c as the initial capital of an investor. The set Π is then the set of all admissible trading strategies.

This problem was solved by Schweizer [7] under some additional restrictions, namely: for an arbitrary n, n = 1, …, N, X_{n} is square integrable and satisfies the so-called nondegeneracy condition. Moreover, π is admissible if and only if for each n, n = 0, 1, …, N, π_{n}ΔX_{n} is square integrable.

Melnikov and Nechaev [6] improved Schweizer’s results; they removed the non-degeneracy condition and they do not require the square integrability of π_{n}ΔX_{n} for all n, but they still require X_{n} ∈ L^{2}(P), n = 0, …, N.

Unlike the approaches of those papers, in the present work the dynamic programming method is used, which enables us to derive relatively simple, one-step backward equations for random variables a_{n}, b_{n}, defining π_{n}^{∗}, while Schweizer’s

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

or Melnikov and Nechaev’s recurrent equations for quantities β_{n}, ρ_{n} defining π_{n}^{∗} involve all previous β_{i} from n + 1 to N. Moreover, the requirement of the square integrability of X_{n}, n = 0, 1, …, is removed. The dynamic programming method has already been applied to the mean-variance hedging problem in [3], [5], but they do not cover our case as [3] deals with the diffusion model, while in [5] the asset process X, which is a semimartingale, and the filtration F are supposed to be continuous.

The paper is organized as follows: in Section 2 we introduce the value function V (n, x) corresponding to (1.1). Proposition 2.1 gives us the main working tool, the Bellman equation. In Section 3 we formulate and prove the main theorem. Section 4 is dedicated to the discussion of the obtained results. Finally, Appendix contains some results, used throughout the paper.

2. Backward Equation for V (n, x)

First let us introduce some conventions and notation:

a) If A is an empty set, then let

∑_{a∈A} Y_{a} = 0, ∏_{a∈A} Y_{a} = 1.

b) The uncertainty \frac{0}{0} = 0.

c) Relations between random variables are understood in the a.s. sense.

d) Π(n, N) denotes the set of π ∈ Π such, that π_{i} = 0 for i ≤ n.

Let us define the value function V (n, x) corresponding to (1.1) as

V (n, x) = essinf_{π∈Π(n,N)} E((H − x − ∑_{i=n+1}^{N} π_{i}ΔX_{i})^{2} | F_{n}).

Note that V (N, x) = (H − x)^{2}.

Proposition 2.1. The function V (n, x) satisfies the recurrent equation

V (n − 1, x) = essinf_{π∈Π(n−1,N)} E(V (n, x + π_{n}ΔX_{n})|F_{n−1})  (2.1)

with the boundary condition V (N, x) = (H − x)^{2}.

Proof. Due to Proposition A1 of Appendix for every fixed π̂ ∈ Π(n − 1, N) we have

V (n − 1, x) ≤ E(V (n, x + π̂_{n}ΔX_{n})|F_{n−1}).

Therefore, taking essinf, we obtain

V (n − 1, x) ≤ essinf_{π∈Π(n−1,N)} E(V (n, x + π_{n}ΔX_{n})|F_{n−1}).  (2.2)

Conversely (π̂ = (π̂_{n}) is fixed),

E(V (n, x + π̂_{n}ΔX_{n})|F_{n−1})

= E( essinf_{π∈Π(n−1,N)} E((H − x − π̂_{n}ΔX_{n} − ∑_{i=n+1}^{N} π_{i}ΔX_{i})^{2} | F_{n}) | F_{n−1})

≤ E(E((H − x − ∑_{i=n}^{N} π̂_{i}ΔX_{i})^{2} | F_{n}) | F_{n−1})

= E((H − x − ∑_{i=n}^{N} π̂_{i}ΔX_{i})^{2} | F_{n−1}).

Taking essinf of both sides, we get

essinf_{π∈Π(n−1,N)} E(V (n, x + π_{n}ΔX_{n})|F_{n−1})

≤ essinf_{π∈Π(n−1,N)} E((H − x − ∑_{i=n}^{N} π_{i}ΔX_{i})^{2} | F_{n−1}) = V (n − 1, x)

which together with (2.2) proves (2.1). □

3. Main Result

After above preliminary result, we are ready to formulate the main theorem.

Theorem 1. Assume that E((ΔX_{n})^{2}|F_{n−1}) < ∞ for all 1 ≤ n ≤ N. Then the value function V (n, x) is a square trinomial in x,

V (n, x) = a_{n}x^{2} + 2b_{n}x + c_{n},

where the F_{n}-measurable random variables a_{n}, b_{n} and c_{n} satisfy the backward recurrent equations

a_{n} = E(a_{n+1}|F_{n}) − \frac{(E(a_{n+1}ΔX_{n+1}|F_{n}))^{2}}{E(a_{n+1}(ΔX_{n+1})^{2}|F_{n})} ,  (3.1)
b_{n} = E(b_{n+1}|F_{n}) − \frac{E(a_{n+1}ΔX_{n+1}|F_{n})E(b_{n+1}ΔX_{n+1}|F_{n})}{E(a_{n+1}(ΔX_{n+1})^{2}|F_{n})} ,  (3.2)
c_{n} = E(c_{n+1}|F_{n}) − \frac{(E(b_{n+1}ΔX_{n+1}|F_{n}))^{2}}{E(a_{n+1}(ΔX_{n+1})^{2}|F_{n})} ,  (3.3)

with the boundary conditions

a_{N} = 1, b_{N} = −H, c_{N} = H^{2}.

Moreover, an optimal strategy π^{∗} = (π_{n}^{∗}) is given by

π_{n}^{∗} = −\frac{E(b_{n}ΔX_{n}|F_{n−1})}{E(a_{n}(ΔX_{n})^{2}|F_{n−1})} − (c + ∑_{i=1}^{n−1} π_{i}^{∗}ΔX_{i})\frac{E(a_{n}ΔX_{n}|F_{n−1})}{E(a_{n}(ΔX_{n})^{2}|F_{n−1})} . (3.4)

Proof. For n = N we have V (N, x) = (H − x)^{2}; hence V (N, x) is a square trinomial in x and a_{N} = 1, b_{N} = −H, c_{N} = H^{2}. Suppose that for some n, V (n, x) is a square trinomial in x

V (n, x) = a_{n}x^{2} + 2b_{n}x + c_{n},

and 0 ≤ a_{n} ≤ 1. Since V (n, x) ≥ 0, this implies that b_{n}^{2} ≤ a_{n}c_{n}, whence

(ω : a_{n} = 0) ⊂ (ω : b_{n} = 0).

Moreover, by the Hölder inequality

(E(a_{n}ΔX_{n}|F_{n−1}))^{2} = (E(\sqrt{a_{n}}\sqrt{a_{n}}ΔX_{n}|F_{n−1}))^{2}
≤ E(a_{n}|F_{n−1})E(a_{n}(ΔX_{n})^{2}|F_{n−1}),  (3.5)

i.e.,

(ω : E(a_{n}(ΔX_{n})^{2}|F_{n−1}) = 0) ⊂ (ω : E(a_{n}ΔX_{n}|F_{n−1}) = 0).  (3.6)

If we denote by A the set (ω : E(a_{n}(ΔX_{n})^{2}|F_{n−1}) = 0), then

E[a_{n}(ΔX_{n})^{2}I_{A}] = E[E(a_{n}(ΔX_{n})^{2}|F_{n−1})I_{A}] = 0,

whence a_{n}(ΔX_{n})^{2}I_{A} = 0. This implies that b_{n}(ΔX_{n})^{2}I_{A} = 0, and therefore b_{n}ΔX_{n}I_{A} = 0. Taking conditional expectation we get E(b_{n}ΔX_{n}|F_{n−1})I_{A} = 0, whence we finally obtain

(ω : E(a_{n}(ΔX_{n})^{2}|F_{n−1}) = 0) ⊂ (ω : E(b_{n}ΔX_{n}|F_{n−1}) = 0).  (3.7)

Therefore essinf w.r.t. π ∈ Π(n − 1, N) in

E(V (n, x + π_{n}ΔX_{n})|F_{n−1})
= E(a_{n}(x + π_{n}ΔX_{n})^{2} + 2b_{n}(x + π_{n}ΔX_{n}) + c_{n}|F_{n−1})
= E(a_{n}x^{2} + 2b_{n}x + c_{n}|F_{n−1}) + π_{n}^{2}E(a_{n}(ΔX_{n})^{2}|F_{n−1})
+ 2π_{n}(xE(a_{n}ΔX_{n}|F_{n−1}) + E(b_{n}ΔX_{n}|F_{n−1}))]  (3.8)

is attained for

π_{n} = −\frac{E(b_{n}ΔX_{n}|F_{n−1})}{E(a_{n}(ΔX_{n})^{2}|F_{n−1})} − x \frac{E(a_{n}ΔX_{n}|F_{n−1})}{E(a_{n}(ΔX_{n})^{2}|F_{n−1})}.  (3.9)

Note that on the set A, π_{n} can be chosen arbitrarily, e.g., it can be set to be 0. The agreement \frac{0}{0} = 0 allows one to write this in compact form (3.9). After substituting π_{n} defined by the above formula in (3.8) and using (2.1), we obtain

V (n − 1, x) = [E(a_{n}|F_{n−1}) − \frac{(E(a_{n}ΔX_{n}|F_{n−1}))^{2}}{E(a_{n}(ΔX_{n})^{2}|F_{n−1})}]x^{2}

+ 2[E(b_{n}|F_{n−1}) − \frac{E(a_{n}ΔX_{n}|F_{n−1})E(b_{n}ΔX_{n}|F_{n−1})}{E(a_{n}(ΔX_{n})^{2}|F_{n−1})}]x

+ E(c_{n}|F_{n−1}) − \frac{(E(b_{n}ΔX_{n}|F_{n−1}))^{2}}{E(a_{n}(ΔX_{n})^{2}|F_{n−1})} .

Denoting

a_{n−1} = E(a_{n}|F_{n−1}) − \frac{(E(a_{n}ΔX_{n}|F_{n−1}))^{2}}{E(a_{n}(ΔX_{n})^{2}|F_{n−1})} ,
b_{n−1} = E(b_{n}|F_{n−1}) − \frac{E(a_{n}ΔX_{n}|F_{n−1})E(b_{n}ΔX_{n}|F_{n−1})}{E(a_{n}(ΔX_{n})^{2}|F_{n−1})} ,
c_{n−1} = E(c_{n}|F_{n−1}) − \frac{(E(b_{n}ΔX_{n}|F_{n−1}))^{2}}{E(a_{n}(ΔX_{n})^{2}|F_{n−1})} ,

we obtain

V (n − 1, x) = a_{n−1}x^{2} + 2b_{n−1}x + c_{n−1}.

Using the definition of a_{n−1} and (3.5) it is easy to check that 0 ≤ a_{n−1} ≤ 1. Therefore (3.6), (3.7) are satisfied for k = n−1 as well and, consequently, using the induction, for all k = 0, …, N. Hence the reasoning used to find essinf is true for all k.

Bellman’s principle suggests us that a natural candidate for an optimal solution is (3.4), where a_{n}, b_{n} satisfy (3.1)–(3.2) with initial conditions a_{N} = 1, b_{N} = −H. Let us show that X_{N}^{π^{∗}} = ∑_{i=1}^{N} π_{i}^{∗}ΔX_{i} is square integrable (i.e., π^{∗} is admissible). For this it is sufficient to check that a_{n}(X_{n}^{π^{∗}})^{2} = a_{n}(∑_{i=1}^{n} π_{i}^{∗}ΔX_{i})^{2} is integrable for all n. Since a_{N} = 1, this will entail the square integrability of X_{N}^{π^{∗}}. Obviously, a_{0}(X_{0}^{π^{∗}})^{2} is integrable (as a_{0} ≤ 1 and X_{0}^{π^{∗}} = c). If we prove the equality

E[a_{n}(X_{n}^{π^{∗}})^{2}] = E[a_{n−1}(X_{n−1}^{π^{∗}})^{2}] + E[\frac{(E(b_{n}ΔX_{n}|F_{n−1}))^{2}}{E(a_{n}(ΔX_{n})^{2}|F_{n−1})}],  (3.10)

then by induction it is sufficient to show that for all n,

\frac{(E(b_{n}ΔX_{n}|F_{n−1}))^{2}}{E(a_{n}(ΔX_{n})^{2}|F_{n−1})}

is integrable. But this directly follows if we note that (c_{n}, F_{n}) is a submartingale since c_{n} = V (n, 0) and for arbitrary x, (V (n, x), F_{n}) is a submartingale, whence

c_{n} ≤ E(c_{N}|F_{n}) = E(H^{2}|F_{n}),

and since for all n we have |b_{n}| ≤ \sqrt{a_{n}c_{n}}, by the Hölder inequality

\frac{(E(b_{n}ΔX_{n}|F_{n−1}))^{2}}{E(a_{n}(ΔX_{n})^{2}|F_{n−1})} ≤ \frac{(E(\sqrt{a_{n}c_{n}}ΔX_{n}|F_{n−1}))^{2}}{E(a_{n}(ΔX_{n})^{2}|F_{n−1})} ≤ E(c_{n}|F_{n−1}).

Now let us verify (3.10). We have

E[a_{n}(X_{n}^{π^{∗}})^{2}] = E[a_{n}(X_{n−1}^{π^{∗}} + π_{n}^{∗}ΔX_{n})^{2}]

= E[a_{n}(X_{n−1}^{π^{∗}})^{2} + 2π_{n}^{∗}a_{n}ΔX_{n}X_{n−1}^{π^{∗}} + a_{n}(π_{n}^{∗}ΔX_{n})^{2}]

= E[E(a_{n}|F_{n−1})(X_{n−1}^{π^{∗}})^{2} + 2π_{n}^{∗}E(a_{n}ΔX_{n}|F_{n−1})X_{n−1}^{π^{∗}}

+ (π_{n}^{∗})^{2}E(a_{n}(ΔX_{n})^{2}|F_{n−1})]

= E[E(a_{n}|F_{n−1})(X_{n−1}^{π^{∗}})^{2} − 2[\frac{E(b_{n}ΔX_{n}|F_{n−1})E(a_{n}ΔX_{n}|F_{n−1})}{E(a_{n}(ΔX_{n})^{2}|F_{n−1})}

+ \frac{(E(a_{n}ΔX_{n}|F_{n−1}))^{2}}{E(a_{n}(ΔX_{n})^{2}|F_{n−1})}X_{n−1}^{π^{∗}}]X_{n−1}^{π^{∗}}

+ \frac{(E(b_{n}ΔX_{n}|F_{n−1}))^{2}}{E(a_{n}(ΔX_{n})^{2}|F_{n−1})} + 2\frac{E(b_{n}ΔX_{n}|F_{n−1})E(a_{n}ΔX_{n}|F_{n−1})}{E(a_{n}(ΔX_{n})^{2}|F_{n−1})}X_{n−1}^{π^{∗}}

+ \frac{(E(a_{n}ΔX_{n}|F_{n−1}))^{2}}{E(a_{n}(ΔX_{n})^{2}|F_{n−1})}(X_{n−1}^{π^{∗}})^{2}]

= E[(E(a_{n}|F_{n−1}) − \frac{(E(a_{n}ΔX_{n}|F_{n−1}))^{2}}{E(a_{n}(ΔX_{n})^{2}|F_{n−1})})(X_{n−1}^{π^{∗}})^{2}]

+ E[\frac{(E(b_{n}ΔX_{n}|F_{n−1}))^{2}}{E(a_{n}(ΔX_{n})^{2}|F_{n−1})}],

which is the desired result. Hence X_{N}^{π^{∗}} is square integrable.

The optimality of π^{∗} can be easily derived using the optimality criterion (Proposition A2), equations (3.1)–(3.2) and the fact that V (n, x) is a square trinomial (one should substitute the expressions for V (n, c+∑_{i=1}^{n} π_{i}^{∗}ΔX_{i}) and V (n− 1, c+∑_{i=1}^{n−1} π_{i}^{∗}ΔX_{i}) in the hypothetical equality E(V (n, c+∑_{i=1}^{n} π_{i}^{∗}ΔX_{i})|F_{n−1})= V (n − 1, c + ∑_{i=1}^{n−1} π_{i}^{∗}ΔX_{i})). □

4. Remarks and Applications

Remark 4.1. Let us show how the obtained results are related to the solutions previously proposed in the literature. In order to make comparisons valid, suppose that X_{n} is square integrable for all n. Recall from [6] that an optimal solution is given by

π_{n}^{∗} = ρ_{n} − (c + ∑_{i=1}^{n−1} π_{i}^{∗}ΔX_{i})β_{n},  (4.1)

where the predictable processes β = (β_{k}) and ρ = (ρ_{k}) are defined by the backward equations

β_{n} = \frac{E(ΔX_{n} ∏_{i=n+1}^{N}(1 − β_{i}ΔX_{i})|F_{n−1})}{E((ΔX_{n})^{2} ∏_{i=n+1}^{N}(1 − β_{i}ΔX_{i})|F_{n−1})} ,  (4.2)

ρ_{n} = \frac{E(HΔX_{n} ∏_{i=n+1}^{N}(1 − β_{i}ΔX_{i})|F_{n−1})}{E((ΔX_{n})^{2} ∏_{i=n+1}^{N}(1 − β_{i}ΔX_{i})|F_{n−1})} .  (4.3)

The link between a = (a_{k}), b = (b_{k}) and β = (β_{k}), ρ = (ρ_{k}) is quite simple, namely

β_{n} = \frac{E(a_{n}ΔX_{n}|F_{n−1})}{E(a_{n}(ΔX_{n})^{2}|F_{n−1})} ,

ρ_{n} = −\frac{E(b_{n}ΔX_{n}|F_{n−1})}{E(a_{n}(ΔX_{n})^{2}|F_{n−1})} .

Indeed, since a_{N} = 1, b_{N} = −H, these relations are true for n = N. The same fact for general n can be easily obtained using induction. Conversely, a_{n} and b_{n} can be expressed in terms of β_{n+1}, …, β_{N} as

a_{n} = E( ∏_{i=n+1}^{N}(1 − β_{i}ΔX_{i}) | F_{n}),

b_{n} = −(H ∏_{i=n+1}^{N}(1 − β_{i}ΔX_{i}) | F_{n}),

which again can be proved by induction. As it can be seen from (4.2)–(4.3), equations (3.1)–(3.2) provide some advantages for describing an optimal strategy as they are one-step, unlike (4.2)–(4.3), which involve β_{n+1}, …, β_{N}. On the other hand, (4.1) seems simpler than (3.4).

Remark 4.2. Note that the requirement of the square integrability of X_{n}, n = 0, …, N, of [6], [7] in our setting is substituted by a weaker one:

E((ΔX_{n})^{2}|F_{n−1}) < ∞

for all n.

Remark 4.3. It is interesting to determine the value of the shortfall

R^{∗} = E[(H − c − ∑_{i=1}^{N} π_{i}^{∗}ΔX_{i})^{2} ]

associated with the optimal strategy π^{∗}. Noting that

E[(H − c − ∑_{i=1}^{N} π_{i}^{∗}ΔX_{i})^{2} ] = V (0, c),

we obtain

R^{∗} = a_{0}c^{2} + 2b_{0}c + c_{0}.

Remark 4.4. When hedging a contigent claim, we usually are also interested in determining the price of this claim, i.e. we consider the problem

minimize E[(H − x − ∑_{i=1}^{N} π_{i}ΔX_{i})^{2}] over all (x, π) ∈ R × Π.

For every fixed c and arbitrary π ∈ Π we have

E[(H − c − ∑_{i=1}^{N} π_{i}^{∗}(c)ΔX_{i})^{2}] ≤ E[(H − c − ∑_{i=1}^{N} π_{i}ΔX_{i})^{2}],

where π^{∗}(c) is defined by (3.1)–(3.4). Let us minimize the left-hand side by c. Since it is equal to

V (0, c) = a_{0}c^{2} + 2b_{0}c + c_{0},

the minimum is attained for −b_{0}/a_{0} (of course, if a_{0} ≠ 0) and a minimal shortfall is equal to

R^{∗} = −\frac{b_{0}^{2} − a_{0}c_{0}}{a_{0}} .

If a_{0} = 0, then b_{0} = 0 too (see the proof of Theorem 1), which means that an optimal strategy π^{∗}(c) does not depend on the initial capital c; therefore the price can be set to 0 and it can be again written as c^{∗} = −\frac{b_{0}}{a_{0}} . The shortfall in this case equals c_{0}. Note that this result due to Remark 4.1 coincides with the solution proposed in [6]. □

Appendix

Lemma A1. The family of random variables Λ_{n}^{π} = E((H −x−∑_{i=1}^{n} π̂_{i}ΔX_{i}− ∑_{i=n+1}^{N} π_{i}ΔX_{i})^{2}|F_{n}), π ∈ Π(π̂, n, N), possesses the ε-lattice property (for ε = 0). Here Π(π̂, n, N) denotes the set of π ∈ Π such that π_{i} = π̂_{i} for i = 0, 1, …, n.

Proof. Let π^{1}, π^{2} ∈ Π(π̂, n, N) and let us define π^{3} ∈ Π(π̂, n, N) as

π_{i}^{3} = π_{i}^{1}I_{B} + π_{i}^{2}I_{B^{c}},

where B = (ω : Λ_{n}^{π^{1}} ≤ Λ_{n}^{π^{2}}). Then since B ∈ F_{n},

Λ_{n}^{π^{3}} = E((H − x − ∑_{i=1}^{N} π_{i}^{3}ΔX_{i})^{2} | F_{n})

= E((H − x − I_{B} ∑_{i=1}^{N} π_{i}^{1}ΔX_{i} − I_{B^{c}} ∑_{i=1}^{N} π_{i}^{2}ΔX_{i})^{2} | F_{n})

= I_{B}E((H − x − ∑_{i=1}^{N} π_{i}^{1}ΔX_{i})^{2} | F_{n})

+ I_{B^{c}}E((H − x − ∑_{i=1}^{N} π_{i}^{2}ΔX_{i})^{2} | F_{n})

= E((H − x − ∑_{i=1}^{N} π_{i}^{1}ΔX_{i})^{2} | F_{n}) ∧ E((H − x − ∑_{i=1}^{N} π_{i}^{2}ΔX_{i})^{2} | F_{n})

and therefore the family Λ_{n}^{π} has the ε-lattice property. □

Proposition A1. The sequence V (n, x + ∑_{i=1}^{n} π̂_{i}ΔX_{i}), F_{n}) is a submartingale for arbitrary fixed π̂ ∈ Π, x ∈ R.

Proof. We must prove that for any n

E(V (n, x + ∑_{i=1}^{n} π̂_{i}ΔX_{i}) | F_{n−1}) ≥ V (n − 1, x + ∑_{i=1}^{n−1} π̂_{i}ΔX_{i}).

Taking into account the previous lemma and Lemma 16.A.5 of [1], we obtain

E( essinf_{π∈Π(π̂,n,N)} E((H − x − ∑_{i=1}^{n} π̂_{i}ΔX_{i} − ∑_{i=n+1}^{N} π_{i}ΔX_{i})^{2}|F_{n}) | F_{n−1})

= essinf_{π∈Π(π̂,n,N)} E(E((H − x − ∑_{i=1}^{n} π̂_{i}ΔX_{i} − ∑_{i=n+1}^{N} π_{i}ΔX_{i})^{2}|F_{n}) | F_{n−1}))

= essinf_{π∈Π(π̂,n,N)} E((H − x − ∑_{i=1}^{n} π̂_{i}ΔX_{i} − ∑_{i=n+1}^{N} π_{i}ΔX_{i})^{2} | F_{n−1})

≥ essinf_{π∈Π(π̂,n−1,N)} E((H − x − ∑_{i=1}^{n−1} π̂_{i}ΔX_{i} − ∑_{i=n}^{N} π_{i}ΔX_{i})^{2} | F_{n−1})

= V (n − 1, x + ∑_{i=k}^{n−1} π̂_{i}ΔX_{i}),

which is the desired result. □

Proposition A2 (Optimality Principle). π^{∗} ∈ Π is optimal if and only if the sequence (V (n, c + ∑_{i=1}^{n} π_{i}^{∗}ΔX_{i}) , F_{n}) is a martingale.

Proof. Necessity. Suppose that π^{∗} = (π_{n}^{∗}) is optimal. Then since (V (n, c + ∑_{i=1}^{n} π^{∗}ΔX_{i}), F_{n}) is a submartingale, due to Lemma 6.6 of [4] it is sufficient to check the equality

E[V (N, c + ∑_{i=1}^{N} π_{i}^{∗}ΔX_{i})] = E[V (0, c)].

We have

E[V (N, c + ∑_{i=1}^{N} π_{i}^{∗}ΔX_{i})] = E[(H − c − ∑_{i=1}^{N} π_{i}^{∗}ΔX_{i})^{2}].

Noting that the optimality of π^{∗} means

V (0, c) = E[(H − c − ∑_{i=1}^{N} π_{i}^{∗}ΔX_{i})^{2}],

we finally obtain

E[V (N, c + ∑_{i=1}^{N} π_{i}^{∗}ΔX_{i})] = E[V (0, c)].

Sufficiency. Suppose that (V (n, c + ∑_{i=1}^{n} π_{i}^{∗}ΔX_{i}), F_{n}) is a martingale. Then

E(V (N, c + ∑_{i=1}^{N} π_{i}^{∗}ΔX_{i}) | F_{0}) = V (0, c),

whence

E[(H − c − ∑_{i=1}^{N} π_{i}^{∗}ΔX_{i})^{2}] = V (0, c),

which means that π^{∗} is optimal. □

Acknowledgement

The author would like to thank the referee for useful remarks and N. Lazrieva and M. Mania for many helpful comments.

References

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(Received 3.10.2002)

Author’s address:

Department of Applied Mathematics Georgian Technical University 77, Kostava St., Tbilisi 0175 Georgia E-mail: shota@gtu.edu.ge