Optimal Investment of Merton Model for Multi-Investors with Frictions

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Optimal Investment of Merton Model for Multi-Investors with Frictions

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Submitted:

29 May 2023

Posted:

01 June 2023

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Abstract

In this paper, we extend the Merton model of investment in discrete time to the cases when there is a finite number of investors and the market is with frictions represented by convex penalty functions defined for each investor. In the main result of this paper, we proved the existence of optimal strategy of investment by using a new approach based on the formulation of an equivalent general equilibrium economy model.

Keywords:

Subject: Computer Science and Mathematics - Applied Mathematics

1. Introduction

The optimal investment of Merton model introduced in [12,13] has been investigated by researchers and extended in different contexts since it appears. One important extension in continuous time is due to Magill and Constantinides [11], where a linear transaction costs function is used in the context of Merton problem. In discrete time, the study of Merton model with linear transaction costs was developed by Jouini and Kallal in [8]. We can also cite the papers of Shreve and Soner [14] which extended Merton problem by including viscosity theory and Cetin, Jarrow and Protter [3] which studied the Merton model for illiquid markets.

Recently, Chebbi and Soner in [1] extend the Merton model in discrete time and finite horizon to the case of market with frictions represented by a convex penalty function defined for one investor. They proved the existence of an optimal strategy by solving a dynamic optimization problem. Then Ounaies, Bonnisseau, Chebbi and Soner in [15] extend this model to the infinite horizon and and they proved the existence of optimal strategy by an argument of fixed points.

In this paper, we will take this direction of extension in order to prove the existence of optimal strategy in Merton model for market frictions in infinte horizon when there are finite number of investors. our approach is very different and based on constructing an equivalent general equilibrium model with multiple agents. The idea to use the general equilibrium theory is inspired by the paper of Le Van and Dana [10].

Sections of this paper are organized as follow: In Section 2, We give a description of Merton model of investment problem in infinite horizon and with market frictions modeled by convex) penalty functions defined for each investor and constraints conditions about liquidation value is defined consequently.

In Section 3, we construct an equivalent general equilibrium economy model to Merton model of investment.

In Section 4, we prove the the existence of an equilibrium for the model of general equilibrium economy and the optimal strategy of Merton problem of invested will be this obtained equilibrium.

2. The Model

Let

(Ω, F, P)

be the probability space where

Ω = {(R^{N})}^{\infty}

is the space of events

{(ω_{t})}_{t \geq 1}

. For

t \in N^{*}

, let

F_{t} = σ (B_{s}; s \in {1, 2, . . ., t})

be the

σ

-field generated by the canonical mapping process

B_{t} (ω) = ω_{t}, t \geq 1, ω \in Ω

. We denote by

F_{\infty} = σ (⋃_{t \in N} F_{t})

, where

F_{0} = {\emptyset, Ω}

is the trivial

σ

-algebra and by

P : F \to [0, 1]

, the probability measure.

In the discrete time model of this paper, we suppose that the market is with a money market account paying a return

r > 0

and N risky assets that provide a random return of

R = {(R_{t})}_{t \geq 1}

with values in

{[- 1, \infty)}^{N}

that are supposed to be identically and independently distributed over time. We denote by

{(p^{j})}_{1 \leq j \leq N}

, the strictly positive asset price process that will is supposed to satisfy the following condition:

p_{t}^{j} = p_{0}^{j} \prod_{k \geq 1} [1 + R_{k}^{j}] ⟺ R_{t}^{j} = \frac{p_{t}^{j} - p_{t - 1}^{j}}{p_{t - 1}^{j}}, j = 1, \dots, N .

(2.1)

where

p_{0}^{j}

is the initial stock value. The return vector at time t is given by

R_{t} (ω) = B_{t} (ω) = ω_{t}

t \in N^{*}

j = 1, \dots, N .

Then

R_{t}

’s are

F_{t}

-measurable and consequently,

R = {(R_{t})}_{t \geq 1}

is an

{(R)}^{N}

-valued,

F

-adapted process. The process p is an

{(R^{+})}^{N}

-valued

F

-adapted process.

In our multi-investors model, we suppose that there is a finite number m of investors labeled i,

(i = 1, 2, \dots, m)

. Each investor has to choose a portfolio of assets j,

(j = 0, 1, 2, \dots, N)

. We denote by

y = {(y_{i, t}^{j})}_{t \geq 1}

, the individual i’s process of money invested in the j-th stock at any time t prior to the portfolio adjustment. The riskless asset

x = {(x_{i, t})}_{t \geq 1}

will be the process of money invested in the money market account at any time t. Shares are traded at determined price vector

p_{t} = (p_{t}^{1}, \dots, p_{t}^{N})

. For

t \geq 1

, the process

z_{i, t}

will denote the number of shares held by the i-th investor at time t with values in

R^{N}

and we have:

y_{i, t}^{j} = z_{i, t}^{j} p_{t}^{j}, j = 1, \dots, N, i = 1, \dots, m, t \geq 1 .

In our model of markets with frictions, we assume that there is a penalty function

g_{i} : R^{N} \to R_{+}^{N}

for each investor i due to transaction costs. The dynamics of the riskless asset will be as follow:

x_{i, t + 1} = (x_{i, t} - α_{i, t} \cdot 1 - p_{t} g_{i} ((z_{i, t + 1} - z_{i, t})) \cdot 1 - c_{i, t}) (1 + r), t \geq 1,

(2.2)

where the

F

-adapted process

c_{i}

denotes the consumption of the i-th investor, and

α_{i}

is the portfolio adjustment process given by:

α_{i, t}^{j} : = p_{t}^{j} Δ_{t} z_{i}^{j} = p_{t}^{j} (z_{i, t + 1}^{j} - z_{i, t}^{j}), j = 1, \dots, N, t \geq 1 .

(2.3)

Note taht rebalancing of portfolio will occur between time t and time

t + 1

and it is easy to see that:

\begin{matrix} y_{i, t + 1}^{j} = (y_{i, t}^{j} + α_{i, t}^{j}) (1 + R_{t + 1}^{j}), \end{matrix}

(2.4)

and the mark-to-market value is given by:

ω_{i, t} : = x_{i, t} + y_{i, t} \cdot 1 = x_{i, t} + \sum_{j = 1}^{N} y_{i, t}^{j}

3. General equilibrium model of Merton investment problem

Given a portfolio position

(x, y) \in R \times {(R^{+})}^{N}

, the after-liquidation value will be defined as follows:

\begin{matrix} L (x_{i, t}, y_{i, t}) & = & a_{i, t} + b_{i, t} \cdot 1 - p_{t} g_{i} ((z_{i, t + 1} - z_{i, t})) \cdot 1 \\ = & x_{i, t} + p_{t} z_{i, t} \cdot 1 - p_{t} g_{i} ((z_{i, t + 1} - z_{i, t})) \cdot 1 \end{matrix}

(3.1)

and the solvency condition is given by the requirement that

L (x_{i, t}, y_{i, t}) \geq 0

for all

t \geq 1

, P-almost surely. Hence, our optimal investment problem will be formulated by the following optimization problem:

\begin{matrix} Q_{i} (x, y) & : & sup_{(c_{i, t}, z_{i, t})} E [\sum_{t = 0}^{\infty} ρ_{i}^{t} u_{i} (c_{i, t})] \\ s u b j e c t t o & : & x_{i, t} + p_{t} z_{i, t} \cdot 1 - p_{t} g_{i} ((z_{i, t + 1} - z_{i, t})) \cdot 1 \geq 0 a . e . \end{matrix}

where for each investor i,

u_{i}

is the utility function and

ρ_{i}^{t}

is the impatience parameter.

The infinite-horizon sequence of prices and quantities are given by:

(p, {(c_{i}, z_{i})}_{i = 1}^{m})

where, for each

i = 1, \dots, m

(p, c_{i}, z_{i}) = ({(p_{t})}_{t = 0}^{+ \infty}, {(c_{i, t})}_{t = 0}^{+ \infty}, {(z_{i, t})}_{t = 0}^{+ \infty}) \in {(R_{+}^{+ \infty})}^{N} \times R_{+}^{+ \infty} \times {(R_{+}^{+ \infty})}^{N},

Now let

E

be the economy characterized by:

E = (R^{N}, {(u_{i}, ρ_{i}, z_{i, - 1})}_{i = 1}^{m})

Equilibrium of this economy is determined by the set of consumption policies and price processes for which each agent maximizes his/her expected utility. More precisely:

Definition 1.

The process

{({\bar{p}}_{t}, {({\bar{c}}_{i, t}, {\bar{z}}_{i, t})}_{i = 1}^{m})}_{t = 0}^{\infty}

is an equilibrium of the economy

E

if the following conditions are satisfied:

Price positivity: ${\bar{p}}_{t} > 0$ for $t \geq 0 .$
Market clearing: at each $t \geq 0$ ,

$\begin{matrix} \sum_{i = 1}^{m} & {\bar{c}}_{i, t} + p_{t} g_{i} ((z_{i, t + 1} - z_{i, t}) \cdot 1 = ω_{t}, a . e . \\ \sum_{j = 1}^{N} & z_{i, t}^{j} = 1 a . e ., \forall i \in {1, \dots, m}, \\ \sum_{i = 1}^{m} & z_{i, t}^{0} = 0 a . e . \end{matrix}$
Optimal consumption plans: for each i, ${({({\bar{c}}_{i, t}, {\bar{z}}_{i, t})}_{i = 1}^{m})}_{t = 0}^{\infty}$ is a solution of the problem $Q_{i} (x, y)$ .

4. Existence of Equilibrium

We will use the following standard assumptions in order to prove the existence of equilibrium:

- Assumption (H1): For each

i = 1, \dots, m

u_{i}

is continuously differentiable, strictly increasing and concave function satisfying

u_{i} (0) = 0

u_{i}^{'} (0) = \infty

- Assumption (H2): At initial period 0,

z_{i, - 1} \geq 0

, and

z_{i, - 1} \neq 0

for

i = 1, \dots, m

with

\sum_{i = 1}^{m} z_{i, - 1} = 1_{m}

- Assumption (H3):

g_{i} : R^{N} \to R_{+}^{N},

is convex with

g_{i} (0) = 0

and

g_{i} \geq 0

for

i = 1, \dots, m

- Assumption (H4): The utility of each agent i is finite:

\sum_{t = 0}^{\infty} ρ_{i}^{t} u_{i} (c_{i, t}) < \infty .

We now constructed the T-truncated economy

E^{T}

E

in which we suppose that there are no activities from period

T + 1

to the infinity and by using a classical argument, we compactify this economy by using the bounded economy

E_{b}^{T}

E^{T}

in which all random variables are bounded. Consider a finite-horizon bounded economy which goes on for

T + 1

periods:

t = 0, \dots, T

with

B_{c}, B_{z}

defined by:

\begin{matrix} C_{i} & : = & \{(c_{i, 0}, \dots, c_{i, T}) : 0 \leq c_{i, t} \leq B_{c}, \forall t \in {1, \dots, T}\} = {[0, B_{c}]}^{T + 1}; \\ Z_{i} & : = & \{(z_{i, 1}^{j}, \dots, z_{i, T}^{j}) : 0 \leq z_{i, t}^{j} \leq B_{z}, \forall t \in {1, \dots, T}\} = {[0, B_{z}]}^{T} . \end{matrix}

The solvency set is given by:

U_{i}^{T} (x, y) : = \{(c_{i}, z_{i}) \in C_{i} \times Z_{i} : x_{i, t} + p_{t} z_{i, t} \cdot 1 - p_{t} g_{i} (z_{i, t + 1} - z_{i, t}) \cdot 1 \geq 0, P - a . s\} .

Now, we define the economy

E_{b}^{T, ϵ}

, for each

ϵ > 0

such that

m ϵ < 1

, by adding

ϵ

units for each agent at date 0. This condition assure the non-emptiness of the solvency set. Thus, the feasible set of each agent i will be:

\begin{matrix} U_{i}^{T, ϵ} (x, y) & : = & {(c_{i}, z_{i}) \in R_{+}^{T + 1} \times {(R_{+}^{T + 1})}^{N} : \\ (x_{i, 0} + p_{0} (z_{i, 0} + ϵ) \cdot 1 - p_{0} g_{i} (- (z_{i, 0} + ϵ)) \cdot 1 - c_{i, 0}) (1 + r) \geq 0, \\ f o r e a c h 1 \leq t \leq T : x_{i, t} + p_{t} z_{i, t} \cdot 1 - p_{t} g_{i} (z_{i, t + 1} - z_{i, t}) \cdot 1 \geq 0, P - a . s .} \end{matrix}

\begin{matrix} L_{i}^{T, ϵ} (x, y) & : = & {(c_{i}, z_{i}) \in R_{+}^{T + 1} \times {(R_{+}^{T + 1})}^{N} : \\ (x_{i, 0} + p_{0} (z_{i, 0} + ϵ) \cdot 1 - p_{0} g_{i} (- (z_{i, 0} + ϵ)) \cdot 1 - c_{i, 0}) (1 + r) > 0, \\ f o r e a c h 1 \leq t \leq T : x_{i, t} + p_{t} z_{i, t} \cdot 1 - p_{t} g_{i} (z_{i, t + 1} - z_{i, t}) \cdot 1 > 0, P - a . s .} \end{matrix}

Lemma 4.1.

The set

L_{i}^{T, ϵ} (x, y)

is non empty, for

t = 0, \dots, T

Proof.

Indeed,

\begin{matrix} L & (x_{i, 1}, y_{i, 1}) \\ = & L ((x_{i, 0} + p_{0} (z_{i, 0} + ϵ) \cdot 1 - p_{0} g_{i} (- (z_{i, 0} + ϵ)) \cdot 1 - c_{i, 0}) (1 + r), (y_{i, 0}^{j ϵ} + α_{i, 0}^{j ϵ}) (1 + R_{1}^{j})) \\ = & L ((x_{i, 0} + p_{0} (z_{i, 0} + ϵ) \cdot 1 - p_{0} g_{i} (- (z_{i, 0} + ϵ)) \cdot 1 - c_{i, 0}) (1 + r), 0) \\ = & (x_{i, 0} + p_{0} (z_{i, 0} + ϵ) \cdot 1 - p_{0} g_{i} (- (z_{i, 0} + ϵ)) \cdot 1 - c_{i, 0}) (1 + r) \geq 0 \end{matrix}

Now, since

ϵ, (z_{i, 0} + ϵ) > 0

, we can select

c_{i, 0} \in (0, B_{c})

and

z_{i, 0} \in (0, B_{z})

such that

(x_{i, 0} + p_{0} (z_{i, 0} + ϵ) \cdot 1 - p_{0} g_{i} (- (z_{i, 0} + ϵ)) \cdot 1 - c_{i, 0}) (1 + r) > 0

□

Lemma 4.2.

The set

U_{i}^{T} (x, y)

has convex values.

Proof.

Now we want to show that

U (x, y)

is convex. Take

(c_{i, t}^{k}, α_{i, t}^{k}) \in U (x^{k}, y^{k})

, for

k = 1, 2

and

t \geq 1

. For

λ \in [0, 1]

, we note by

{\bar{c}}_{i, t} = λ c_{i, t}^{1} + (1 - λ) c_{i, t}^{2}

and similarly

{\bar{x}}_{i, t}

{\bar{z}}_{i, t}

. We have:

\begin{matrix} L ({\bar{x}}_{i, t}, {\bar{y}}_{i, t}) & = & {\bar{x}}_{i, t} + {\bar{p}}_{t} {\bar{z}}_{i, t} \cdot 1 - {\bar{p}}_{t} g ({\bar{z}}_{i, t + 1} - {\bar{z}}_{i, t}) \cdot 1 \\ = & λ (x_{i, t}^{1} + {\bar{p}}_{t} z_{i, t}^{1} \cdot 1) + (1 - λ) (x_{i, t}^{2} + {\bar{p}}_{t} z_{i, t}^{2} \cdot 1) - {\bar{p}}_{t} g ({\bar{z}}_{i, t + 1} - {\bar{z}}_{i, t}) \cdot 1 \\ \geq & {\bar{p}}_{t} [λ g (z_{i, t + 1}^{1} - z_{i, t}^{1}) \cdot 1 + (1 - λ) g (z_{i, t + 1}^{2} - z_{i, t}^{2}) \cdot 1 - g ({\bar{z}}_{i, t + 1} - {\bar{z}}_{i, t}) \cdot 1] \\ \geq & 0 \end{matrix}

since g is convex and

(x_{i, t}^{k}, y_{i, t}^{k}) \in \bar{L}

for

k = 1, 2

and

t \geq 1

□

For simplicity, we denote

U_{i} = C_{i} \times Z_{i}

Lemma 4.3.

L_{i}^{T, ϵ} (x, y)

is lower semi-continuous correspondence on

U_{i}

and

U_{i}^{T, ϵ} (x, y)

is upper semi-continuous with compact convex values.

Proof.

Since

L_{i}^{T, ϵ} (x, y)

is non-empty and has open graph, then it is lower semi-continuous correspondence. Since

U_{i}

is compact and the correspondence

U_{i}^{T, ϵ} (x, y)

has a closed graph, then

U_{i}^{T, ϵ} (x, y)

is upper semi-continuous with compact values.

□

Definition 2.

The stochastic process

{({\bar{p}}_{t}, {({\bar{c}}_{i, t}, {\bar{z}}_{i, t})}_{i = 1}^{m})}_{t = 0}^{T}

is an equilibrium of the economy

E_{b}^{T}

if it satisfies the following conditions:

Price positivity: ${\bar{p}}_{t} > 0$ for $t = 0, 1, \dots, T$
Market clearing:

$\begin{matrix} \sum_{i = 1}^{m} {\bar{c}}_{i, 0} + p_{0} g_{i} (- ({\bar{z}}_{i, 0} + ϵ)) & = & \sum_{i = 1}^{m} x_{i, 0} + p_{0} ({\bar{z}}_{i, 0}^{j} + ϵ) \cdot \vec{1}, a . e . \\ \sum_{i = 1}^{m} {\bar{c}}_{i, t} + p_{t} g_{i} ({\bar{z}}_{i, t + 1} - {\bar{z}}_{i, t}) & = & \sum_{i = 1}^{m} x_{i, t} + {\bar{p}}_{t} {\bar{z}}_{i, t} \cdot 1, a . e . \end{matrix}$
Optimal consumption plans: for each i, ${({\bar{c}}_{i, t}, {\bar{z}}_{i, t})}_{t = 1}^{T}$ is a solution of the maximization problem of agent i with the feasible set $U_{i}^{T, ϵ} (x, y)$ such that:

$Q_{i}^{T, ϵ} (x, y) : sup_{(c_{i, t}, z_{i, t})} E [\sum_{t = 0}^{T} ρ^{t} u_{i} (c_{i, t})] .$

For

i = 0, \dots, m

, consider an element

h = (h_{i})

defined on

X : = B \times \prod_{i = 1}^{m} U_{i}

h_{i} = \{\begin{matrix} p & for i = 0 \\ (c_{i}, z_{i}) & for i = 1, \dots, m \end{matrix}

where

B = {p \in R^{N} | ∥ p ∥ \leq 1}

Now let

φ_{0}

be the correspondence defined by:

φ_{0} : \prod_{i = 1}^{m} U_{i} \to 2^{B}

\begin{matrix} φ_{0} ({(h_{i})}_{i = 0}^{m}) & : = & arg max_{p \in B} {(\sum_{i = 1}^{m} c_{i, 0} + p_{0} g_{i} (- (z_{i, 0} + ϵ)) \cdot 1 - x_{i, 0} - p_{0} (z_{i, 0} + ϵ) \cdot 1 \\ + & \sum_{t = 1}^{T} \sum_{i = 1}^{m} c_{i, t} + p_{t} g_{i} (z_{i, t + 1} - z_{i, t}) \cdot 1 - x_{i, t} - p_{t} z_{i, t}^{j} \cdot 1} . \end{matrix}

and for each

i = 1, \dots, m

, consider:

φ_{i} : B \to 2^{U_{i}}

φ_{i} (p) : = {arg max}_{(c_{i}, z_{i}) \in U (x, y)} E [\sum_{t = 0}^{T} ρ_{i}^{t} u_{i} (c_{i, t})] .

Lemma 4.4.

The correspondence

φ_{i}

is upper semi-continuous with non-empty, convex, compact valued for each

i = 1, \dots, m

Proof.

This is a direct consequence of the Maximum Theorem. □

According to the Kakutani Theorem, there exists

(\bar{p}, ({\bar{c}}_{i, t}, {\bar{z}}_{i, t}))

such that

\begin{matrix} \bar{p} & \in & φ_{0} ({({\bar{c}}_{i}, {\bar{z}}_{i})}_{i = 1}^{m}) \end{matrix}

(4.1)

\begin{matrix} ({\bar{c}}_{i}, {\bar{z}}_{i}) & \in & φ_{i} (\bar{p}) . \end{matrix}

(4.2)

For simplicity, we denote by

\begin{matrix} {\bar{E}}_{t} & = & \sum_{i = 1}^{m} {\bar{c}}_{i, t} - x_{i, t}, t \geq 0 \\ {\bar{F}}_{0} & = & \sum_{i = 1}^{m} g_{i} (- ({\bar{z}}_{i, 0}^{j} + ϵ)) - ({\bar{z}}_{i, 0}^{j} + ϵ) \cdot 1 \\ {\bar{F}}_{t} & = & \sum_{i = 1}^{m} g_{i} ({\bar{z}}_{i, t + 1} - {\bar{z}}_{i, t}) - {\bar{z}}_{i, t}^{j} \cdot 1, t \geq 1 \end{matrix}

Lemma 4.5.

Under Assumptions (H1), (H2) and (H3), there exists an equilibrium for the finite-horizon bounded ϵ-economy

E_{b}^{T, ϵ}

Proof.

We start proving that

{\bar{E}}_{t} + \bar{p_{t}} {\bar{F}}_{t} = 0

and

{\bar{p}}_{t} > 0

for

t = 0, \dots, T .

Indeed, From (4.1), one can easily check that for every

p \in B

, we have:

\sum_{t = 0}^{T} (p_{t} - \bar{p_{t}}) {\bar{F}}_{t} \leq 0 .

(4.3)

We recall the solvency constraint,

x_{i, t} + {\bar{p}}_{t} z_{i, t} \cdot 1 - {\bar{p}}_{t} g_{i} (z_{i, t + 1} - z_{i, t}) \cdot 1 \geq 0

Moreover, the value of an agent’s consumption cannot exceed the value of his wealth and the following inequality will be satisfied:

\begin{matrix} x_{i, t} + {\bar{p}}_{t} {\bar{z}}_{i, t} \cdot 1 - {\bar{p}}_{t} g_{i} ({\bar{z}}_{i, t + 1} - {\bar{z}}_{i, t}) \cdot 1 & \geq & {\bar{c}}_{i, t} \\ x_{i, t} - {\bar{c}}_{i, t} + {\bar{p}}_{t} {\bar{z}}_{i, t} \cdot 1 - {\bar{p}}_{t} g_{i} ({\bar{z}}_{i, t + 1} - {\bar{z}}_{i, t}) \cdot 1 & \geq & 0 \end{matrix}

(4.4)

By summing the inequality (4.4) over i, we obtain that, for each t:

\begin{matrix} \sum_{i = 1}^{m} x_{i, t} - {\bar{c}}_{i, t} + {\bar{p}}_{t} [\sum_{i = 1}^{m} {\bar{z}}_{i, t}^{j} \cdot 1 - g_{i} ({\bar{z}}_{i, t + 1} - {\bar{z}}_{i, t}) \cdot 1] & \geq & 0 \\ {\bar{E}}_{t} + \bar{p_{t}} {\bar{F}}_{t} & \leq & 0 \end{matrix}

(4.5)

{\bar{p}}_{t} = 0

, we deduce that

{\bar{c}}_{i, t} = B_{c} > ω_{i, t}

. Therefore for all t,

\sum_{i = 1}^{m} {\bar{c}}_{i, t} > \sum_{i = 1}^{m} x_{i, t}

, which contradicts (4.5). Hence, we obtain as a result,

{\bar{p}}_{t} > 0

Since prices are strictly positive and the utility functions are strictly increasing, all budget constraints are binding. By summing over i) at date t, we obtain:

{\bar{E}}_{t} + \bar{p_{t}} {\bar{F}}_{t} = 0 .

Hence, the optimality of

({\bar{c}}_{i}, {\bar{z}}_{i})

is from (4.2). □

Lemma 4.6.

Suppose Assumptions (H1), (H2) and (H3) are satisfied, then there exists an equilibrium for the finite-horizon bounded economy

E_{b}^{T}

Proof.

We have proved that for each

ϵ = \frac{1}{n} > 0

, where n is an integer and large enough, there exists an equilibrium denoted:

e q u i (n) : = {(\bar{p} (n), {({\bar{c}}_{i, t} (n), {\bar{z}}_{i, t} (n))}_{i = 1}^{m})}_{t = 0}^{T};

for the economy,

E_{b}^{T, ϵ_{n}}

. Since prices and allocations are bounded, there exists a sub-sequence

(n_{1}, n_{2}, \dots,)

such that

e q u i (n_{s})

converges. without loss of generality, we can assume that

(\bar{p} (n), {({\bar{c}}_{i} (n), {\bar{z}}_{i} (n))}_{i = 1}^{m}) \to (\bar{p}, {({\bar{c}}_{i}, {\bar{z}}_{i})}_{i = 1}^{m})

when n tends to infinity. Moreover, by taking limit of market clearing conditions of the

E_{b}^{T, ϵ_{n}}

, we obtain the corresponding conditions of the bounded truncated economy

E_{b}^{T}

. □

Remark 1.

It should be noticed that at equilibrium, we have

{\bar{p}}_{0} > 0

according to (2.1).

Lemma 4.7.

For each i,

({\bar{c}}_{i}, {\bar{z}}_{i})

is optimal.

Proof.

Since

\sum_{i = 1}^{m} z_{i, - 1}^{j} = 1

, for all

j \in {1, \dots, N}

, there exists an agent i such that

z_{i, - 1} > 0

. According to Remark 1, we have

L_{i}^{T} (x, y) \neq \emptyset

. We now prove the optimality of

({\bar{c}}_{i}, {\bar{z}}_{i})

Let

(c_{i}, z_{i})

be a feasible allocation of the maximization problem of agent i with the feasible set

U_{i}^{T} (\bar{x}, \bar{y})

. We should prove that

E [\sum_{t = 0}^{T} ρ_{i}^{t} u_{i} (c_{i, t})] \leq E [\sum_{t = 0}^{T} ρ_{i}^{t} u_{i} ({\bar{c}}_{i, t})]

Since

L_{i}^{T} (\bar{x}, \bar{y}) \neq \emptyset

, there exists

{(h)}_{h \geq 0}

and

(c_{i}^{h}, z_{i}^{h}) \in L_{i}^{T} (\bar{x}, \bar{y})

such that

(c_{i}^{h}, z_{i}^{h})

converges to

(c_{i}, z_{i})

. Then, for each i, we have

x_{i, t} + {\bar{p}}_{t} z_{i, t}^{h} \cdot \vec{1} - {\bar{p}}_{t} g_{i} (z_{i, t + 1}^{h} - z_{i, t}^{h}) > 0 .

Fix h. Let

n_{0}

be high enough such that for every

n \geq n_{0}

(c_{i}^{h}, z_{i}^{h}) \in U_{i}^{T, \frac{1}{n}} (\bar{x} (n), \bar{y} (n))

. Then

E [\sum_{t = 0}^{T} ρ_{i}^{t} u_{i} (c_{i, t}^{h})] \leq E [\sum_{t = 0}^{T} ρ_{i}^{t} u_{i} ({\bar{c}}_{i, t} (n))] .

Tend n tend to infinity, we obtain

E [\sum_{t = 0}^{T} ρ_{i}^{t} u_{i} (c_{i, t}^{h})] \leq E [\sum_{t = 0}^{T} ρ_{i}^{t} u_{i} ({\bar{c}}_{i, t})] .

Let tends h to infinity, we obtain

E [\sum_{t = 0}^{T} ρ_{i}^{t} u_{i} (c_{i, t})] \leq E [\sum_{t = 0}^{T} ρ_{i}^{t} u_{i} ({\bar{c}}_{i, t})]

We have just showed that

({\bar{c}}_{i}, {\bar{z}}_{i})

is an optimal solution. We now prove that

{\bar{p}}_{t} > 0

for every t. Indeed, if

{\bar{p}}_{t} = 0

, the optimality of

({\bar{c}}_{i}, {\bar{z}}_{i})

implies that

{\bar{c}}_{i, t} = B_{c} > x_{i, t},

contradiction.

□

After proving the existence of the equilibrium when

ϵ

tends to 0, we deduce that this equilibrium holds for the truncated unbounded economy.

Lemma 4.8.

An equilibrium for

E_{b}^{T}

is an equilibrium for

E^{T}

Proof.

Let

{({\bar{p}}_{t}, {({\bar{c}}_{i, t}, {\bar{z}}_{i, t})}_{i = 1}^{m})}_{t = 0}^{T}

be an equilibrium of

E_{b}^{T}

. Note that

z_{i, T + 1} = 0

for every

i = 1, \dots T

. We can see that conditions

(i)

and

(i i)

in Definition (2) are satisfied. We will show that condition

(i i i)

is also verified. Let

a_{i} : = {({\bar{c}}_{i, t}, {\bar{z}}_{i, t})}_{t = 0}^{T}

be a feasible plan of agent i. Suppose that

\sum_{t = 0}^{T} ρ_{i}^{t} u_{i} (c_{i, t}) > \sum_{t = 0}^{T} ρ_{i}^{t} u_{i} ({\bar{c}}_{i, t})

. For each

γ \in (0, 1)

, we define

a_{i} (γ) : = γ a_{i} + (1 - γ) {\bar{a}}_{i}

. By definition of

E_{b}^{T}

,we can choose

γ

sufficiently close to 0 such that

a_{i} (γ) \in C_{i} \times Z_{i}

. It is clear that

a_{i} (γ)

is a feasible allocation. By the concavity of the utility function, we have

\begin{matrix} \sum_{t = 0}^{T} ρ_{i}^{t} u_{i} (c_{i, t} (γ)) & \geq & γ \sum_{t = 0}^{T} ρ_{i}^{t} u_{i} (c_{i, t}) + (1 - γ) \sum_{t = 0}^{\infty} ρ_{i}^{t} u_{i} ({\bar{c}}_{i, t}) \\ > & \sum_{t = 0}^{T} ρ_{i}^{t} u_{i} ({\bar{c}}_{i, t}) \end{matrix}

We deduce that:

E [\sum_{t = 0}^{T} ρ_{i}^{t} u_{i} (c_{i, t} (γ))] > E [\sum_{t = 0}^{T} ρ_{i}^{t} u_{i} ({\bar{c}}_{i, t})]

which contradicts the optimality of

{\bar{a}}_{i}

We denote by

({\bar{p}}^{T}, {({\bar{c}}_{i}^{T}, {\bar{z}}_{i}^{T})}_{i = 1}^{m})

an equilibrium of the T-truncated economy

E^{T}

. Since

∥ {\bar{p}}_{t} ∥ \leq 1

, for every

t \leq T

{\bar{c}}_{i}^{T} \leq B_{c}

and

\sum_{i = 1}^{m} {\bar{z}}_{i}^{T} = 1

.Thus, we can assume that:

({\bar{p}}^{T}, {({\bar{c}}_{i}^{T}, {\bar{z}}_{i}^{T})}_{i = 1}^{m}) ⟶ (\bar{p}, {({\bar{c}}_{i}, {\bar{z}}_{i})}_{i = 1}^{m})

when T goes to infinity.

One can easily check that all markets clear.

Now we can give the main result of this paper:

Theorem 4.1.

If hypothesis (H1), (H2),(H3) and (H4) are satisfied, then there exists an equilibrium of the infinite horizon economy

E

Proof.

We have proved previously that for each

T \geq 1

, there exists an equilibrium for the economy

E^{T}

. Let

(c_{i}, z_{i})

be a feasible allocation of the problem

Q_{i} (\bar{p}, \bar{z})

. We will prove that

E [\sum_{t = 0}^{\infty} ρ_{i}^{t} u_{i} (c_{i, t})] \leq E [\sum_{t = 0}^{\infty} ρ_{i}^{t} u_{i} ({\bar{c}}_{i, t})]

We define

{(c_{i}^{'}, z_{i}^{'})}_{t = 0}^{T}

as follows:

\begin{matrix} z_{i, t}^{'} & = & z_{i, t} i f t \leq T - 1, \\ c_{i, t}^{'} & = & c_{i, t} i f t \leq T - 1, \\ c_{i, t} & = & z_{i, t} = 0 i f t > T \\ x_{i, T} + {\bar{p}}_{T} z_{i, T}^{'} - {\bar{p}}_{T} g_{i} (- z_{i, T}^{'}) & = & x_{i, T} + {\bar{p}}_{T} z_{i, T} - {\bar{p}}_{T} g_{i} (- z_{i, T}) \end{matrix}

We can see that

{(c_{i}^{'}, z_{i}^{'})}_{t = 0}^{T} \in U_{i}^{T} (\bar{x}, \bar{y})

Since

L_{i}^{T} (\bar{x}, \bar{y}) \neq \emptyset

, there exists a sequence

({(c_{i}^{n}, z_{i}^{n})}_{t = 0}^{T})_{n = 0}^{\infty} \in L_{i}^{T} (\bar{x}, \bar{y})

with

z_{i, T + 1}^{n} = 0

and this sequence converges to

{(c_{i}^{'}, z_{i}^{'})}_{t = 0}^{T}

when n tends to infinity . We have

x_{i, t}^{n} + {\bar{p}}_{t} z_{i, t}^{n} - {\bar{p}}_{t} g_{i} (z_{i, t + 1}^{n} - z_{i, t}^{n}) > 0 .

We can choose

s_{0}

high enough such that

s_{0} > T

and for every

s \geq s_{0}

, we have

x_{i, t}^{n} + {\bar{p}}_{t}^{s} z_{i, t}^{n} - {\bar{p}}_{t}^{s} g_{i} (z_{i, t + 1}^{n} - z_{i, t}^{n}) > 0 .

Consequently

{(c_{i}^{n}, z_{i}^{n})}_{t = 0}^{T} \in U_{i}^{T} ({\bar{x}}^{s}, {\bar{y}}^{s})

. Therefore, we get

\sum_{t = 0}^{T} ρ_{i}^{t} u_{i} (c_{i, t}^{n}) \leq \sum_{t = 0}^{s} ρ_{i}^{t} u_{i} ({\bar{c}}_{i, t}^{s})

. Tends s to infinity, we obtain

\sum_{t = 0}^{T} ρ_{i}^{t} u_{i} (c_{i, t}^{n}) \leq \sum_{t = 0}^{\infty} ρ_{i}^{t} u_{i} ({\bar{c}}_{i, t})

. Now, if we tend n tend to infinity, we obtain

\sum_{t = 0}^{T} ρ_{i}^{t} u_{i} (c_{i, t}) \leq \sum_{t = 0}^{\infty} ρ_{i}^{t} u_{i} ({\bar{c}}_{i, t})

for every T. Consequently:

\sum_{t = 0}^{T - 1} ρ_{i}^{t} u_{i} (c_{i, t}) \leq \sum_{t = 0}^{\infty} ρ_{i}^{t} u_{i} ({\bar{c}}_{i, t}) .

Let T tend to infinity, we obtain

\sum_{t = 0}^{\infty} ρ_{i}^{t} u_{i} (c_{i, t}) \leq \sum_{t = 0}^{\infty} ρ_{i}^{t} u_{i} ({\bar{c}}_{i, t}) .

Then,

E [\sum_{t = 0}^{\infty} ρ_{i}^{t} u_{i} (c_{i, t})] \leq E [\sum_{t = 0}^{\infty} ρ_{i}^{t} u_{i} ({\bar{c}}_{i, t})] .

Hence, we have proved the optimality of

({\bar{c}}_{i}, {\bar{z}}_{i})

. Note that prices

{\bar{p}}_{t}

are strictly positive since the utility function of agent i is strictly increasing.

□

The obtained equilibrium is the expected optimal strategy of Merton investment problem in the case of multi-investors and in markets with frictions formulated by a penalty functions for every investor due to loss in trading. Our model and main result extends models and results obtained by Chebbi and Soner in [1] and by Ounaies, Bonnisseau, Chebbi and Soner in [15]

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research, King Saud University for funding through Vice Deanship of Scientific Research Chairs; Research Chair of Financial and Actuarial Studies.

References

S. Chebbi and H.M. Soner, Merton problem in a discrete market with frictions, Nonlinear Analysis: Real World Applications 14 (2013) 179-187. [CrossRef]
G.M. Constantinides, Capital market equilibrium with transaction costs, Journal of Political Economy 94 (1986) 842-862. [CrossRef]
U. Çetin, R. Jarrow and P. Protter, Liquidity risk and arbitrage pricing theory, Finance and Stochastics 8 (2004) 311-341. [CrossRef]
U. Çetin and L.C.G. Rogers, Modelling liquidity effects in discrete time, Mathematical Finance 17 (2007) 15-29. [CrossRef]
M.H.A. Davis and A.R. Norman, Portfolio selection with transaction costs, Mathematics of Operations Research 15(4) (1990) 676-713. [CrossRef]
B. Dumas and E. Luciano, An exact solution to a dynamic portfolio choice problem under transaction costs, Journal of Finance 46 (1991) 577-595. [CrossRef]
S. Goekey and H.M. Soner, Liquidity in a binomial market, Mathematical Finance 22 (2012) 250-276. [CrossRef]
E. Jouini and E. Kallal, Martingales and arbitrage in securities markets with transaction costs, Journal of Economic Theory 66 (1995) 178-197. [CrossRef]
I. Karatzas and S.E. Shreve, Methods of Mathematical Finance, Springer-Verlag, 1998. [CrossRef]
C. Le Van and Rose-Anne Dana, Dynamic programming in economics, Kluer Academic Publishers, 2003.
M.J.P. Magill and G.M. Constantinides, Portfolio selection with transaction costs, Journal of Economic Theory 13 (1976) 254-263.
R.C. Merton, Lifetime portfolio selection under uncertainty: The continuous time case. The review of Economics and Statistics (1969) 247-257. [CrossRef]
R.C. Merton, Optimum consumption and portfolio rules in a continuous time case, Journal of Economic Theory 3 (1971) 373-413. [CrossRef]
S.E. Shreve and H.M. Soner, Optimal investment and consumption with transaction costs, The annals of Applied Probability 4(3) (1994) 609-692. [CrossRef]
S. Ounaies, J-M. Bonnisseau, S. Chebbi and H.M. Soner, Merton problem in an infinite horizon and a discrete time with frictions. Journal of Industrial and Management Optimization 12(4)(2016) 1323-1331. [CrossRef]

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Optimal Investment of Merton Model for Multi-Investors with Frictions

Abstract

1. Introduction

2. The Model

3. General equilibrium model of Merton investment problem

4. Existence of Equilibrium

Acknowledgments

References

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