Sección 10: Políticas, Estimaciones y Errores Contables

In the setting of arbitrage-free but incomplete financial markets models, i.e. the set Me(S)

of equivalent martingale measures does not reduce to a singleton, and since a convex com- bination of two equivalent martingale measures is also an equivalent martingale measure, the set Me(S) is in fact of infinite cardinality. It is very difficult if not impossible to find

the explicit solution to the dual problem (3.42). However, some computations may be carried more or less explicitly in some particular models. Let us consider here the financial market model with Itˆo price processes and Brownian filtration as presented in Section (2.7). Observe that since we assumed the risk-premium process to be bounded, it is arbitrage-free

and in general incomplete. In the notation of Section (2.7) we consider the set including Me(S)

Mloc = {ZTν : ν ∈ K(σ)} ⊃ Me(S) = {ZTν : ν ∈ Km(σ)} .

For any ν ∈ K(σ) and any wealth process Xx_{= x +R αdS, α ∈ A(S), as an application}

of Itˆo formula, we can show that the process Zν_Xx_{is a P -local martingale. Moreover, notice}

that for all ν ∈ K(σ), the bounded process νn _{= ν1}

|ν|≤n belongs to Km(σ) (remember

Remark 2.7.4 where we have seen that any bounded process is in Km(σ)), and Zν n

T converges

a.s. to Z_Tν. Therefore, Me(S) ⊂ Mloc ⊂ D and from part (4) of Kramkov-Schachermayer

Theorem (3.4.1), we conclude that ˜

v(y) = inf

ν∈K(σ) E[ eU (yZ ν

T)], y > 0 . (4.5)

The motivation and advantage of introducing the set Mlocis that it is explicit (contrary to

the set D), completely parametrized by the set of controls ν ∈ K(σ), and does not involve any assumptions about the martingale integrability as in the case of Km(σ), so that the

stochastic control methods may be used to find a solution ˜νy _{in K(σ) to ˜v(y) in (4.5). In}

fact, if we make the additional assumption that the function ξ ∈ R 7→ eU (eξ) is convex,

which holds true if for example x ∈ (0, ∞) 7→ xU0(x) is increasing (both the logarithm and power utility functions satisfy the latter condition), then it is proved in Karatzas et al. [KLSX91] that for all y > 0, the dual problem ˜v(y) admits a solution Zνˆy

T ∈

Mloc. Moreover, we prove that for all ν ∈ K(σ) such that E[

RT

0 |νt|

2_{dt] = ∞, we have}

E[ eU (yZν

T)] = ∞, thus in the dual problem (4.5), we can restrict ourselves to taking the

infimum over K2(σ) = {ν ∈ K(σ) : E[

RT

0 |νt|

2_{dt] < ∞}, and furthermore ˆν}y _{∈ K} 2(σ).

Note that this solution Z_Tνˆyˆ does not belong (in general) to the set Me(S). From the

Kramkov-Schachermayer Theorem (3.4.1), the solution to the dual problem is then given by

ˆ

X_Tx = I(ˆyZ_Tνˆyˆ)

where ˆy > 0 is the solution to argmin_y>0[˜v(y) + xy] and satisfying E[Z_TνˆyˆI(ˆyZ_Tνˆyˆ)] = x .

Recall that the function I(·) (the continuous, strictly decreasing inverse of the marginal utility function U0 on (0, ∞)) maps (0, ∞) onto itself, hence the optimal wealth process

ˆ

Xx _{is nonnegative. Moreover we notice that the process Z}ˆνˆy_Xˆx _{is a nonnegative P -local}

martingale, hence a supermartingale with the property that E[Zνˆyˆ

T XˆTx] = x. Therefore, it

is a true martingale, and we finally have determined the optimal wealth process ˆXx _as

ˆ X_tx = E Z ˆ νyˆ T Zνˆyˆ t I(ˆyZ_Tˆνyˆ)Ft, 0 ≤ t ≤ T .

We apply now the results described above to two examples of utility functions.

Example 4.2.1 (Logarithmic utility function) In this example we choose for our op- timization problem (3.7) the logarithmic utility function U (x) = ln(x), x > 0 which plays a special role in portfolio choice, for which we have

I(y) = 1

y and U (y) = − ln(y) − 1,e y > 0. For all ν ∈ K2(σ), we have

E[ eU (yZ_Tν)] = − ln(y) − 1 + 1 2E  Z T 0 (|λs|2+ |νs|2)ds, y > 0.

Therefore, the solution to the dual problem (4.5) is reached for ν = 0 (independently of y) and correspond to Z_T0. Moreover, we have

E[Z_T0I(ˆyZ_T0)] = E[Z_T0 1 ˆ yZ0

T

] = x ,

thus the Lagrange multiplier is ˆy = _x1. The optimal wealth process ˆXx _{for v(x), also known}

as the “growth optimal portfolio”, is described explicitly by ˆ X_tx = E Z 0 T Z0 t 1 ˆ yZ0 T  Ft = x Z0 t , 0 ≤ t ≤ T.

In order to obtain the optimal control ˆα we apply Itˆo lemma to the above equation, and we make the identification with the dynamics d ˆXx

t = ˆαtdSt. In a financial market model

written in the “Dol´eans-Dade exponential” form

dSt= St(µtdt + σtdWt),

we find the optimal portfolio weight ˆπt, i.e. the optimal proportion of wealth invested in S:

ˆ πt:= ˆ αtSt ˆ Xx t = µt σ2 t .

This solution is called a myopic solution, i.e. it depends only on the local behavior of the price process, in accordance with terminology adopted by Merton.

Example 4.2.2 (Power utility function) We consider again the case of the power util- ity function U (x) = x

p

p , x > 0 with p ∈ (0, 1) which exhibits constant relative risk aversion (CRRA), for which we have

I(y) = yp−11 and U (y) =_e y −q

q , y > 0 , q = p 1 − p. For every ν ∈ K(σ), we have

E[ eU (yZ_Tν)] = y −q q E[(Z ν T) −q ], y > 0.

From the above equation, we notice that the solution to the dual problem ˜v(y) is independent of y and is a solution to the following problem

inf ν∈K(σ) E[(Z ν T) −q ] . (4.6)

In a Markovian setting, such as a stochastic volatility model, an explicit solution to the above stochastic control problem can be derived by using the traditional dynamic program- ming approach. In a more general setting of Itˆo processes, some methods of Backward SDEs may be used (see [MT03] and the reference given there). If ˆν denote the solution to equation (4.6), the optimal wealth process is given by

ˆ X_tx = x E[(Zˆν T)−q] E (Z ˆ ν T)−q Zνˆ t  Ft , 0 ≤ t ≤ T .

Complements of Integration

Let us denote by (Ω, F , F = (Ft)t∈[0,T ], P ) a filtered probability space and L1(Ω, F , P ) the

set of integrable random variables.

A.1

Uniform Integrability

Definition A.1.1 (Uniformly integrable random variables) Let (fi)i∈I be a family of ran-

dom variables in L1(Ω, F , P ) . We say that (fi)i∈I is uniformly integrable if

lim

x→∞sup_i∈I E[|fi|1|fi|≥x] = 0. (A.1)

We note that any family of random variables, bounded by a fixed integrable random variable (in particular any finite family of random variables in L1_{(Ω, F , P ) is uniformly integrable.}

The following result extends the dominated convergence theorem.

Theorem A.1.2 Let (fn)n≥1 be a sequence of random variables in L1(Ω, F , P ) converging

a.s to a random variable f . Then f is integrable and the convergence of fn to f holds in

L1_{(Ω, F , P ) if and only if the sequence (f}

n)n≥1 is uniformly integrable. When the random

variables fn are nonnegative, this is equivalent to

lim

n→∞E[fn] = E[f ] (A.2)

The following corollary is used in the proof of Theorem 3.2.3.

Corollary A.1.3 Let (fn)n≥1 be a sequence of nonnegative random variables bounded in

L1_{(Ω, F , P ), i.e. sup}

such that limn→∞E[fn] = E[f ] + δ with δ > 0. Then, there exists a subsequence (fnk)k≥1

of (fn)n≥1 and a disjoint sequence (Ak)k≥1 of (Ω, F ) such that

E[fnk1Ak] ≥

δ

2, ∀k ≥ 1. (A.3)

Proof. A proof may be found in [P09].

The following result, due to la Vall´ee-Poussin, gives a practical condition for proving the uniform integrability.

Theorem A.1.4 (la Vall´ee-Poussin) Let (fi)i∈I be a family of random variables in L1(Ω, F , P ).

The following assertions are equivalent (i) (fi)i∈I is uniformly integrable

(ii) There exists a nonnegative function ϕ defined on R+, limx→∞ ϕ(x)

x = ∞, such that

sup

i∈I

E[ϕ(|fi|)] < ∞.

In practice, we often use the implication (ii) =⇒ (i). For example, by taking ϕ(x) = x2_,

we see that any family of random variables bounded in L2 _{is uniformly integrable.}