Optimal consumption and investment with bounded downside risk measures for logarithmic utility functions

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downside risk measures for logarithmic utility functions

Claudia Kluppelberg, Serguei Pergamenchtchikov

To cite this version:

Claudia Kluppelberg, Serguei Pergamenchtchikov. Optimal consumption and investment with

bounded downside risk measures for logarithmic utility functions. Walter de Gruyter, pp.428, 2009,

Advanced Financial Modelling, H. Albrecher, W. Rungalder and W. Schachermayer. �hal-00454078�

Optimal consumption and investment with bounded downside risk measures for

logarithmic utility functions

Claudia Kl¨uppelberg and Serguei Pergamenchtchikov

Abstract. We investigate optimal consumption problems for a Black-Scholes market under uniform restrictions on Value-at-Risk and Expected Shortfall for logarithmic utility functions. We find the so- lutions in terms of a dynamic strategy in explicit form, which can be compared and interpreted. This paper continues our previous work, where we solved similar problems for power utility functions.

Key words. Black-Scholes model, Capital-at-Risk, Expected Shortfall, logarithmic utility, optimal consumption, portfolio optimization, utility maximization, Value-at-Risk

AMS classification. primary: 91B70, 93E20, 49K30; secondary: 49L20, 49K45.

1 Introduction

One of the principal questions in mathematical finance is the optimal investment/con- sumption problem for continuous time market models. By applying results from sto- chastic control theory, explicit solutions have been obtained for some special cases (see e.g. Karatzas and Shreve [9], Korn [11] and references therein).

With the rapid development of the derivatives markets, together with margin tradings on certain financial products, the exposure to losses of investments into risky assets can be considerable. Without a careful analysis of the potential danger, the investment can cause catastrophic consequences such as, for example, the recent crisis in the “Soci´et´e G´en´erale”.

To avoid such situations the Basel Committee on Banking Supervision in 1995 sug- gested some measures for the assessment of market risks. It is widely accepted that the Value-at-Risk (VaR) is a useful summary risk measure (see, Jorion [7] or Dowd [4]). We recall that the VaR is the maximum expected loss over a given horizon period at a given confidence level. Alternatively, the Expected Shortfall (ES) or Tail Condi- tion Expectation (TCE) measures also the expected loss given the confidence level is violated.

In order to satisfy the Basel commitee requirements, portfolios have to control the level of VaR or (the more restrictive) ES throughout the investment horizon. This leads to stochastic control problems under restrictions on such risk measures.

Our goal in this paper is the optimal choice of a dynamic portfolio subject to a risk

Second author: This work was supported by the European Science Foundation through the AMaMeF programme.

limit specified in terms of VaR or ES uniformly over horizon time interval

[0, T]

. In Kl¨uppelberg and Pergamenshchikov in [10] we considered the optimal invest- ment/consumption problem with uniform risk limits throughout the investment hori- zon for power utility functions. In that paper also some interpretation of VaR and ES besides an account of the relevant literature can be found. Our results in [10] have in- teresting interpretations. We have, for instance, shown that for power utility functions with exponents less than one, the optimal constrained strategies are riskless for suffi- ciently small risk bounds: they recommend consumption only. On the contrary, for the (utility bound) of a linear utility function the optimal constrained strategies recommend to invest everything into risky assets and consume nothing.

In this paper we investigate the optimal investment/consumption problem for loga- rithmitic utility functions again under constraints on uniform versions of VaR and ES over the whole investment horizon

[0, T]

. Using optimization methods in Hilbert func- tional spaces, we find all optimal solutions in explicit form. It turns out that the optimal constrained strategies are the unconstrained ones multiplied by some coefficient which is less then one and depends on the specific constraints.

Consequently, we can make the main recommendation: To control the market risk throughout the investment horizon

[0, T]

restrict the optimal unconstrained portfolio allocation by specific multipliers (given in explicit form in (3.6) for the VaR constraint and in (3.26) for the ES constraint).

Our paper is organised as follows. In Section 2 we formulate the problem. We de- fine the Black-Scholes model for the price processes and present the wealth process in terms of an SDE. We define the cost function for the logarithmic utility function and present the admissible control processes. We also present the unconstrained con- sumption and investment problem of utility maximization for logarithmic utility. In Sections 3 and 3.2 we consider the constrained problems. Section 3 is devoted to a risk bound in terms of Value-at-Risk, whereas Section 4.1 discusses the consequences of a risk bound in terms of Expected Shortfall. Auxiliary results and proofs are postponed to Section 4. We start there with material needed for the proofs of both regimes, the Value-at-Risk and the ES risk bounds. In Section 4.1 all proofs of Section 3 can be found, and in Section 4.1 all proofs of Section 4.1. Some technical lemmas postponed to the Appendix, again divided in two parts for the Value-at-Risk regime and the ES regime.

2 Formulating the problem

2.1 The model and first results

We work in the same framework of self-financing portfolios as in Kl¨uppelberg and

Pergamenshchikov in [10], where the financial market is of Black-Scholes type consist-

ing of one riskless bond and several risky stocks on the interval

[0, T]

. Their respective

prices

S₀ = (S₀(t))0≤t≤T

and

S_i = (S_i(t))0≤t≤T

for

i= 1, . . . , d

evolve according to

the equations:







d

S₀(t) = r_tS₀(t)

d

t , S₀(0) = 1,

d

S_i(t) = S_i(t)µ_i(t)

d

t + S_i(t)Pd

j=1 σ_ij(t)

d

W_j(t), S_i(0)>0.

(2.1)

Here

W_t = (W₁(t), . . . , W_d(t))^′

is a standard

d

-dimensional Wiener process in

R^d

;

r_t ∈ R

is the riskless interest rate;

µ_t = (µ₁(t), . . . , µ_d(t))^′

is the vector of stock- appreciation rates and

σ_t = (σ_ij(t))1≤i,j≤d

is the matrix of stock-volatilities. We as- sume that the coefficients

(r_t)_0≤t≤T

,

(µ_t)_0≤t≤T

and

(σ_t)_0≤t≤T

are deterministic cadlag functions. We also assume that the matrix

σ_t

is non degenerated for all

0≤t≤T

.

We denote by

_Ft =σ{W_s, s≤t}

,

t ≥0

, the filtration generated by the Brownian motion (augmented by the null sets). Furthermore,

_{| · |}

denotes the Euclidean norm for vectors and the corresponding matrix norm for matrices and prime denotes the transposed. For

(y_t)_0≤t≤T

square integrable over the fixed interval

[0, T]

we define

kykT = (RT

0 |y_t|²

d

t)^1/2

.

The portfolio process

(π_t= (π₁(t), . . . , π_d(t))^′)_0≤t≤T

represents the fractions of the wealth process invested into the stocks. The consumption rate is denoted by

(v_t)_0≤t≤T

. Then (see [10] for details) the wealth process

(X_t)0≤t≤T

is the solution to the SDE

d

X_t = X_t(r_t +y^′_tθ_t−v_t)

d

t +X_ty^′_t

d

W_t, X₀ = x >0,

(2.2) where

θ_t=σ_t⁻¹(µ_t−r_t1)

with

¹= (1, . . . ,1)^′∈R^d,

and we assume that

_Z

T 0

|θ_t|²

d

t < ∞.

The control variables are

y_t =σ_t^′π_t ∈ R^d

and

v_t≥ 0

. More precisely, we define the

(Ft)_0≤t≤T

-progressively measurable control process as

ν= (y_t, v_t)t≥0

, which satisfies

Z T

0 |y_t|²

d

t < ∞

and

Z T

0

v_t

d

t < ∞

a.s.. (2.3) In this paper we consider logarithmitic utility functions. Consequently, we assume throughout that

Z T 0

(lnv_t)₋

d

t <∞

a.s., (2.4) where

(a)₋ =−min(a,0)

.

To emphasize that the wealth process (2.2) corresponds to some control process

ν

we write

X^ν

. Now we describe the set of control processes.

Definition 2.1 A stochastic control process

ν = (ν_t)0≤t≤T = ((y_t, v_t))0≤t≤T

is called

admissible, if it is

(Ft)0≤t≤T

-progressively measurable with values in

R^d×R+

, satis-

fying integrability conditions (2.3)–(2.4) such that the SDE (2.2) has a unique strong

a.s. positive continuous solution

(X_t^ν)0≤t≤T

for which

E

Z T 0

(ln(vtX_t^ν))₋

d

t+ (lnX_T^ν)₋

!

<∞.

We denote by

_V

the class of all admissible control processes.

For

ν ∈ V

we define the cost function

J(x, ν) :=E_x

Z T 0

ln v_tX_t^ν

d

t + lnX_T^ν

!

.

(2.5)

Here

^E_x

is the expectation operator conditional on

X₀^ν=x

.

We recall a well-known result, henceforth called the unconstrained problem:

maxν∈V J(x, ν).

(2.6)

To formulate the solution we set

ω(t) =T−t+ 1

and

_br_t=r_t + |θ_t|²

2 , 0≤t≤T .

Theorem 2.2 (Karatzas and Shreve [9], Example 6.6, p. 104) The optimal value of

J(x, ν)

is given by

maxν∈V J(x, ν) = J(x, ν^∗) = (T+ 1) ln x T+ 1 +

Z T 0

ω(t)br_t

d

t .

The optimal control process

ν^∗= (y^∗_t, v^∗_t)_0≤t≤T ∈ V

is of the form

y^∗_t = θ_t

and

v_t^∗ = 1

ω(t),

(2.7)

where the optimal wealth process

(X_t^∗)0≤t≤T

is given as the solution to d

X_t^∗ = X_t^∗ r_t +|θ_t|² − v^∗_t

d

t + X_t^∗θ^′_t

d

W_t, X₀^∗ = x ,

(2.8) which is

X_t^∗ = xT+ 1−t

T + 1 exp Z ^t

0 br_u

d

u + Z t

0

θ_u^′

d

W_u .

Note that the optimal solution (2.7) of problem (2.6) is deterministic, and we denote in the following by

_U

the set of deterministic functions

ν = (y_t, v_t)0≤t≤T

satisfying conditions (2.3) and (2.4).

For the above result we can state that

maxν∈V J(x, ν) = max

ν∈U J(x, ν).

Intuitively, it is clear that to construct financial portfolios in the market model (2.1) the investor can invoke only information given by the coefficients

(r_t)_0≤t≤T

,

(µ_t)_0≤t≤T

and

(σ_t)_0≤t≤T

which are deterministic functions.

Then for

ν∈ U

, by Itˆo’s formula, equation (2.2) has solution

X_t^ν = xEt(y)e^{Rt−Vt+(y,θ)t},

with

R_t=Rt

0r_u

d

u

,

V_t=Rt

0v_u

d

u

,

(y, θ)_t=Rt

0y_u^′ θ_u

d

u

and the stochastic exponential

Et(y) = exp Z ^t

0

y_u^′

d

W_u−1 2

Z t

0 |yu|²

d

u .

Therefore, for

ν ∈ U

the process

(X_t^ν)_0≤t≤T

is positive, continuous and satisfies

sup

0≤t≤T

E|lnX_t^ν| < ∞.

This implies that

_{U ⊂ V}

. Moreover, for

ν∈ U

we can calculate the cost function (2.5) explicitly as

J(x, ν) = (T + 1) lnx+ Z T

0

ω(t)

r_t+y^′_tθ_t−1 2|y_t|²

d

t +

Z T 0

(lnv_t−V_t)

d

t −V_T.

(2.9)

3 Optimization with constraints: main results

3.1 Value-at-Risk constraints

As in Kl¨uppelberg and Pergamenchtchikov [10] we use as risk measures the modifica- tions of Value-at-Risk and Expected Shortfall introduced in Emmer, Kl¨uppelberg and Korn [5], which reflect the capital reserve. For simplicity, in order to avoid non-relevant cases, we consider only

0< α <1/2

.

Definition 3.1 [Value-at-Risk (VaR)]

For a control process

ν

and

0< α≤1/2

define the Value-at-Risk (VaR) by

VaR_t(ν, α) :=x e^Rt−Q_t, t≥0,

where for

t≥0

the quantity

Q_t = inf{z ≥0 : P X_t^ν ≤ z

≥ α}

is the

α

-quantile of

X_t^ν

.

Note that for every

ν ∈ U

we find

Q_t = xexp

R_t−V_t+ (y, θ)_t−1

2kyk²_t− |q_α|kykt

,

(3.1)

where

q_α

is the

α

-quantile of the standard normal distribution.

We define the level risk function for some coefficient

0< ζ <1

as

ζ_t = ζ x e^Rt, t∈[0, T].

(3.2) The coefficient

ζ ∈(0,1)

introduces some risk aversion behaviour into the model.

In that sense it acts similarly as a utility function does. However,

ζ

has a clear interpre- tation, and every investor can choose and understand the influence of the risk bound

ζ

as a proportion of the riskless bond investment.

We consider the maximization problem for the cost function (2.9) over strategies

ν ∈ U

for which the Value-at-Risk is bounded by the level function (3.2) over the interval

[0, T]

, i.e.

maxν∈U J(x, ν)

subject to

sup

0≤t≤T

VaR_t(ν, α)

ζ_t ≤ 1.

(3.3)

To formulate the solution of this problem we define

G(u, λ) :=

Z T 0

(ω(t) +λ)²

(λ|q_α|+u(ω(t) +λ))²|θ_t|²

d

t , u≥0, λ≥0.

(3.4) Moreover, for fixed

λ >0

we denote by

ρ(λ) = inf{u≥0 : G(u, λ)≤1},

(3.5) if it exists, and set

ρ(λ) = +∞

otherwise. For a proof of the following lemma see A.1.

Lemma 3.2 Assume that

_|q_α|>kθkT >0

and

0≤λ≤λ_max= k₁+q

k₂ q_α²− kθk²_T +k²₁ q_α²− kθk²_T ,

where

k₁=k√

ωθk²_T

and

k₂ =kωθk²_T

. Then the equation

G(·, λ) = 1

has the unique positive solution

ρ(λ)

. Moreover,

ρ(λ)<∞

for all

0≤λ≤λ_max

, and

ρ(λ_max) = 0

.

Now for

λ≥0

fixed and

0≤t≤T

we define the weight function

τ_λ(t) = ρ(λ)(ω(t) +λ)

λ|q_α|+ρ(λ)(ω(t) +λ).

(3.6) Here we set

τ_λ(·)≡1

for

ρ(λ) = +∞

. It is clear, that for every fixed

λ≥0

,

0≤τ_λ(T)≤τ_λ(t)≤1, 0≤t≤T .

(3.7) To take the VaR constraint into account we define

Φ(λ) =|q_α|kτ_λθkT+1

2kτ_λθk²_T − k√τ_λθk²_T.

Denote by

Φ⁻¹

the inverse of

Φ

, provided it exists. A proof of the following lemma is

given in A.1.

Lemma 3.3 Assume that

_kθkT >0

and

0< ζ <1−e^{−|qα|kθkT+kθk2T/2}.

(3.8) Then for all

0≤a≤ −ln(1−ζ)

the inverse

Φ⁻¹(a)

exists. Moreover,

0≤Φ⁻¹(a)< λ_max

for

0< a≤ −ln(1−ζ)

and

Φ⁻¹(0) =λ_max

.

Now set

φ(κ) := Φ⁻¹

ln1−κ 1−ζ

, 0≤κ≤ζ ,

(3.9)

and define the investment strategy

e

y_t^κ:=θ_tτφ(κ)(t), 0≤t≤T .

(3.10) To introduce the optimal consumption rate we define

v_t^κ= κ

T−tκ

(3.11)

and recall that for

κ=κ₀= T T+ 1

the function

v_t^κ

coincides with the optimal unconstrained consumption rate

1/ω(t)

as defined in (2.7).

It remains to fix the parameter

κ

. To this end we introduce the cost function

Γ(κ) = ln(1−κ) +Tlnκ+

Z T 0

ω(t)|θ_t|²

τφ(κ)(t)−1 2τ_φ(κ)² (t)

d

t .

(3.12) To choose the parameter

κ

we maximize

Γ

:

γ=γ(ζ) =

argmax

_0≤κ≤ζΓ(κ).

(3.13)

With this notation we can formulate the main result of this section.

Theorem 3.4 Assume that

_kθkT >0

. Then for all

ζ > 0

satisfying (3.8) and for all

0< α <1/2

for which

|q_α| ≥2 (T+ 1)kθkT,

(3.14) the optimal value of

J(x, ν)

for problem (3.3) is given by

J(x, ν^∗) = A(x) + Γ (γ(ζ)),

(3.15) where

A(x) = (T+ 1) lnx+ Z T

0

ω(t)r_t

d

t−T lnT

(3.16)

and the optimal control

ν^∗= (y^∗_t, v_t^∗)_0≤t≤T

is of the form

y^∗_t =ye_t^γ

and

v_t^∗=v^γ_t.

(3.17) The optimal wealth process is the solution of the SDE

d

X_t^∗ = X_t^∗(r_t −v^∗_t + (y_t^∗)^′θ_t)

d

t +X_t^∗(y_t^∗)^′

d

W_t, X₀^∗ = x ,

given by

X_t^∗ = xEt(y^∗)T−γ(ζ)t

T e^{Rt−Vt+(y∗, θ)t}, 0≤t≤T .

The following corollary is a consequence of (2.9).

Corollary 3.5 If

_kθkT = 0

, then for all

0< ζ <1

and for all

0< α <1/2 y_t^∗= 0

and

v_t^∗=v^γ_t

with

γ =

argmax

_0≤κ≤ζ(ln(1−κ) +Tlnκ) = min(κ₀, ζ)

. Moreover, the optimal wealth process is the deterministic function

X_t^∗ = xT−min(κ₀, ζ)t

T e^Rt, 0≤t≤T .

In the next corollary we give some sufficient condition, for which the investment process equals zero (the optimal strategy is riskless). This is the first marginal case.

Corollary 3.6 Assume that

_kθkT >0

and that (3.8) and (3.14) hold. Define

|θ|∞= sup

0≤t≤T |θ_t| <∞.

If

0< ζ < κ₀

and

|q_α| ≥(1 +T)kθkT

1 + ζ(T + 1)|θ|²_∞ (1−ζ)T−ζ

,

(3.18)

then

γ=ζ

and the optimal solution

ν^∗ = (y_t^∗, v^∗_t)_0≤t≤T

is of the form

y_t^∗= 0

and

v_t^∗=v_t^ζ.

Moreover, the optimal wealth process is the deterministic function

X_t^∗ = xT −ζt

T e^Rt, 0≤t≤T .

Below we give some sufficient conditions, for which the solution of optimization problem (3.3) coincides with the unconstrained solution (2.7). This is the second marginal case.

Theorem 3.7 Assume that

ζ >1− 1

T e^{−|qα|kθkT+kθk2T/2}.

(3.19)

Then for all

0 < α < 1/2

for which

_|q_α| ≥ kθkT

, the solution of the optimization

problem (3.3) is given by (2.7)–(2.8).

3.2 Expected Shortfall Constraints

Our next risk measure is an analogous modification of the Expected Shortfall (ES).

Definition 3.8 [Expected Shortfall (ES)]

For a control process

ν

and

0< α≤1/2

define

m_t(ν, α) = E

x X_t^ν|X_t^ν ≤ Q_t

, t≥0,

where

Q_t

is the

α

-quantile of

X_t^ν

given by (3.1). The Expected Shortfall (ES) is then defined as

ES_t(ν, α) = xe^Rt −m_t(ν, α), t≥0.

Again for

ν∈ U

we find

m_t(ν, α) = x F_α(|q_α|+kykt)e^{Rt−Vt+(y,θ)t},

where

F_α(z) = 1 R∞

|q_α|e^−t2/2

d

t Z ∞

z

e^−t2/2

d

t .

(3.20)

We consider the maximization problem for the cost function (2.5) over strategies

ν ∈ U

for which the Expected Shortfall is bounded by the level function (3.2) over the interval

[0, T]

, i.e.

maxν∈U J(x, ν)

subject to

sup

0≤t≤T

ES_t(ν, α)

ζ_t ≤ 1.

(3.21)

We proceed similarly as for the VaR-coinstraint problem (3.3). Define

G₁(u, λ) :=

Z T 0

(ω(t) +λ)²

(λ ι_α(u) +u(ω(t) +λ))²|θ_t|²

d

t , u≥0, λ≥0.

(3.22) where

ι_α(u) = 1

̟(u+|q_α|)−u

with

̟(y) =e^y

2 2

Z ∞ y

e^−t

2

2

d

t .

(3.23) It is well-known and easy to prove that

1 y − 1

y³ ≤ ̟(y) ≤ 1

y, y >0.

(3.24)

This means that

ι_α(u) ≥ |q_α|

for all

u≥0

, which implies for every fixed

λ≥0

that

G₁(u, λ)≤G(u, λ)

for all

u≥0

. Moreover, similarly to (3.5) we define

ρ₁(λ) = inf{u≥0 : G₁(u, λ)≤1}.

(3.25) Since

H

has similar behaviour as

G

, the following lemma is a modification of Lemma 3.2.

Its proof is analogous to the proof of Lemma 3.2.

Lemma 3.9 Assume that

_|q_α|>kθkT >0

and

0≤λ≤λ^′_max= k₁+q

k₂ ψ_α²(0)− kθk²_T +k²₁ ψ_α²(0)− kθk²_T ,

where

k₁

and

k₂

are given in Lemma 3.2. Then the equation

G₁(·, λ) = 1

has the unique positive solution

ρ₁(λ)

. Moreover,

ρ₁(λ)<∞

for

0≤λ≤λ^′_max

and

ρ₁(λ^′_max) = 0

.

Now for

λ≥0

fixed and

0≤t≤T

we define the weight function

ςλ(t) = ρ₁(λ) (ω(t) +λ)

λ ι_α(ρ₁(λ)) +ρ₁(λ) (ω(t) +λ),

(3.26) and we set

ςλ(·)≡1

for

ρ₁(λ) = +∞

. Note that for every fixed

λ≥0

,

0≤ςλ(T)≤ςλ(t)≤1, 0≤t≤T .

(3.27) To take the ES constraint into account we define

Φ₁(λ) = −k√ς_λθk²_T −lnF_α(|q_α|+kς_λθkT).

(3.28) Denote by

Φ⁻¹₁

the inverse of

Φ₁

provided it exists. The proof of the next lemma is given in Section A.2.

Lemma 3.10 Assume that

_kθkT >0

and

0< ζ <1−F_α(|q_α|+kθkT)e^kθk2T .

(3.29) Then for all

0≤a≤ −ln(1−ζ)

the inverse

Φ⁻¹₁

exists and

0 ≤Φ⁻¹₁ (a)< λ_max

for

0< a≤ −ln(1−ζ)

and

Φ⁻¹₁ (0) =λ^′_max

.

Now, similarly to (3.5) we set

φ₁(κ) = Φ⁻¹₁

ln1−κ 1−ζ

, 0≤κ≤ζ ,

(3.30)

and define the investment strategy

e

y_t^1,κ=θtς_φ1(κ)(t), 0≤t≤T .

(3.31) We introduce the cost function

Γ₁(κ) = ln(1−κ) +Tlnκ+ Z T

0

ω(t)|θ_t|²

ς_φ1(κ)(t)−1 2ς_φ²

1(κ)(t)

d

t .

(3.32) To fix the parameter

κ

we maximize

Γ₁

:

γ₁=γ₁(ζ) =

argmax

_0≤κ≤ζΓ₁(κ).

(3.33)

With this notation we can formulate the main result of this section.

Theorem 3.11 Assume that

_kθkT >0

. Then for all

ζ >0

satisfying (3.29) and for all

0< α <1/2

satisfying

|q_α| ≥max(1,2(T+ 1)kθk^T)

(3.34) the optimal value of

J(x, ν)

for the optimization problem (3.21) is given by

J(x, ν^∗) = A(x) + Γ₁(γ₁(ζ)),

where the function

A

is defined in (3.16) and the optimal control

ν^∗ = (y^∗_t, v_t^∗)_0≤t≤T

is of the form (recall the definition of

v^κ_t

in (3.11))

y^∗_t =ye_t^1,γ1

and

v^∗_t =v_t^γ1.

(3.35) The optimal wealth process is the solution to the SDE

d

X_t^∗ = X_t^∗(r_t −v^∗_t + (y_t^∗)^′θ_t)

d

t +X_t^∗(y_t^∗)^′

d

W_t, X₀^∗ = x ,

given by

X_t^∗ = xEt(y^∗)T−γ₁(ζ)t

T e^{Rt−Vt+(y∗, θ)t}, 0≤t≤T .

Corollary 3.12 If

_kθkT = 0

, then the optimal solution of problem (3.21) is given in Corollary 3.5.

Similarly to the optimization problem with VaR constraint we observe two marginal cases. Note that the following corollary is again a consequence of (2.9).

Corollary 3.13 Assume that

_kθkT >0

and that (3.29) and (3.34) hold. Then

γ₁=ζ

and the assertions of Corollary 3.6 hold with

κ1

replaced by

κ1

.

Theorem 3.14 Assume that

_kθk²T/2

?

NO!!!

ζ >1− 1

T+ 1F_α(|q_α|+kθkT)e^kθk2T .

(3.36) Then for all

0< α <1/2

for which

_|q_α|>max(1,kθkT)

the solution of problem (3.21) is given by (2.7)–(2.8).

3.3 Conclusion

If we compare the optimal solutions (3.17) and (3.35) with the unconstrained optimal

strategy (2.7), then the risk bounds forces investors to restrict their investment into the

risk assets by multiplying the unconstrained optimal strategy by the coefficients given

in (3.10) and (3.13) for VaR constraints and (3.30) and (3.33) for ES constraints. The

impact of the risk measure constraints enter into the portfolio process through the risk

level

ζ

and the confidence level

α

.

4 Auxiliary results and proofs

In this section we consider maximization problems with constraints for the two terms of (2.9):

I(V) :=

Z T 0

(lnvt−Vt)

d

t

and

H(y) :=

Z T 0

ω(t)

y^′_tθt−1 2|yt|²

d

t .

(4.1) We start with a result concerning the optimization of

I(·)

, which will be needed to prove results from both Sections 3.1 and 3.2.

Let

^W[0, T]

be the set of differentiable functions

f : [0, T] → R

having positive cadlag derivative

f^˙

satisfying condition (2.4). For

b >0

we define

W_0,b[0, T] ={f ∈W[0, T] : f(0) = 0

and

f(T) =b}.

(4.2) Lemma 4.1 Consider the optimization problem

max

f∈W₀

,b[0,T]I(f).

(4.3)

The optimal value of

I

is given by

I^∗(b) = max

f∈W₀

,b[0,T]I(f) =I(f^∗) = −TlnT − T ln e^b

e^b−1,

(4.4) with optimal solution

f^∗(t) = ln T e^b

T e^b−t(e^b−1), 0≤t≤T .

(4.5) Proof. Firstly, we consider the optimization problem (4.3) in the space

^C2[0, T]

of two times continuously differentiable functions on

[0, T]

:

max

f∈W₀

,b[0,T]∩C²[0,T]I(f),

By variational calculus methods we find that it has solution (4.4); i.e.

max

f∈W

0,b[0,T]∩C²[0,T]I(f) =I(f^∗),

where the optimal solution

f^∗

is given in (4.5).

Take now

f ∈W_0,b[0, T]

and suppose first that its derivative

f˙_min= inf

0≤t≤T

f˙(t) >0.

Let

Υ

be a positive two times differentiable function on

[−1,1]

such that

^R₋₁¹ Υ(z)

d

z= 1

, and set

Υ(z) := 0

for

_|z| ≥1

. We can take, for example,

Υ(z) =







R1 1

−1exp

−₁¹

−υ2

d

υ exp

−1−z¹²

if

_|z| ≤1,

0

if

_|z|>1.

By setting

f^˙(t) = ˙f(0)

for all

t ≤ 0

and

f^˙(t) = ˙f(T)

for all

t ≥ T

, we define an approximating sequence of functions by

υ_n(t) =n Z

R

Υ(n(u−t)) ˙f(u)

d

u .

It is clear that

(υ_n)_n≥1∈C²[0, T]

. Moreover, we recall that

f^˙

is cadlag, which implies that it is bounded on

[0, T]

; i.e.

sup

0≤t≤T

f˙(t) := ˙f_max < ∞,

and its discontinuity set has Lebesgue measure zero. Therefore, the sequence

(υ_n)_n≥1

is bounded; more preceisly,

0<f˙_min≤ υ_n(t) ≤f˙_max < ∞, 0≤t≤T ,

(4.6) and

υ_n→f˙

as

n→ ∞

for Lebesgue almost all

t∈[0, T]

. Therefore, by the Lebesgue convergence theorem we obtain

n→∞lim Z T

0

|υ_n(t)−f˙(t)|

d

t = 0.

Moreover, inequalities (4.6) imply

|lnυ_n| ≤ln

max( ˙f_max,1) +|ln

min( ˙f_min, 1)

|.

Therefore,

f_n(t) =Rt

0 υ_n(u)

d

u

belongs to

Γ_b

n∩C²[0, T]

for

bn:=RT

0 υ_n(u)

d

u

. It is clear that

n→∞lim I(f_n) = I(f)

and

lim

n→∞bn=b .

This implies that

I(f)≤I^∗(b),

where

I^∗(b)

is defined in (4.4).

Consider now the case, where

inf_0≤t≤T f˙(t) = 0

. For

0 < δ < 1

we consider the approximation sequence of functions

fe_δ(t) = max(δ ,f˙(t))

and

f_δ(t) = Z t

0

fe_δ(u)

d

u , 0≤t≤T .

It is clear that

f_δ ∈Γ_b

δ

for

b_δ =RT

0 fe_δ(t)

d

t

. Therefore,

I(f_δ)≤I^∗(b_δ)

. Moreover, in view of the convergence

δ→0lim Z T

0

fe_δ(t)−f˙(t)

d

t= 0

we get

lim sup_δ→0I(f_δ)≤I^∗(b)

. Moreover, note that

|I(f_δ)−I(f)| ≤ Z

A_δ

(lnδ−ln ˙f(t))

d

t +T Z

A_δ

δ−f˙(t)

d

t

≤ Z

A_δ

(ln ˙f(t))₋

d

t +δ TΛ(A_δ),

where

A_δ = {t ∈ [0, T] : 0 ≤ f˙(t) ≤ δ}

and

Λ(A_δ)

is the Lebesgue measuere of

A_δ

. Moreover, by the definition of the

^W[0, T]

in (4.2) the Lebesgue measure of the set

_{t∈ [0, T] : ˙f(t) = 0}

equals to zero and

^R₀^T(ln ˙f_t)₋

d

t < ∞

. This implies that

limδ→0Λ(A_δ) = 0

and hence

δ→0lim I(f_δ) =I(f),

i.e.

I(f)≤I^∗(b)

.

₂

In order to deal with

H

as defined in (4.1) we need some preliminary result. As usual, we denote by

_L2[0, T]

the Hilbert space of functions

y

satisfying the square integrability condition in (2.3).

Define for

y∈ L2[0, T]

with

_kykT >0

y_t=y_t/kykT

and

l_y(h) =ky+hkT− kykT −(y, h)_T.

(4.7) We shall need the following lemma.

Lemma 4.2 Assume that

y∈ L2[0, T]

and

_kykT >0

. Then for every

h∈ L2[0, T]

the function

l_y(h)≥0

.

Proof. Obviously, if

h≡ay

for some

a∈R, thenl_y(h) = (|1 +a| −1−a)kykT ≥0

. Let now

h6≡ay

for all

a∈R

. Then

l_y(h) = 2(y , h)_T +khk²_T ky+hkT+kykT

−(y, h)_T = khk²_T −(y, h)_T((y, h)_T+l_y(h)) ky+hkT +kykT

.

It is easy to show directly that for all

h

ky+hkT +kykT+ (y, h)_T ≥0

with equality if and only if

h≡ay

for some

a≤ −1

. Therefore, if

h6≡ay

, we obtain

l_y(h) = khk²T−(y , h)²_T

ky+hkT +kykT+ (y, h)_T ≥ 0. 2

4.1 Results and proofs of Section 3.1

We introduce the constraint

K: L2[0, T]→R

as

K(y) := 1

2kyk²_T +|q_α| kykT−(y, θ)_T

(4.8) For

0< a≤ −ln(1−ζ)

we consider the following optimization problems

y∈Lmax2[0,T] H(y)

subject to

K(y) =a

(4.9) Proposition 4.3 Assume that the conditions of Lemma 3.3 hold. Then the optimization problem (4.9) has the unique solution

y^∗=ey^a=θtτλ_a(t)

with

λ_a = Φ⁻¹(a)

.

Proof. According to Lagrange’s method we consider the following unconstrained problem

y∈Lmax2[0,T]Ψ(y, λ),

(4.10) where

Ψ(y, λ) =H(y)−λK(y)

and

λ∈R

is the Lagrange multiplier. Now it suffices to find some

λ ∈ R

for which the problem (4.10) has a solution, which satisfies the constraint in (4.9). To this end we represent

Ψ

as

Ψ(y, λ) = Z T

0

(ω(t) +λ)

y_t^′θ_t−1 2|y_t|²

d

t −λ|q_α| kykT.

It is easy to see that for

λ < 0

the maximum in (4.10) equals

+∞

; i.e. the problem (4.9) has no solution. Therefore, we assume that

λ≥0

. First we calculate the Fr´echet derivative; i.e. the linear operator

D_y(·, λ) :L2[0, T]→R

defined for

h∈ L2[0, T]

as

D_y(h, λ) = lim

δ→0

Ψ(y+δh, λ)−Ψ(y, λ)

δ .

For

_kykT >0

we obtain

D_y(h, λ) = Z T

0

(d_y(t, λ))^′h_t

d

t

with

d_y(t, λ) = (ω(t) +λ)(θ_t−y_t)−λ|q_α|y_t.

If

_kykT = 0

, then

D_y(h, λ) = Z T

0

(ω(t) +λ)θ_t^′h_t

d

t−λ|q_α| khkT.

Define now

∆_y(h, λ) = Ψ(y+h, λ)−Ψ(y, λ)−D_y(h, λ).

(4.11)

We have to show that

∆_y(h, λ≤0

for all

y, h∈ L2[0, T]

. Indeed, if

_kykT = 0

then

∆_y(h, λ) =−1 2

Z T 0

(ω(t) +λ)|h_t|²

d

t ≤ 0.

If

_kykT >0

, then

∆_y(h, λ) =−1 2

Z T 0

(ω(t) +λ)|h_t|²

d

t −λ|q_α|l_y(h)≤0,

by Lemma 4.2 for all

λ≥0

and for all

y, h∈ L2[0, T]

.

To find the solution of the optimization problem (4.10) we have to find

y ∈ L2[0, T]

such that

D_y(h, λ) = 0

for all

h∈ L2[0, T].

(4.12) First notice that for

_kθkT >0

, the solution of (4.12) can not be zero, since for

y= 0

we obtain

D_y(h, λ)<0

for

h=−θ

. Consequently, we have to find an optimal solution to (4.12) for

y

satisfying

_kykT >0

. This means we have to find a non-zero

y ∈ L2[0, T]

such that

d_y(t, λ) = 0.

One can show directly that for

0 ≤λ ≤λ_max

the unique solution of this equation is given by

y_t^λ:=θ_tτ_λ(t),

(4.13)

where

τλ(t)

is defined in (3.6). It remains to choose the Lagrage multiplier

λ

so that it satisfies the constraint in (4.9). To this end note that

K(y^λ) = Φ(λ).

Under the conditions of Lemma 3.3 the inverse of

Φ

exists. Thus the function

y^λa 6≡0

with

λ_a = Φ⁻¹(a)

is the solution of the problem (4.9).

₂

We are now ready to proof the main results in Section 3.1. The auxiliary lemmas are proved in A.1.

Proof of Theorem 3.4. In view of the representation of the cost function

J(x, ν)

in the form (2.9), we start to maximize

J(x, ν)

by maximizing

I

over all functions

V

. To

???

this end we fix the last value of the consumption process, by setting

κ= 1−e^−VT

. By Lemma 4.1 we find that

I(V)≤I(V^κ) =−TlnT +Tlnκ ,

where

V_t^κ= Z t

0

v^κ(t)

d

t= ln T

T −κt, 0≤t≤T .

(4.14)

Define now

L_t(ν) = (y, θ)_t − 1

2kyk²_t−V_t− |q_α| kykt, 0≤t≤T ,

and note that condition (3.3) is equivalent to

0≤t≤Tinf L_t(ν) ≥ ln (1−ζ).

(4.15) Firstly, we consider the bound in (4.15) only at time

t=T

:

L_T(ν) ≥ ln (1−ζ).

Recall definition (4.8) of

K

and choose the function

V

as

V^κ

as in (4.14). Then we can rewrite the bound for

LT(ν)

as a bound for

K

and obtain

K(y) ≤ ln1−κ

1−ζ, 0≤κ≤ζ .

To find the optimal investment strategy we need to solve the optimization problem (4.9) for

0≤a≤ln((1−κ)/(1−ζ))

. By Proposition 4.3 for

0< a≤ −ln(1−ζ)

y∈L2[0,T]max, K(y)=a H(y) =H(ye^a) :=C(a),

(4.16) where the solution

y_e^a

is defined in Proposition 4.3. Note that the definitions of the functions

H

and

_ey^a

imply

C(a) = Z T

0

ω(t)

τλ_a(t)−1 2τ_λ²

a(t)

|θ_t|²

d

t

with

λ_a= Φ⁻¹(a).

To consider the optimization problem (4.9) for

a= 0

we observe that

K(y)≥ kykT (|q_α| − kθkT) +1

2kyk²_T ≥0,

provided that

_|q_α| > kθkT

(which follows from (3.14)). Thus, there exists only one function for which

K(y) = 0

, namely

y≡0

. Furthermore, by Lemma 3.2

ρ(λ_max) = 0

and, therefore, definition (3.6) implies

τλmax(·)≡0, y^λmax≡0

and

Φ(λ_max) = 0.

(4.17) This means that

λ_max = Φ⁻¹(0)

and

y^Φ−1(0) = 0

; i.e.

y^λa

with

λ_a = Φ⁻¹(a)

is the solution of the optimization problem (4.9) for all

0 ≤a ≤ −ln(1−ζ)

. Now we calculate the derivative of

C(a)

:

d

d

aC(a) = ˙λ_a Z T

0

ω(t) 1−τλ_a(t)

|θ_t|²

∂τλ(t)

∂λ |λ=λ_a

d

t ,

Since

λ^˙_a = 1/˙Φ(λ_a)

, by Lemma A.1, the derivative of

C(a)

is positive. Therefore,

0≤a≤ln((1−κ)/(1−ζmax ))C(a) = C

ln1−κ 1−ζ

,

and we choose

a= ln((1−κ)/(1−ζ)

in (4.16).

Now recall the definitions (3.10) and (3.11) and set

ν^κ = (ye_t^κ, v_t^κ)_0≤t≤T

. Thus for

ν∈ U

with

V_T =−ln(1−κ)

we have

J(x, ν) ≤J(x, ν^κ) =A(x) + Γ(κ).

It is clear that (3.13) gives the optimal value for the parameter

κ

.

To finish the proof we have to verify condition (4.15) for the strategy

ν^∗

defined in (3.17). Indeed, we have

L_t(ν^∗) = (y^∗, θ)_t−1

2ky^∗k²_t− |q_α| ky^∗kt− Z t

0

v_s^∗

d

s

=:− Z t

0

g(u)

d

u− Z t

0

v^∗_s

d

s ,

where

g(t) =τ_t^∗|θ_t|²

|q_α|χ(t)−1 + τ_t^∗ 2

and

χ(t) = τ_t^∗ 2qRt

0(τ_s^∗)²|θ_s|²

d

s .

We recall

φ(κ)

from (3.9) and

κ1

from (3.13), then

τ_t^∗=τυ1(t)

with

υ₁=φ(γ).

Definition (3.6) implies

χ(t)≥ τυ1(T)

2τυ1(0)kθkT ≥ 1 +υ₁

2kθkT(1 +T+υ₁)≥ 1 2kθkT(1 +T).

Therefore, condition (3.14) guarantees that

g(t)≥0

for

t≥0

, which implies

L_t(ν^∗)≥L_T(ν^∗) = ln(1−ζ).

This concludes the proof of Theorem 3.4.

₂

Proof of Corollary 3.6. Consider now the optimization problem (3.13). To solve it we have to find the derivative of the integral in (3.12)

E(κ) :=

Z T 0

ω(t)|θ_t|²

τφ(κ)(t)−1

2τ_φ(κ)² (t)

d

t .

Indeed, we have with

φ(κ)

as in (3.9),

E(κ) =˙ Z T

0

ω(t)|θ_t|² 1−τφ(κ)(t) ∂

∂κτφ(κ)(t)

d

t .

Defining

τ₁(t, φ(κ)) :=^∂τ_∂λ^λ(t)|λ=φ(κ)

we obtain

∂

∂κτφ(κ)(t) =τ₁(t, φ(κ))

d

d

κφ(κ).

(4.18)

Therefore,

E(κ) =˙ − 1

1−κB(φ(κ))

with

B(λ) = RT

0 ω(t)|θ_t|² (1−τλ(t))τ₁(t, λ)

d

t

˙Φ(λ) .

Define

_bτ(t, λ) :=|qα|τλ(t)/kτλθkT

. Then, in view of Lemma A.1, we have

τ₁(t, λ)≤0

and, therefore, taking representation (A.1) into account we obtain

B(λ) = RT

0 ω(t)|θ_t|² (1−τ(t, λ))|τ₁(t, λ)|

d

t RT

0 bτ(t, λ)|τ₁(t, λ)|

d

t .

Moreover, using the lower bound (A.2) we estimate

B(λ)< (1 +T)²|θ|²_∞kθkT

|q_α| −(T+ 1)kθkT

=:Bmax.

(4.19)

Condition (3.18) for

0< ζ < κ₀

implies that

B_max≤

1 ζ −1

T−1.

Thus for

0≤κ≤ζ < κ₀

we obtain

˙Γ(κ) > T

κ − 1

1−κ(1 +B_max) ≥ T ζ − 1

1−ζ(1 +Bmax) ≥ 0.

This implies

γ = ζ

and, therefore,

a(γ) := ln((1−γ)/(1−ζ) = 0

, which implies also by Lemma 3.3 that

φ(a(γ)) = λ_max

. Therefore, we conclude from (4.17) that

y_t^∗=τλmax(t)θ_t= 0

for all

0≤t≤T

.

₂

Proof of Theorem 3.7. It suffices to verify condition (4.15) for the strategy

ν^∗ = (y_t^∗, v_t^∗)_0≤t≤T

with

y_t^∗ = θ_t

and

v_t^∗ = 1/ω(t)

for

t ∈ [0, T]

. It is easy to show that condition (3.19) implies that

L_T(ν^∗) ≥ln(1−ζ)

. Moreover, for

0 ≤t ≤ T

we can represent

L_t(ν^∗)

as

L_t(ν^∗) =− Z t

0

g_s^∗

d

s− Z t

0

v^∗_s

d

s ,

where

g_t^∗= |q_α|

kθkt

−1 |θ_t|²

2 ≥

|q_α| kθkT

−1 |θ_t|²

2 ≥0

since we have assumed that

_|qα| ≥ kθkT

. Therefore,

L_t(ν^∗)

is decreasing in

t

; i.e.

L_t(ν^∗)≥L_T(ν^∗)

for all

0≤t≤T

. This implies the assertion of Theorem 3.7.

₂

4.2 Results and proofs of Section 3.2 Next we introduce the constraint

K₁(y) :=−(y, θ)_T −f_α(kykT)

(4.20) with

fα(x) := lnF_α(|qα|+x)

and

F_α

introduced in (3.20).

For

0< a≤ −ln(1−ζ)

we consider the following optimization problems

y∈Lmax2[0,T] H(y)

subject to

K₁(y) =a .

(4.21) The following result is the analog of Proposition 4.3.

Proposition 4.4 Assume that the conditions of Lemma 3.10 hold. Then the optimiza- tion problem (4.21) has the unique solution

y_t^∗=ye^1,a_t =θtςλ_1,a(t)

with

λ_1,a= Φ⁻¹₁ (a)

. Proof. As in the proof of Proposition 4.3 we use Lagrange’s method. We consider the unconstrained problem

y∈Lmax2[0,T]Ψ₁(y, λ),

(4.22) where

Ψ₁(y, λ) =H(y)−λK₁(y)

and

λ≥0

is the Lagrange multiplier. Taking into account the defining

f_α

in (4.20), we obtain the representation

Ψ₁(y, λ) = Z T

0

(ω(t) +λ)θ^′_ty_t −ω(t) 2 |y_t|²

d

t+λ f_α(kykT).

Its Fr´echet derivative is given by

D_1,y(h, λ) = lim

δ→0

Ψ₁(y+δh, λ)−Ψ₁(y, λ)

δ .

It is easy to show directly that for

_kykT >0 D_1,y(h, λ) =

Z T 0

(d_1,y(t, λ))^′h_t

d

t ,

where

d_1,y(t, λ) = (ω(t) +λ)θ_t−ω(t)y_t+λf˙_α(kykT)y_t,

and

f^˙_α(·)

denotes the derivative of

f_α(·)

.

If

_kykT = 0

, then

D_1,y(h, λ) = Z T

0

(ω(t) +λ)θ_t^′h_t

d

t+λf˙_α(0)khkT.

We set now

∆_1,y(h, λ) = Ψ₁(y+h, λ)−Ψ₁(y, λ)−D_1,y(h, λ),

(4.23)