Optimal investment and consumption for financial markets with jumps under transaction costs

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Optimal investment and consumption for financial markets with jumps under transaction costs

Sergei Egorov, Serguei Pergamenchtchikov

To cite this version:

Sergei Egorov, Serguei Pergamenchtchikov. Optimal investment and consumption for financial markets

with jumps under transaction costs. 2021. �hal-03265239�

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Optimal investment and consumption for financial markets with jumps under transaction costs

Sergei Egorov · Serguei Pergamenchtchikov

Received: date / Accepted: date

Abstract We consider a portfolio optimization problem for financial markets described by semi-martingales with independent increments and jumps defined through L´evy processes.

For this problem we show the corresponding verification theorem and construct the opti- mal consumption/investment strategies. For the power utility functions we find the optimal strategies in the explicit form and then we apply these strategies to markets with transaction costs. Based on the Leland - Lepinette approach we develop asymptotic optimal investment and consumption method when the number of portfolio revision tends to infinity. Finally, we cared out Monte Carlo simulations to illustrate numerically the obtained theoretical results in practice.

MSC:primary 60P05, secondary 62G05

Keywords: Financial markets; Optimal investment/consumption problem; Stochastic con- trol; Dynamical programming; Hamilton–Jacobi–Bellman equation.

Research was supported by RSF, project no 20-61-47043 (Tomsk State University) Sergei Egorov

Laboratoire de Math´ematiques Raphael Salem, UMR 6085 CNRS – Universit´e de Rouen Normandie, France e-mail: sergei.egorov@univ-rouen.fr

Serguei Pergamenshchikov

Laboratoire de Math´ematiques Raphael Salem, UMR 6085 CNRS – Universit´e de Rouen Normandie, France and

International Laboratory of Statistics of Stochastic Processes and Quantitative Finance, National Research Tomsk State University,

e-mail: serge.pergamenshchikov@univ-rouen.fr

1 Introduction

1.1 Motivations

In this paper, we consider a portfolio optimization problem for L´evy financial markets with non-random time-dependent coefficients. Such problems are very popular in the stochas- tic financial markets theory. Beginning with the classic work of Merton, where the opti- mal investment problem for Black-Scholes models was first studied, interest in these prob- lems is constantly growing to the present day (see, for example, [11,4,5,17] and the refer- ences therein). It should be emphasized that the financial markets defined by the continu- ous stochastic processes similar to the geometric Brownian motion are very limited for the practical applications and they do not allow us to describe situations of abrupt, impulsive changes in price processes observed during of the crises and instability in financial mar- kets. It seems that for the first time an optimization portfolio problem for financial markets with jumps was studied in the paper [8], in which the authors using the stochastic Pontrya- gin maximum principle constructed optimal investment/consumption strategies. Later, these problems were studied for more complex market models and in different settings: in [14, 9,24] the authors considered maximization utility problems in general semi-martingale set- tings, in [25,4,2,19] such problems were considered for stochastic volatility markets, in [6, 10] for the markets defined by the affine processes, in [12,13,3,7] the authors considered the portfolio optimization problems with constraints. In [5,17] the authors considered pure consumption problem on L´evy markets with infinite time horizon under proportional trans- action costs where they used geometric approach and viscosity solutions in similar spirit as it was done in [1].

The main goal of our work is to study the classical investment and consumption problem on the finite time interval[0, T]for the financial market model with jumps under transaction costs. Moreover, we are interested to find the optimal solutions in the explicit form and illustrate their behavior by the Monte - Carlo method.

1.2 Main investments

Based on stochastic dynamic programming and Leland - Lepinette approach, we develop a portfolio optimization method for Levy-type financial markets with transaction costs. To this end, first, we deduce and study the Hamilton–Jacobi–Bellman (HJB) equation. The challenge here is that we could not use directly the classical HJB analysis method from [11], which was due to the additional integral term corresponding to jumps in the market model. Therefore, we need to develop a special analytical tool to analyse this equation and to construct optimal strategies. Similar to [12,13,4,2] we study this problem through the verification theorem method. So, in this paper, probably for the first time, we show a spe- cial verification theorem in a non-Gaussian financial markets framework. Then, using this theorem we construct optimal strategies, and, finally, for the power utility functions we pro- vide the solutions for such optimization problems in an explicit form. Moreover, to take into account transaction costs in the optimization problems we use the Leland method (see, for example, [15]) in the difference from the geometric approach developed in [5,17] for L´evy markets. More precisely, using the explicit form for the obtained optimal strategies we construct their discretized versions and, then study the asymptotic behavior as the number of revisions tends to infinity. We provide the conditions on transaction costs for which the discretized strategy is asymptotically optimal, i.e. when the objective function tends to its

optimal value. It turns out, that in the case of large transaction costs, to obtain the optimal- ity property one needs to use the Lepinette approach developed for the asymptotic hedging problem in [16], i.e. one needs to do portfolio revisions not uniformly as in the Leland approach, but in time moments of special power form. Finally, we illustrate the obtained theoretical results by the numeric simulations.

1.3 Organisation of paper

In Section2we describe the problem and all necessary definitions. In Section3we study optimal control problem for stochastic systems with jumps through the verification theorem method. In Section4we state the main results. In Section5we the optimal strategies. In Section6we give the results of Monte-Carlo simulations. All main results are shown in Section7. All auxiliary tool is given in AppendixA.

2 Problem

In this paper, we consider optimal investment and consumption problems for financial mar- kets of L´evy type with time-dependent coefficients. More precisely, we consider the financial market on the time interval[0, T]which consists of a risk-free asset (bond)(B_t)_0≤t≤T and mrisky assets (stocks)(S_i(t))_0≤t≤T,1≤i≤m, defined as







dB_t=r_tB_tdt , B0= 1,

dS_i(t) =µ_i(t)S_i(t) dt+S_i(t−) dLe_i(t),

(2.1) where the interest rater_tand the drifts(µ_j(t))_1≤j≤mare non random[0, T]→Rintegrated functions, i.e.

Z T 0

|r_t|+

m

X

i=1

|µ_i(t)|

!

dt <∞. (2.2)

We assume, that the random market sources(eL_i(t))_0≤t≤T are represented as

Le_i(t) =

m

X

j=1 t

Z

0

σ_ij(u) dW_j(u) +ς_ij(u) dL_j(u)

, (2.3)

in which the volatilitiesσ_t= σ_ij(t)

1≤i,j≤mandς_t= ς_ij(t)

1≤i,j≤marem×mnon random matrices such that for any1≤i, j≤m,0≤t≤T

Z T 0

σ²_ij(u) du <∞ and 0≤ς_ij(t)≤1. (2.4) Moreover, we assume, that(W₁(t))_0≤t≤T,...,(W_m(t))_0≤t≤Tare independent standard Brow- nian motions and(L₁(t))_0≤t≤T, . . . ,(L_m(t))_0≤t≤Tare independent pure jumps L´evy pro- cesses, i.e.

L_j(t) = Z t

0

Z

R∗

y ν_j(ω; dy ,du)−e^νj(dy ,du)

, (2.5)

whereν_j(ω; dy ,du)is random jump measure with compensator^νej(dy ,du) =Π_j(dy) du andΠ_j(·)is the corresponding L´evy measure onR∗ =R^{\ {}0}. It should be noted that in case of independent L´evy processes,∆L_j(t)∆L_i(t) = 0, for anyi6=j. First, to provide the positivity for the risky price process (2.1) we assume, thatΠ_j(]− ∞,−1]) = 0. Moreover, we assume also that

Π_j |ln(1 +y)|1_{{−1<y<−1/2}}

<∞ and Π_j(y²)<∞, (2.6) where1_Ais an indicator function of the setA. These conditions are technical and used in the market model to guarantee existence of optimal trading strategy. In the sequel we set µ_t = (µ₁(t), . . . , µ_m(t))

0

,W_t = (W₁(t), . . . , W_m(t))⁰andL_t = (L₁(t), . . . , L_m(t))⁰, where the prime⁰ denotes the transposition. Everywhere bellow we use natural filtration F = (F_t)_0≤t≤T, i.e.F_t=σ{W_u, L_u : 0≤u≤t}. Similar to [12] in this paper we use the fractional strategies defined as

θ_i(t) = ^αi(t)S_i(t)

X_t and c_t= ^ζt

X_t, (2.7)

whereα_i(t)is the amount of investment intoi-th risky asset purchased by an investor at the time momenttandX_tis the corresponding wealth process withβ_tbond units defined as

X_t=

m

X

i=1

α_i(t)S_i(t) +β_tB_t, (2.8) using (2.7) and (2.8) can be proved the following equality1−Pm

i=1θ_i(t) = β_tB_t/X_t. Moreover, the processζ_tin (2.7) is the consumption intensity, i.e. it is non negative inte- grable process for which the integralRt

0ζ_sdsis the total amount of capital consumed by the investor on time interval[0, t]. Now using the self-financing-consumption principle for the wealth process (2.8) (see, for example, [11]) and the definition (2.1), we obtain that







dX_t=X_t(r_t+θ_t⁰µˇ_t−c_t) dt+X_t−θ_t−⁰ deL_t, X₀=x >0,

(2.9)

whereµˇ_t = (µ_t−r_te_m),e_m = (1, . . . ,1)⁰ ∈ R^mandθ_t = (θ₁(t), . . . , θ_m(t))⁰ and Le_t = (eL₁(t), . . . ,Le_m(t))⁰. It should be noted that to provide the positivity of portfolio valueX_tthe jump sizes of the process_Lˇ_t=Rt

0 θ_u−⁰ ς_u−dL_uhave to be more than−1, i.e.

∆_Lˇ_t>−1. To this end we assume that the financial strategyθ_t= (θ₁(t), . . . , θ_m(t))⁰is a c`adl`ag process with values in the set[0,1]^msuch that, for any fixed0≤t≤T, almost sure Pm

j=1θ_j(t)≤1. In the sequel we denote by Θ=







θ= (θ₁, . . . , θ_m)⁰∈[0,1]^m :

m

X

j=1

θ_j≤1







. (2.10)

Using this set we introduce admissible strategies.

Definition 2.1. A stochastic processυ = (θ_t, c_t)_0≤t≤T is called admissible if the first component(θ_t)_0≤t≤T is a predictable process with values in the set(2.10)and the process (c_t)_0≤t≤T is adapted, non negative integrated on[0, T], for which the equation(2.9)has unique strong solution such thatX_t−>0andX_t>0a.s. for0≤t≤T.

We denote byVthe set of admissible strategies. It should be noted that for anyυ ∈ Vthe wealth process (2.9) can be represented as

X_t=x e

R_t

0(r_s+θ_s⁰µˇ_s) ds−^R₀^tc_sdsE_t(V) and the Dol´ean exponential

E_t(V) = e^Vt−12hVit Y

0≤s≤t

(1 +∆V_s)e^−∆Vs, (2.11)

whereV_t=Rt

0θ⁰_sdeL_sandhVi_t=

t

R

0

θ⁰_sσ_sσ⁰_sθ_sds.

Now to formulate an optimal consumption and investment problem we introduce for some0< γ <1, the objective function as

J(x, υ) :=E_x





T

Z

0

ζ_t^γdt+ (X_T)^γ₊



, (2.12)

whereE_xis the conditional expectation givenX₀=x and(x)₊is the positive part ofx, i.e.(x)₊= max(x,0)andζ_t=c_tX_tis the intensity of consumption.

The goal is to maximize the objective function on the setV, i.e. to find a strategyυ^∗∈ V, such that

J(x, υ^∗) = sup

υ∈V

J(x, υ) =:J^∗(x). (2.13) According to the dynamic programming principle to study this problem we have to study the value functions defined on the interval[t, T]as

J^∗(t, x) := sup

υ∈V

J(t, x, υ), (2.14)

where forυ∈ V

J(t, x, υ) :=E_x,t





T

Z

t

(c_uX_u)^γdu+X_T^γ



.

HereE_x,t is the conditional expectation with respect toX_t = x. Note that in our case J(t, x, υ)can be equal to∞for some strategyυ.

Remark 2.2. It should be noted that the definition(2.12)for the objective function can be applied to any financial strategies not only for admissible ones, i.e. not only to strategies with positive wealth processes. Indeed, in practice, the capital can be negative for a short period of time, such as for markets with transaction costs. In such cases, it is also necessary to calculate the objective function and compare the strategies between them.

Remark 2.3. Note that the optimal consumption and investment problem without terminal functional for the model(2.1)is studied in [8] through the maximum Pontryagin princi- ple. The similar problems were considered in [23,24] on the basis of the dual problems approach.

3 Verification theorem

In this section we will generalize the controlled process (2.9) and the definition of admissible strategies. For such framework, we show some verification theorem. Let’s define diffusion jumps controlled processX_twith values in an open convex setX ⊆Rand control process υ_twith values in closed setK ⊆R^m. LetX = (X_t)_0≤t≤T,X_t, X_t−∈ Xbe c`adl`ag process of form







dX_t=a(t, X_t, υ_t) dt+b⁰(t, X_t−, υ_t−) deL_t, X₀=x∈ X,

(3.1)

where the processLe_t∈ R^mis defined in (2.3) with the L´evy measuresΠ_jsatisfying the second condition in (2.6), i.e.Π_j(y²)< ∞. The functionsa : [0, T]× X × K →Rand b:X × K →R^mare non random, continuous and such that, for any non randomv∈ K, the equation (3.1) withυ_t≡vhas an unique strong solution for whichX_t∈ X andX_t−∈ X on the time interval[0, T]and

Z T 0

|a(t, X_t,v)|+|b(t, X_t,v)|²

dt <∞ a.s..

Definition 3.1. A stochastic processυ= (υ_t)_0≤t≤T is called admissible if it is(F_t)_0≤t≤T -adapted, has c`adl`ag trajectories, takes values in the setK, and such that the equation(3.1) on time interval[0, T]has an unique strong solution for whichX_t, X_t− ∈ X on the time interval[0, T]and

T

Z

0

|a(t, X_t, υ_t)|+|b(t, X_t, υ_t)|²

dt <∞ a.s.. (3.2)

We denote byV the set of all admissible strategies. Note that the conditions on functions a andbimplyV 6= ∅, at least the strategyυ_t ≡ v ∈ V. Now we fix utility functions U₁ : [0, T]× X × K →R+andU₂ : X →R+and, then, for any0< t≤T, we set the objective functions as

J(t, x, υ) :=E_t,x





T

Z

t

U₁(u, X_u, υ_u) du+U₂(X_T)



.

Our goal is to find an admissible strategyυ^∗∈ V, such that, for any0≤t < T, J^∗(t, x) := sup

υ∈V

J(t, x, υ) =J(t, x, υ^∗). (3.3) To apply the dynamic programming method, we will need to introduce the Hamilton func- tion. To do this, for any[0, T]× X →Rfunctiong(t, x), twice continuously differentiable inxand continuously differentiated int, such that

sup

x∈X

|g(t, x)|

1 +|x| <∞, (3.4)

we define

H₀(t, x, g,v) =a(t, x,v)g_x(t, x) +1

2^gxx(t, x)trσ_t⁰b(t, x,v)b⁰(t, x,v)σ_t

+U₁(t, x,v) +g(t, x,v), (3.5)

whereg(t, x,v) =

m

P

i=1

R

R∗

Υ_g(t, x, b⁰(t, x,v)ς_i,ty)Π_i(dy),ς_i,t= (ς_1i(t), . . . , ς_mi(t))⁰and Υ_g(t, x, v) = (g(t, x+v)−g(t, x)−g_x(t, x)v)1_{{x+v∈X }}

forv∈R∗. Notationsg_t,g_x,g_xx mean corresponding derivatives of functiong(t, x). Remark 3.2. It should be emphasized that one needs to introduce the indicator1_{{x+v∈X }} in the termΥ(t, x, v)since there are no assumptions on measuresΠ_i(·)or functiongto keep the sumx+vin the setX, forvfromR∗. As it is shown in appendixA.1the function gis bounded and its Lebesgue integral will be used as jumps compensator for the process g(t, X_t).

Now, for anyx∈ Xand0≤t≤T, we set the Hamilton function as H(t, x, g) := sup

v∈K

H₀(t, x, g,v). (3.6)

Now we introduce the following Hamilton–Jacobi–Bellman equation as







z_t(t, x) +H(t, x, z) = 0, t∈[0, T], z(T, x) =U₂(x), x∈ X.

(3.7) Next, we need the following conditions.

H₁) There exists solutionz∈ C^1,2([0, T]× X,R)of the equation(3.7)such that, inf

0≤t≤T inf

x∈X z(t, x)>−∞ and sup

0≤t≤T

sup

x∈X

|z(t, x)|

1 +|x| <∞. (3.8) H₂) There exists[0, T]× X → Kmeasurable functionv₀, such that for the solution z=z(t, x)of the equation(3.7)the Hamilton functionH(t, x, z) =H₀(t, x, z,v₀(t, x)).

H₃) For anyx∈ X, there exists an unique almost surely solutionX^∗= (X_t^∗)_0≤t≤T with values in the setXandX_t−^∗ ∈ X of the equation

dX_t^∗=a^∗(t, X_t^∗) dt+ (b^∗(t−, X_t−^∗ ))⁰deL_t, X₀^∗=x , (3.9) wherea^∗(t, x) =a(t, x,v₀(t, x))andb^∗(t, x) =b(x,v₀(t, x)). Moreover, the process v₀= (v₀(t, X_t^∗))_0≤t≤T is admissible, i.e. belongs toV.

H₄) For any0≤t≤T andx∈ X, E_t,x sup

t≤u≤T

|z(u, X_u^∗)|<∞. (3.10)

Using the approach proposed in [12], we show the following verification theorem.

Theorem 3.3. Suppose conditionsH₁)–H₄)are hold. Then, for any0 ≤ t ≤ T and x∈ X,

z(t, x) =J^∗(t, x) =J(t, x, υ^∗),

where the optimal strategyυ^∗ = (υ^∗_s)_t≤s≤T,υ_s^∗ = v₀(s, X_s^∗)is determined in terms of H₂)–H₄)and the functionJ^∗(t, x)is defined in(3.3).

Remark 3.4. Here(3.8)andH₄)are technical conditions. In particular, the first condition in(3.8)will be used to apply Fatou’s lemma for the limit transition in conditional expec- tations, the last condition provides the finiteness almost sure for the jumps compensator of z(t, X_t)andH₄)will be used to apply dominated convergence theorem to prove optimality of the processυ^∗.

Remark 3.5. Note, that we don’t assume the uniqueness of a solution for the equation(3.7).

But if conditions of Theorem3.3hold thenz(t, x)will be the unique solution of the equation (3.7), since the supremumJ(t, x, υ^∗)is always unique.

Now we apply Theorem3.3to the problem (2.13). In this case the controlled process driven by (2.9) with state spaceX =]0,∞[, admissible strategyυ = (θ_t, c_t)_0≤t≤T with values inK= Θ×R+, where the setΘdefined by (2.10). So, to study the optimal con- sumption and investment problem (2.14) we will apply Theorem3.3for the utility func- tionsU₁(x,v) = (xc)^γ andU₂(x) = x^γ, wherex ∈ X,v = (θ,c) ∈ K, the vector θ = (θ₁, . . . , θ_m)⁰and0 < γ <1. First note that, the process (2.9) can be obtained as a special case of the model (3.1)

a(t, x,v) =x(r_t+θ

0

ˇ

µ_t−c) and b(t, x,v) =x θ . (3.11) In this case one can check directly, that the HJB equation (3.7) has the following form











z_t(t, x) +r_tx z_x(t, x) + max

θ∈ΘΓ(t, x, z, θ) + (1−γ) γ

z_x(t, x) _1−γ^γ

= 0, z(T, x) =x^γ,

(3.12)

whereΓ(t, x, z, θ) =x z_x(t, x)θ⁰ˇµ_t+x²z_xx(t, x)θ⁰σ_tσ_t⁰θ/2 +z(t, x, θ)andz(t, x, θ)is defined bygin (3.5) forg=z. Moreover, according to ConditionH₂)to find an optimal control functionv₀= (θ₀,c₀)one needs to chooseθ₀=θ₀(t, x, z)andc₀=c₀(t, x, z)as

θ₀= arg max

θ∈ΘΓ(t, x, z, θ) and c₀= 1 x

z_x(t, x) γ

1/(γ−1)

. (3.13)

Using here the Fourier separation variables method we can conclude, that the solution of the HJB equation has the following form

z(t, x) =A(t)x^γ. (3.14)

It should be noted, that if we substitutezwith the form (3.14) in (3.12), we obtain that the functionΓ(t, x, z, θ) =A(t)x^γF(t, θ), where

F(t, θ) =γ θ⁰µˇ_t+ ^γ(γ−1)

2 trθ⁰σ_tσ⁰_tθ+

m

X

i=1

Z

R_∗

ρ

θ⁰ς_i,ty

Π_i(dy), (3.15)

whereρ(x) = (1 +x)^γ −1−γxandς_i,t = (ς_1i(t), . . . , ς_mi(t))⁰ isi-th column of the matrixς_t. Therefore, from (3.13) we obtain that in this case for the HJB solution (3.14)

θ₀=θ₀(t) = arg max

θ∈ΘF(t, θ) and c₀=c₀(t) =





T

Z

t

Ψ_t,udu+Ψ_t,T





−1

, (3.16)

whereΨ_t,u=e

R_u t h^∗

sds

andh^∗_s= (F(s, θ^∗_s) +γ r_s)/(1−γ). From (2.9) we obtain, that the corresponding wealth process(X_t^∗)_0≤t≤T is defined as

X_t^∗=x e

R_t 0(r_s+(θ^∗

s)⁰ˇµ_s) ds−^R₀^tc^∗

sds

E_t(V^∗) (3.17)

andE_t(V^∗)is the Dol´ean exponential (2.11) for the processV_t^∗ = Rt

0(θ^∗_s)⁰deL_swith its quadratic characteristichV^∗i_t=

t

R

0

(θ_s^∗)⁰σ_sσ_s⁰θ^∗_sds.

Remark 3.6. If there exists many solutions in for the maximization problem of the function F(t,·)we chose any point.

4 Main results

Fist we study the strategy (3.16).

Theorem 4.1. The strategyυ^∗ = (θ^∗_t, c^∗_t)_0≤t≤T defined in(3.16), i.e.θ_t^∗ =θ₀andc^∗_t = c₀(t), is a solution for the problem(2.13). Moreover, for any0 ≤t < T, optimal value function(2.14)is given

J^∗(t, x) =J(t, x, υ^∗) =x^γ





T

Z

t

Ψ_t,udu+Ψ_t,T





1−γ

. (4.1)

The proof is given in section7.

Remark 4.2. Note that for the homogenous market model(2.1), i.e. for the constant coeffi- cientsr_t, µ_t, σ_t, ς_tthe optimal strategy(3.16)is obtained in [23].

Now we consider the optimization problem (2.1) for the markets with transaction costs on the basis of the Leland’s approach proposed in [15] for hedging problems. In this case we use the optimal strategy

α^∗_t= α^∗₁(t), . . . , α^∗_m(t)⁰

=

θ₁^∗(t)X_t^∗ S₁(t) ^{, . . . ,}

θ^∗_m(t)X_t^∗ S_m(t)

⁰ ,

β_t^∗=

(1−Pm

j=1θ^∗_j(t))X_t^∗

B_t and ζ_t^∗=c^∗_tX_t^∗, (4.2)

where the fractional strategyθ_t^∗= (θ^∗₁(t), . . . , θ^∗_m(t))⁰is given in (3.13) and the correspond- ing wealth processX_t^∗is calculated as

X_t^∗=x+

t

Z

0

α^∗_u−⁰ dS_u+

t

Z

0

β^∗_udB_u−

t

Z

0

ζ_u^∗du . (4.3)

According to the Leland approach we can do onlynportfolio revisions of the strategy (4.2) on the interval[0, T]at the moments0< t₁< . . . < t_n=T, i.e. we set

α⁽_tⁿ⁾=

n

X

k=1

α^∗_tk−11_]tk−1,tk], β_t⁽ⁿ⁾=

n

X

k=1

β^∗_tk−11_]tk−1,tk]

and ζ_t⁽ⁿ⁾=

n

X

k=1

ζ_t^∗_k−11_]tk−1,tk]. (4.4)

Moreover, for each revision the investors must pay the transaction costs proportionally to the volume of trade at the momentt_kdefined as

κ

m

X

j=1

S_t

k|α⁽_jⁿ⁾(t_k)−α⁽_jⁿ⁾(t_k−1)|=κ

m

X

j=1

S_t

k|α^∗_j(t_k−1)−α^∗_j(t_k−2)|,

where by the conventionα^∗_j(t) = 0fort <0. Therefore, the total transaction costs on the interval[0, T]is presented as

D_n=κ

m

X

j=1 n

X

k=1

S_j(t_k)|α⁽_jⁿ⁾(t_k)−α⁽_jⁿ⁾(t_k−1)|, (4.5) whereκ >0is a proportional transaction coefficient which is assumed to be a function of the revisions numbersn, i.e.κ=κ_n. Note that, in this caseα⁽_u−ⁿ⁾=α⁽_uⁿ⁾. Therefore, to take into account transaction costs we defined the wealth as

X_T⁽ⁿ⁾=x+

T

Z

0

(α⁽_uⁿ⁾)⁰dS_u+

T

Z

0

β⁽_uⁿ⁾dB_u−

T

Z

0

ζ_u⁽ⁿ⁾du−D_n. (4.6)

To study properties of the strategyψ⁽ⁿ⁾= (α⁽_tⁿ⁾, β⁽_tⁿ⁾, ζ_t⁽ⁿ⁾)_0≤t≤T we set

J(x, ψ⁽ⁿ⁾) =Ex





T

Z

0

ζ_u⁽ⁿ⁾

γ

du+ X_T⁽ⁿ⁾

γ +



. (4.7)

Now we study asymptotic properties of the strategyψ⁽ⁿ⁾asn→ ∞. To do this we need to assume the following conditions.

A₁)In the model(2.1)the functionsr_t,µ_tandσ_tare bounded on the interval[0, T].

A₂)The functionsr_t, µ_t, σ_t, ς_tin the model(2.1)are such that the optimal strategy (3.13)satisfies the following inequality

sup

n≥1

sup

0<t₁<...<t_n=T

Pn j=1|θ^∗_t

j −θ_t^∗

j−1| 1 +Pn

j=1

pt_j−t_j−1 <∞. (4.8) Note thatA₂)holds true if the optimal strategy (3.13) is1/2-Holder function, i.e.

sup

0≤s,t≤T

|θ_t^∗−θ^∗_s|

p|t−s| <∞. (4.9)

Theorem 4.3. Assume that ConditionsA₁)–A₂)hold true and the transaction coefficient is a function ofn, i.e.κ= κ_n such thatκ_n = o

n^−1/2

, asn → ∞. Then the strategy (4.4)-(4.6)with the revision moments(t_j=jT /n)_1≤j≤nis asymptotically optimal, i.e.

n→∞lim J(x, ψ⁽ⁿ⁾) =J^∗(x), (4.10) where the optimal functionalJ^∗(x)is defined in(2.13).

Now we study the optimization problem (2.1) for the markets with large transaction costs, i.e. in the case when the normalized transaction coefficient^√nκ_n does not go to zero as n→ ∞. We assume only thatκ=κ_n=o(1), i.e.κ_n→0asn→ ∞. In this case we need to change the strategy (4.4) making use of Lepinette approach proposed in [16] and used then for the markets with jumps in [21]. According to this approach we do the portfolio revisions at the points

t_j=t^∗(u_j)T , u_j= ^j

n and t^∗(u) =u^q, (4.11) where the powerq≥1is a function ofn, i.e.q=q_nsuch that

n→∞lim q_n= +∞ and lim

n→∞

q_n n +κ_n

r n q_n

= 0. (4.12)

Note that, ifκ_n→0asn→ ∞, then we can take, for example,q_n=^√n+κ_nn.

Theorem 4.4. Assume that the conditionsA₁)–A₂)hold true and the transaction coeffi- cient is a function ofn, i.e.κ=κ_nsuch thatκ_n=o(1)asn→ ∞. Then the strategy(4.4) -(4.6)with the revision time moments defined in(4.11)-(4.12)is asymptotically optimal, i.e. satisfies the property(4.10).

Moreover, we need to find sufficient conditions providing ConditionA₂).

Proposition 4.5. Assume that, in the model(2.1)m= 1, the functionsr_t,µ_t,σ_tandς_tare continuously differentiable andς_t>0for all0≤t≤T. Then ConditionsA₁)–A₂)hold true.

Remark 4.6. It should be noted that ConditionA₂)hold for anym ≥1for the homoge- neous market(2.1), i.e. for the constant parametersr_t, µ_t, σ_t, ς_t.

5 Properties of the optimal strategies(3.16)-(3.17)

In this section we study the Do´ean exponentialsE_t(V)defined in (2.11).

Proposition 5.1. For any non random measurable[0, T]→Θfunctionθ= (θ_t)_0≤t≤T the Dol´ean exponential(2.11)is square integrated martingale.

Proof. First note, that the processV_t =Rt

0θ⁰_sdLe_sis square integrated martingale. There- fore, taking into account thatdE_t(V) =E_t−(V) dV_t, to show this lemma it suffices to check thatsup_0≤t≤TEE_t²(V)<∞. To this end we represent the Dol´ean exponential in the mul- tiplicative form, i.e.

E_t(V) = e

R_t 0θ⁰

sσ_sdW_s−¹₂^R₀^tθ⁰

sσ_sσ⁰

sθ_sds

E_t^(d), (5.1)

whereE_t^(d) = e

R_t

0eς⁰_sdL_s+P 0≤s≤t

ln(1+eς_s⁰∆L_s)−eς⁰_s∆L_s

is the jump Dol´ean exponential and the non random functions^ςes = ς_s⁰θ_s = (e^ς1,s, . . . ,e^ςm,s)⁰ ∈ [0,1]^m. Note here, that this exponential can be represented as productionE_t^(d) = Qm

j=1E_j,t^(d) of the independent exponentials

E_j,t^(d)= e

R_t

0ςe_j,s⁰ dL_j(s)+P 0≤s≤t

ln

1+eς_j,s⁰ ∆L_j(s)

−eς⁰_j,s∆L_j(s)

. Therefore, to prove this lemma it suffices to show that

max

1≤j≤m sup

0≤t≤T

E(E_j,t^(d))²<∞. (5.2) To this end, note that we can write that

lnE_j,t^(d)=Z_j,t+

t

Z

0

Π_j g(e^ςj,sy)−e^ςj,sy

ds , (5.3)

where the processZ_j,t=Rt 0

R

R∗

g e^ςj,sy

dν_j,ν_j(ω; dy ,ds) = (ν_j−ν˜_j)(ω; dy ,ds)and g(x) = ln(1 +x). Note here, that for any0≤ς≤1

Π_j(|g(ςy)−ςy|)≤Π_j 1_{y<−1/2}(1 +|g(y)| + 2Π_j(1_{|y|≤1/2}y²) +Π_j 1_{y>1/2}(y+g(y)

. Using here the conditions (2.6) we get that

max

1≤j≤m sup

0≤ς≤1

Π_j(|g(ςy)−ςy|)<∞. (5.4) Therefore, for (5.2) it suffices to show, that

max

1≤j≤m sup

0≤t≤T

Ee^2Zj,t<∞. (5.5)

It is clear that a.s.

Z_j,t= lim

δ→0

Z_j,t^(δ) and Z_j,t^(δ)=

t

Z

0

Z

R_∗

g e^ςj,sy dν⁽_j^δ),

whereν⁽_j^δ)(ω; dy ,ds) =1_{|y|>δ}ν_j(ω; dy ,ds). Note, that we can represent the function eς_j,sas a limit in the Lebesgue measure on the interval[0, t]of the piece functions

ςe_j,s⁽ⁿ⁾=

n

X

l=1

c⁽_j,lⁿ⁾1_]t

l−1,t_l], 0≤c⁽_j,lⁿ⁾≤1 and t_l= ^l

nt . (5.6)

It should be emphasized, that in view of the boundedness of the functionse^ςj,sand^ςe

(n) j,s we can conclude through the dominated convergence theorem that

n→∞lim Z t

0

|^ςej,s−e^ς

(n)

j,s|ds= 0. (5.7)

Now, setting

Z_j,t^(δ,n)=

t

Z

0

Z

|y|>δ

g

eς_j,s⁽ⁿ⁾y

dν_j, we can obtain that

E ^Z

(δ)

j,t −Z_j,t^{(δ,n) ≤}2

t

Z

0

Z

|y|>δ

∆⁽_jⁿ⁾(y, s)Π(dy) ds (5.8)

and∆⁽_jⁿ⁾(y, s) =

^g e^ςj,sy

−g

ςe_j,s⁽ⁿ⁾y

. Note here, that for0< ε <1−δand for any y >−1 +εwe get that

∆⁽_jⁿ⁾(y, s)≤1

ε|^ςej,s−e^ς

(n) j,s||y|.

Moreover, note, that fory > −1we have∆⁽_jⁿ⁾(y, s)≤ 2|ln(1 +y)|. Therefore, we can estimate from above the integral in the right side of the inequality (5.8) as

2t Z

{−1<y<−1+ε}

|ln(1 +y)|Π(dy) +U_∗(ε)

t

Z

0

|e^ςj,s−^ςe

(n) j,s|ds , where

U_∗(ε) =1 ε





 Z

{−1+ε<y<−δ}

|y|Π(dy) + Z

{y>δ}

y Π(dy)





^. Now, taking into account the limit (5.7), we obtain that for any0< ε <1−δ

lim sup

n→∞

E ^Z

(δ)

j,t −Z_j,t^{(δ,n) ≤}4t

Z

{−1<y<−1+ε}

|g(y)|Π(dy).

Letting hereε→0, we obtain through the condition (2.6) P− lim

n→∞Z_j,t^(δ,n)=Z_j,t^(δ). This implies by the Fatou lemma

Ee^2Zj,t ≤lim inf

δ→0 lim inf

n→∞ Ee^2Z

(δ,n)

j,t . (5.9)

Note here, that

Ee^2Z

(δ,n) j,t =

n

Y

l=1

Ee^2ηl and η_l=

t_l

Z

t_l−1

Z

R∗

g

c⁽_j,lⁿ⁾y

1_{|y|>δ}dν_j.

We calculate directly, that

Ee^2ηl=e^{(tl−tl−1)$}

(δ,n) j,l ,

where

$⁽_j,l^δ,n)= Z

|y|>δ

e^2g

c⁽ⁿ⁾

j,l y

−1−2g

c⁽_j,lⁿ⁾y

Π_j(dy). Using the conditions (2.6) we can estimate this term as

$_j,l^(δ,n)≤2 Z

−1<y<−1/2

(1 +|g(y)|)Π_j(dy)

+ 2e Z

|y|≤1/2

g²(y)Π_j(dy) + Z

y>1/2

1 +y²

Π_j(dy) :=$^∗<∞.

Therefore, using this estimate in (5.9), we get that Ee^2Z

(δ,n) j,t ≤e^t$

∗

<∞. This implies the upper bound (5.5). Hence Proposition5.1.

Proposition 5.2. For any non random measurable[0, T]→Θfunctionθ= (θ_t)_0≤t≤T, the Dol´ean exponential(2.11)is strictly positive, i.e.inf_0≤t≤TE_t(V)>0a.s.

Proof. First note, that the functionsσ_jiare square integrated and, therefore, through Doob’s inequality (A.3) we get

E max

0≤t≤T





t

Z

0

θ_u⁰σ_udW_u





2

≤4

T

Z

0

θ_u⁰σ_uσ⁰_uθ_udu≤4

T

Z

0

trσ_uσ_u⁰ du <∞,

i.e.max_0≤t≤T|Rt

0θ⁰_uσ_udW_u| < ∞a.s. This means, that in view of the representations (5.1) and (5.3) and the upper bound (5.4) to prove this proposition it suffices to show that

max

1≤j≤m sup

0≤t≤T

|Z_j,t|<∞ a.s. (5.10)

Indeed, we can represent this process as

Z_j,t=Z_j,t⁽¹⁾+Z_j,t⁽²⁾, where

Z_j,t⁽¹⁾=

t

Z

0

Z

R_∗

1_{y<−1/2}g ^ςej,sy

dν_j and Z_j,t⁽²⁾=

t

Z

0

Z

R_∗

1_{y≥−1/2}g e^ςj,sy dν_j.

Note here, that in view of the condition (2.6) we get, that for any0≤t≤T

|Z_j,t⁽¹⁾| ≤ X

0≤s≤T

1_{∆L

j(s)<−1/2}|g(∆L_j(s))| +Π_j |g(y)|1_{{−1<y<−1/2}}

<∞ a.s..

Moreover, we estimate the termZ_j,t⁽²⁾from above through the inequality (A.2) withp= 2 and using again the conditions (2.6), i.e.

E sup

0≤t≤T

(Z_j,t⁽²⁾)²≤_Cˇ₂

T

Z

0

Z

R∗

1_{y≥−1/2}g² e^ςj,sy

dsΠ_j(dy)

≤_Cˇ_{2T Πj |g}(y)|1_{y≥−1/2}

<∞. Hence Proposition5.2.

Proposition 5.3. For any0≤t≤Tandx >0 E_t,x sup

t≤s≤T

(X_s^∗)²<∞. (5.11)

Moreover,

inf

0≤t≤T X_t^∗>0 a.s. (5.12)

Proof. Indeed, note that, from (3.17) it is easy to deduce thatX_t^∗ ≤ CE_t(V^∗)for some C >0. Therefore, Doob’s martingale inequality (A.3) and Proposition5.1imply

E_t,x sup

t≤s≤T

(X_s^∗)²≤CE sup

t≤s≤T

E_t²(V^∗)≤4CEE_T²(V^∗) <∞

and we get the bound (5.11). Moreover, using again the representation (3.17) we obtain through Proposition5.2the lower bound (5.12).

Now we need to study properties of the discrete investment strategy (4.4).

Proposition 5.4. If the conditions(2.2),(2.4)and(2.6)hold true, then max

1≤j≤msup

n≥1

sup

0<t₁<...<t_n=T

E

Z T 0

α⁽_jⁿ⁾(t) dS_j(t)

<∞

and

sup

n≥1

sup

0<t₁<...<t_n=T

E

Z T 0

β_t⁽ⁿ⁾dB_t

<∞.

Proof. First note, that the stock price can be represented as S_j(t) =S₀ exp

Z t 0

µ_j(u) du

E_t(eL_j) (5.13)

whereE_t(eL_j)is defined in (2.11) with V_t = Le_j andhVi_t = Pm l=1

Rt

0 σ²_j,l(s) ds. Note now, that for any fixed0 ≤u≤T the processEb_u,t(eL_j) = E_t(eL_j)/E_u(eL_j)is the Dol´ean exponential foru ≤t≤ T which is independent fromF_uand, therefore, Proposition5.1 yields

max

1≤j≤m sup

0≤u<t≤T

EEb_u,t² (eL_j) = max

1≤j≤m sup

0≤u<t≤T

E

Eb_u,t² (eL_j)|F_u

<∞.