Sergei Egorov
email: sergei.egorov@univ-rouen.fr

Serguei Pergamenchtchikov

Serguei Pergamenshchikov
email: serge.pergamenshchikov@univ-rouen.fr

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

Keywords: primary 60P05, secondary 62G05 Financial markets, Optimal investment/consumption problem, Stochastic control, Dynamical programming, Hamilton-Jacobi-Bellman equation

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

For this problem we show the corresponding verification theorem and construct the optimal 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.

1 Introduction

Motivations

In this paper, we consider a portfolio optimization problem for Lévy financial markets with non-random time-dependent coefficients. Such problems are very popular in the stochastic financial markets theory. Beginning with the classic work of Merton, where the optimal investment problem for Black-Scholes models was first studied, interest in these problems is constantly growing to the present day (see, for example, [START_REF] Karatzas | Methods of Mathematical Finance[END_REF][START_REF] Delong | Optimal investment and consumption in a Black-Scholes market with stochastic coefficients driven by a non-diffusion process[END_REF][START_REF] De Valére | Consumption-investment optimization problem in a Lvy financial model with transaction costs[END_REF][START_REF] Lépinette | Consumption-investment optimization problem in a Lévy financial model with transaction costs (hal-00931665)[END_REF] and the references therein). It should be emphasized that the financial markets defined by the continuous 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 markets. It seems that for the first time an optimization portfolio problem for financial markets with jumps was studied in the paper [START_REF] Framstad | Sufficient Stochastic Maximum Principle for the Optimal Control of Jump Diffusions and Applications to Finance[END_REF], in which the authors using the stochastic Pontryagin maximum principle constructed optimal investment/consumption strategies. Later, these problems were studied for more complex market models and in different settings: in [START_REF] Kramkov | The condition on the Asymptotic Elasticity of Utility Functions and Optimal Investment in Incomplete Markets[END_REF][START_REF] Kallsen | Optimal portfolios for exponential Lévy processes[END_REF][START_REF] Nutz | The Bellman equation for power utility maximization with semimartingales[END_REF] the authors considered maximization utility problems in general semi-martingale settings, in [START_REF] Zariphopoulou | A solution approach to valuation with unhedgeable risks[END_REF][START_REF] Delong | Optimal investment and consumption in a Black-Scholes market with stochastic coefficients driven by a non-diffusion process[END_REF][START_REF] Berdjane | Optimal consumption and investment for markets with randoms coefficients[END_REF][START_REF] Liu | Dynamic asset allocation with event risk[END_REF] such problems were considered for stochastic volatility markets, in [START_REF] Duffie | Affine processes and applications in finance[END_REF][START_REF] Kallsen | Utility maximization in affine stochastic volatility models[END_REF] for the markets defined by the affine processes, in [START_REF] Klüppelberg | Optimal consumption and investment with bounded downside risk for power utility functions[END_REF][START_REF] Klüppelberg | Optimal consumption and investment with bounded downside risk measures for logarithmic utility functions[END_REF][START_REF] Chouaf | Optimal investment with bounded VaR for power utility functions[END_REF][START_REF] Emmer | Optimal portfolios when stock prices follow an exponential Lévy process[END_REF] the authors considered the portfolio optimization problems with constraints. In [START_REF] De Valére | Consumption-investment optimization problem in a Lvy financial model with transaction costs[END_REF][START_REF] Lépinette | Consumption-investment optimization problem in a Lévy financial model with transaction costs (hal-00931665)[END_REF] the authors considered pure consumption problem on Lévy markets with infinite time horizon under proportional transaction costs where they used geometric approach and viscosity solutions in similar spirit as it was done in [START_REF] Benth | Optimal portfolio management rules in a non-Gaussian market with durability and intertemporal substitution[END_REF].

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.

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 [START_REF] Karatzas | Methods of Mathematical Finance[END_REF], 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 [START_REF] Klüppelberg | Optimal consumption and investment with bounded downside risk for power utility functions[END_REF][START_REF] Klüppelberg | Optimal consumption and investment with bounded downside risk measures for logarithmic utility functions[END_REF][START_REF] Delong | Optimal investment and consumption in a Black-Scholes market with stochastic coefficients driven by a non-diffusion process[END_REF][START_REF] Berdjane | Optimal consumption and investment for markets with randoms coefficients[END_REF] we study this problem through the verification theorem method. So, in this paper, probably for the first time, we show a special 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 provide 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, [START_REF] Leland | Option Pricing and Replication with Transactions Costs[END_REF]) in the difference from the geometric approach developed in [START_REF] De Valére | Consumption-investment optimization problem in a Lvy financial model with transaction costs[END_REF][START_REF] Lépinette | Consumption-investment optimization problem in a Lévy financial model with transaction costs (hal-00931665)[END_REF] for Lévy 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 optimality property one needs to use the Lepinette approach developed for the asymptotic hedging problem in [START_REF] Lépinette | Modified Leland's strategy for constant transaction costs rate[END_REF], 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.

Organisation of paper

In Section 2 we describe the problem and all necessary definitions. In Section 3 we study optimal control problem for stochastic systems with jumps through the verification theorem method. In Section 4 we state the main results. In Section 5 we the optimal strategies. In Section 6 we give the results of Monte-Carlo simulations. All main results are shown in Section 7. All auxiliary tool is given in Appendix A.

Problem

In this paper, we consider optimal investment and consumption problems for financial markets of Lévy 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 m risky assets (stocks)

(S i (t)) 0≤t≤T , 1 ≤ i ≤ m, defined as    dB t = r t B t dt , B 0 = 1 , dS i (t) = µ i (t) S i (t) dt + S i (t-) d L i (t) , (2.1)
where the interest rate r t and the drifts (µ j (t)) 1≤j≤m are non random [0, T ] → R integrated functions, i.e.

T 0 |r t | + m i=1 |µ i (t)| dt < ∞ .
(2.2)

We assume, that the random market sources ( L i (t)) 0≤t≤T are represented as

L i (t) = m j=1 t 0 σ ij (u) dW j (u) + ς ij (u) dL j (u) , (2.3)
in which the volatilities

σ t = σ ij (t)
1≤i,j≤m

and ς t = ς ij (t) 1≤i,j≤m
are m × m non random matrices such that for any

1 ≤ i, j ≤ m, 0 ≤ t ≤ T T 0 σ 2 ij (u) du < ∞ and 0 ≤ ς ij (t) ≤ 1 .
(2.4) Moreover, we assume, that (W 1 (t)) 0≤t≤T ,...,(W m (t)) 0≤t≤T are independent standard Brownian motions and (L 1 (t)) 0≤t≤T , . . . , (L m (t)) 0≤t≤T are independent pure jumps Lévy processes, i.e.

L j (t) = t 0 R * y ν j (ω; dy , du) -ν j (dy , du) , (2.5)
where ν j (ω; dy , du) is random jump measure with compensator ν j (dy , du) = Π j (dy) du and Π j (•) is the corresponding Lévy measure on R * = R \ {0} . It should be noted that in case of independent Lévy processes, ∆L j (t) ∆L i (t) = 0 , for any i = j. First, to provide the positivity for the risky price process (2.1) we assume, that Π j (] -∞, -1]) = 0. Moreover, we assume also that

Π j | ln(1 + y)|1 {-1<y<-1/2} < ∞ and Π j (y 2 ) < ∞ , (2.6)
where 1 A is an indicator function of the set A. These conditions are technical and used in the market model to guarantee existence of optimal trading strategy. In the sequel we set

µ t = (µ 1 (t) , . . . , µ m (t)) , W t = (W 1 (t) , . . . , W m (t)) and L t = (L 1 (t) , . . . , L m (t))
, where the prime 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 [START_REF] Klüppelberg | Optimal consumption and investment with bounded downside risk for power utility functions[END_REF] in this paper we use the fractional strategies defined as

θ i (t) = α i (t) S i (t) X t and c t = ζ t X t , (2.7)
where α i (t) is the amount of investment into i-th risky asset purchased by an investor at the time moment t and X t is the corresponding wealth process with β t bond units defined as

X t = m i=1 α i (t) S i (t) + β t B t , (2.8)
using (2.7) and (2.8) can be proved the following equality 1 -m i=1 θ i (t) = β t B t /X t . Moreover, the process ζ t in (2.7) is the consumption intensity, i.e. it is non negative integrable process for which the integral t 0 ζ s ds is 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, [START_REF] Karatzas | Methods of Mathematical Finance[END_REF]) and the definition (2.1), we obtain that

   dX t = X t (r t + θ t μt -c t ) dt + X t-θ t-d L t , X 0 = x > 0 , (2.9)
where μt = (µ t -r t e m ) , e m = (1, . . . , 1) ∈ R m and θ t = (θ 1 (t), . . . , θ m (t)) and L t = ( L 1 (t) , . . . , L m (t)) . It should be noted that to provide the positivity of portfolio value X t the jump sizes of the process Ľt = t 0 θ u-ς u-dL u have to be more than -1 , i.e. ∆ Ľt > -1 . To this end we assume that the financial strategy θ t = (θ 1 (t) , . . . , θ m (t)) is a càdlàg process with values in the set [0, 1] m such that, for any fixed 0 ≤ t ≤ T , almost sure m j=1 θ j (t) ≤ 1 . In the sequel we denote by

Θ =    θ = (θ 1 , . . . , θ m ) ∈ [0, 1] m : m j=1 θ j ≤ 1    . (2.10)
Using this set we introduce admissible strategies. Definition 2.1. A stochastic process υ = (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 that X t-> 0 and X t > 0 a.s. for 0 ≤ t ≤ T .

We denote by V the set of admissible strategies. It should be noted that for any υ ∈ V the wealth process (2.9) can be represented as

X t = x e t 0 (r s +θ s μs ) ds-t 0 c s ds E t (V )
and the Doléan exponential

E t (V ) = e V t -1 2 V t 0≤s≤t (1 + ∆V s ) e -∆V s , (2.11)
where

V t = t 0 θ s d L s and V t = t 0 θ s σ s σ s θ s ds.
Now to formulate an optimal consumption and investment problem we introduce for some 0 < γ < 1, the objective function as

J(x, υ) := E x   T 0 ζ γ t dt + (X T ) γ +   , (2.12)
where E x is the conditional expectation given X 0 = x and (x) + is the positive part of x , i.e. (x) + = max(x, 0) and ζ t = c t X t is the intensity of consumption.

The goal is to maximize the objective function on the set V, i.e. to find a strategy υ * ∈ 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 for υ ∈ V

J(t, x, υ) := E x,t   T t (c u X u ) γ du + X γ T   .
Here E x,t is the conditional expectation with respect to X t = x . Note that in our case J(t, x, υ) can be equal to ∞ for some strategy υ .

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 [START_REF] Framstad | Sufficient Stochastic Maximum Principle for the Optimal Control of Jump Diffusions and Applications to Finance[END_REF] through the maximum Pontryagin principle. The similar problems were considered in [START_REF] Nutz | Optimal consumption and investment with power utility[END_REF][START_REF] Nutz | The Bellman equation for power utility maximization with semimartingales[END_REF] on the basis of the dual problems approach.

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 process X t with values in an open convex set X ⊆ R and control process

υ t with values in closed set K ⊆ R m . Let X = (X t ) 0≤t≤T , X t , X t-∈ X be càdlàg process of form    dX t = a(t, X t , υ t ) dt + b (t, X t-, υ t-) d L t , X 0 = x ∈ X , (3.1)
where the process L t ∈ R m is defined in (2.3) with the Lévy measures Π j satisfying the second condition in (2.6), i.e. Π j (y 2 ) < ∞. The functions a : [0, T ] × X × K → R and b : X × K → R m are non random, continuous and such that, for any non random v ∈ K, the equation (3.1) with υ t ≡ v has an unique strong solution for which X t ∈ X and X t-∈ X on the time interval [0, T ] and

T 0 |a(t, X t , v)| + |b(t, X t , v)| 2 dt < ∞ a.s.. Definition 3.1. A stochastic process υ = (υ t ) 0≤t≤T is called admissible if it is (F t ) 0≤t≤T
-adapted, has càdlàg trajectories, takes values in the set K , and such that the equation (3.1) on time interval [0, T ] has an unique strong solution for which X t , X t-∈ X on the time interval [0, T ] and

T 0 |a(t, X t , υ t )| + |b(t, X t , υ t )| 2 dt < ∞ a.s.. (3.2)
We denote by V the set of all admissible strategies. Note that the conditions on functions a and b imply V = ∅ , at least the strategy υ t ≡ v ∈ V . Now we fix utility functions U 1 : [0, T ] × X × K → R + and U 2 : X → R + and, then, for any 0 < t ≤ T , we set the objective functions as

J(t, x, υ) := E t,x   T t U 1 (u, X u , υ u ) du + U 2 (X T )   .
Our goal is to find an admissible strategy υ * ∈ V, such that, for any 0 ≤ 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 function. To do this, for any [0, T ] × X → R function g(t, x) , twice continuously differentiable in x and continuously differentiated in t, such that

sup x∈X |g(t, x)| 1 + |x| < ∞ , (3.4)
we define

H 0 (t, x, g, v) = a(t, x, v) g x (t, x) + 1 2 g xx (t, x) tr σ t b(t, x, v) b (t, x, v) σ t + U 1 (t, x, v) + g(t, x, v) , (3.5)
where

g(t, x, v) = m i=1 R * Υ g (t,
x, b (t, x, v) ς i,t y) Π i (dy), ς i,t = (ς 1i (t) , . . . , ς mi (t)) and

Υ g (t, x, v) = (g(t, x + v) -g(t, x) -g x (t, x) v) 1 {x+v∈X } for v ∈ R * .
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) = U 2 (x) , x ∈ X . (3.7)
Next, we need the following conditions. (

3.8)

H 2 ) There exists [0, T ] × X → K measurable function v 0 , such that for the solution z = z(t, x) of the equation (3.7) the Hamilton function H(t, x, z) = H 0 (t, x, z, v 0 (t, x)).

H 3 ) For any x ∈ X , there exists an unique almost surely solution X * = (X * t ) 0≤t≤T with values in the set X and X * t-∈ X of the equation

dX * t = a * (t, X * t ) dt + (b * (t-, X * t-)) d L t , X * 0 = x , (3.9
)

where a * (t, x) = a(t, x, v 0 (t, x)) and b * (t, x) = b(x, v 0 (t, x)). Moreover, the process v 0 = (v 0 (t, X * t )) 0≤t≤T is admissible, i.e. belongs to V . H 4 ) For any 0 ≤ t ≤ T and x ∈ X , E t,x sup t≤u≤T |z(u, X * u )| < ∞ .
(3.10)

Using the approach proposed in [START_REF] Klüppelberg | Optimal consumption and investment with bounded downside risk for power utility functions[END_REF], we show the following verification theorem.

Theorem 3.3. Suppose conditions H 1 ) -H 4 ) are hold. Then, for any

0 ≤ t ≤ T and x ∈ X , z(t, x) = J * (t, x) = J(t, x, υ * ) ,
where the optimal strategy

υ * = (υ * s ) t≤s≤T , υ * s = v 0 (s, X * s ) is determined in terms of H 2 ) -H 4 ) and the function J * (t, x) is defined in (3.3).
Remark 3.4. Here (3.8) and H 4 ) 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 expectations, the last condition provides the finiteness almost sure for the jumps compensator of z(t, X t ) and H 4 ) will be used to apply dominated convergence theorem to prove optimality of the process υ * . Remark 3.5. Note, that we don't assume the uniqueness of a solution for the equation (3.7). But if conditions of Theorem 3.3 hold then z(t, x) will be the unique solution of the equation (3.7), since the supremum J(t, x, υ * ) is always unique. Now we apply Theorem 3.3 to the problem (2.13). In this case the controlled process driven by (2.9) with state space X =]0, ∞[, admissible strategy υ = (θ t , c t ) 0≤t≤T with values in K = Θ × R + , where the set Θ defined by (2.10). So, to study the optimal consumption and investment problem (2.14) we will apply Theorem 3.3 for the utility functions

U 1 (x, v) = (x c) γ and U 2 (x) = x γ , where x ∈ X , v = (θ, c) ∈ K, the vector θ = (θ 1 , . . . , θ m ) and 0 < 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 μt -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 t x z x (t, x) + max θ∈Θ Γ (t, x, z, θ) + (1 -γ) γ z x (t, x) γ 1-γ = 0 , z(T, x) = x γ , (3.12)
where Γ (t, x, z, θ) = x z x (t, x) θ μt + x 2 z xx (t, x) θ σ t σ t θ/2 + z(t, x, θ) and z(t, x, θ) is defined by g in (3.5) for g = z. Moreover, according to Condition H 2 ) to find an optimal control function v 0 = (θ 0 , c 0 ) one needs to choose θ 0 = θ 0 (t, x, z) and c 0 = c 0 (t, x, z) as

θ 0 = arg max θ∈Θ Γ (t, x, z, θ) and c 0 = 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 substitute z with the form (3.14) in (3.12), we obtain that the function

Γ (t, x, z, θ) = A(t)x γ F (t, θ),
where

F (t, θ) = γ θ μt + γ (γ -1) 2 tr θ σ t σ t θ + m i=1 R * ρ θ ς i,t y Π i (dy) , (3.15)
where ρ(x) = (1 + x) γ -1 -γx and ς i,t = (ς 1i (t) , . . . , ς mi (t)) is i-th column of the matrix ς t . Therefore, from (3.13) we obtain that in this case for the HJB solution (3.14)

θ 0 = θ 0 (t) = arg max θ∈Θ F (t, θ) and c 0 = c 0 (t) =   T t Ψ t,u du + Ψ t,T   -1 , (3.16)
where

Ψ t,u = e u t h * s ds and h * 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 t 0 (r s +(θ * s ) μs ) ds-t 0 c * s ds E t (V * ) (3.17)
and E t (V * ) is the Doléan exponential (2.11) for the process

V * t = t 0 (θ * s ) d L s with its quadratic characteristic V * t = t 0 (θ * s ) σ s σ s θ * s ds.
Remark 3.6. If there exists many solutions in for the maximization problem of the function F (t, •) we chose any point.

Main results

Fist we study the strategy (3.16).

Theorem

J * (t, x) = J(t, x, υ * ) = x γ   T t Ψ t,u du + Ψ t,T   1-γ . (4.1)
The proof is given in section 7.

Remark 4.2. Note that for the homogenous market model (2.1), i.e. for the constant coefficients r t , µ t , σ t , ς t the optimal strategy (3.16) is obtained in [START_REF] Nutz | Optimal consumption and investment with power utility[END_REF]. Now we consider the optimization problem (2.1) for the markets with transaction costs on the basis of the Leland's approach proposed in [START_REF] Leland | Option Pricing and Replication with Transactions Costs[END_REF] for hedging problems. In this case we use the optimal strategy

α * t = α * 1 (t), . . . , α * m (t) = θ * 1 (t) X * t S 1 (t) , . . . , θ * m (t) X * t S m (t) , β * t = (1 -m j=1 θ * j (t)) X * t B t and ζ * t = c * t X * t , (4.2)
where the fractional strategy

θ * t = (θ * 1 (t), . . . , θ * m (t)
) is given in (3.13) and the corresponding wealth process X * t is calculated as

X * t = x + t 0 α * u- dS u + t 0 β * u dB u - t 0 ζ * u du . (4.3)
According to the Leland approach we can do only n portfolio revisions of the strategy (4.2) on the interval [0, T ] at the moments 0 < t 1 < . . . < t n = T , i.e. we set

α (n) t = n k=1 α * t k-1 1 ]t k-1 ,t k ] , β (n) t = n k=1 β * t k-1 1 ]t k-1 ,t k ] and ζ (n) t = n k=1 ζ * t k-1 1 ]t k-1 ,t k ] . (4.4)
Moreover, for each revision the investors must pay the transaction costs proportionally to the volume of trade at the moment t k defined as

κ m j=1 S t k |α (n) j (t k ) -α (n) j (t k-1 )| = κ m j=1 S t k |α * j (t k-1 ) -α * j (t k-2 )| ,
where by the convention α * j (t) = 0 for t < 0. Therefore, the total transaction costs on the interval [0, T ] is presented as

D n = κ m j=1 n k=1 S j (t k )|α (n) j (t k ) -α (n) j (t k-1 )| , (4.5)
where κ > 0 is a proportional transaction coefficient which is assumed to be a function of the revisions numbers n, i.e. κ = κ n . Note that, in this case α

(n) u-= α (n) u .
Therefore, to take into account transaction costs we defined the wealth as

X (n) T = x + T 0 (α (n) u ) dS u + T 0 β (n) u dB u - T 0 ζ (n) u du -D n . (4.6)
To study properties of the strategy ψ (n) = (α

(n) t , β (n) t , ζ (n)
t ) 0≤t≤T we set

J(x, ψ (n) ) = Ex   T 0 ζ (n) u γ du + X (n) T γ +   . (4.7)
Now we study asymptotic properties of the strategy ψ (n) as n → ∞. To do this we need to assume the following conditions.

A 1 ) In the model (2.1) the functions r t , µ t and σ t are bounded on the interval [0, T ].

A 2 ) The functions r t , µ t , σ t , ς t in the model (2.1) are such that the optimal strategy (3.13) satisfies the following inequality

sup n≥1 sup 0<t 1 <...<t n =T n j=1 |θ * t j -θ * t j-1 | 1 + n j=1 t j -t j-1 < ∞ . ( 4
lim n→∞ J(x, ψ (n) ) = J * (x) , (4.10)
where the optimal functional J * (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 that κ = κ n = o(1), i.e. κ n → 0 as n → ∞. In this case we need to change the strategy (4.4) making use of Lepinette approach proposed in [START_REF] Lépinette | Modified Leland's strategy for constant transaction costs rate[END_REF] and used then for the markets with jumps in [START_REF] Nguyen | Approximate hedging with constant proportional transaction costs in financial markets with jumps[END_REF]. 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 power q ≥ 1 is a function of n, i.e. q = q n such that

lim n→∞ q n = +∞ and lim n→∞ q n n + κ n n q n = 0 . (4.12)
Note that, if κ n → 0 as n → ∞, then we can take, for example,

q n = √ n + κ n n.
(V ) = E t-(V ) dV t , to show this lemma it suffices to check that sup 0≤t≤T E E 2 t (V ) < ∞.
To this end we represent the Doléan exponential in the multiplicative form, i.e.

E t (V ) = e t 0 θ s σ s dW s -1 2 t 0 θ s σ s σ s θ s ds E (d) t , (5.1)
where E (d) t = e t 0 ς s dL s + 0≤s≤t ln(1+ ς s ∆L s )-ς s ∆L s is the jump Doléan exponential and the non random functions ς s = ς s θ s = ( ς 1,s , . . . , ς m,s ) ∈ [0, 1] m . Note here, that this exponential can be represented as production

E (d) t = m j=1 E (d) j,t of the independent exponentials E (d)
j,t = e t 0 ς j,s dL j (s)+ 0≤s≤t ln 1+ ς j,s ∆L j (s) -ς 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 (d) j,t ) 2 < ∞ .
(5.2)

To this end, note that we can write that

ln E (d) j,t = Z j,t + t 0 Π j g( ς j,s y) -ς j,s y ds , (5.3)
where the process Z j,t = t 0 R * g ς j,s y dν j , ν j (ω; dy , ds) = (ν j -νj )(ω; dy , ds) and g(x) = ln(1 + x). Note here, that for any 0 ≤ ς ≤ 1

Π j (|g(ςy) -ςy|) ≤ Π j 1 {y<-1/2} (1 + |g(y)| + 2Π j (1 {|y|≤1/2} y 2 ) + 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 E e 2Z j,t < ∞ .

(5.5)

It is clear that a.s.

Z j,t = lim δ→0 Z (δ) j,t and Z (δ) j,t = t 0 R * g ς j,s y dν (δ) j ,
where ν (δ) j (ω; dy , ds) = 1 {|y|>δ} ν j (ω; dy , ds). Note, that we can represent the function ς j,s as a limit in the Lebesgue measure on the interval [0, t] of the piece functions

ς (n) j,s = n l=1 c (n) j,l 1 ]t l-1 ,t l ] , 0 ≤ c (n) j,l ≤ 1 and t l = l n t .
(5.6)

It should be emphasized, that in view of the boundedness of the functions ς j,s and ς (n) j,s we can conclude through the dominated convergence theorem that

lim n→∞ t 0 | ς j,s -ς (n) j,s |ds = 0 .
(5.7)

Now, setting

Z (δ,n) j,t = t 0 |y|>δ g ς (n) j,s y dν j ,
we can obtain that

E Z (δ) j,t -Z (δ,n) j,t ≤ 2 t 0 |y|>δ ∆ (n) j (y, s) Π(dy) ds (5.8)
and

∆ (n) j (y, s) = g ς j,s y -g ς (n)
j,s y . Note here, that for 0 < ε < 1 -δ and for any y > -1 + ε we get that

∆ (n) j (y, s) ≤ 1 ε | ς j,s -ς (n) j,s ||y| .
Moreover, note, that for y > -1 we have

∆ (n) j (y, s) ≤ 2| ln(1 + y)|.
Therefore, we can estimate from above the integral in the right side of the inequality (5.8) as

2 t {-1<y<-1+ε} | ln(1 + y)| Π(dy) + U * (ε) t 0 | ς j,s -ς (n) j,s |ds , where U * (ε) = 1 ε    {-1+ε<y<-δ} |y| Π(dy) + {y>δ} y Π(dy)    .
Now, taking into account the limit (5.7), we obtain that for any

0 < ε < 1 -δ lim sup n→∞ E Z (δ) j,t -Z (δ,n) j,t ≤ 4 t {-1<y<-1+ε}
|g(y)| Π(dy) .

Letting here ε → 0, we obtain through the condition (2.6)

P -lim n→∞ Z (δ,n) j,t = Z (δ) j,t .
This implies by the Fatou lemma

E e 2Z j,t ≤ lim inf δ→0 lim inf n→∞ E e 2Z (δ,n) j,t
.

(5.9)

Note here, that

E e 2Z (δ,n) j,t = n l=1 E e 2η l and η l = t l t l-1 R * g c (n) j,l y 1 {|y|>δ} dν j .
We calculate directly, that

E e 2η l = e (t l -t l-1 ) (δ,n) j,l
, where

(δ,n) j,l = |y|>δ e 2g c (n) j,l y -1 -2g c (n) j,l y Π j (dy) .
Using the conditions (2.6) we can estimate this term as

(δ,n) j,l ≤ 2 -1<y<-1/2 (1 + |g(y)|) Π j (dy) + 2e |y|≤1/2 g 2 (y) Π j (dy) + y>1/2 1 + y 2 Π j (dy) := * < ∞ .
Therefore, using this estimate in (5.9), we get that

E e 2Z (δ,n) j,t ≤ e t * < ∞ .
This implies the upper bound (5.5). Hence Proposition 5.1.

Proposition 5.2. For any non random measurable [0, T ] → Θ function θ = (θ t ) 0≤t≤T , the Doléan exponential (2.11) is strictly positive, i.e. inf 0≤t≤T E t (V ) > 0 a.s.

Proof. First note, that the functions σ ji are square integrated and, therefore, through Doob's inequality (A.3) we get

E max 0≤t≤T   t 0 θ u σ u dW u   2 ≤ 4 T 0 θ u σ u σ u θ u du ≤ 4 T 0 tr σ u σ u du < ∞ , i.e. max 0≤t≤T | t 0 θ u σ u dW 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 (1) j,t + Z (2) j,t , where Z (1)
j,t = t 0 R * 1 {y<-1/2} g ς j,s y dν j and Z (2) j,t = t 0 R * 1 {y≥-1/2} g ς j,s y dν j .
Note here, that in view of the condition (2.6) we get, that for any

0 ≤ t ≤ T |Z (1) j,t | ≤ 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 term Z

(2) j,t from above through the inequality (A.2) with p = 2 and using again the conditions (2.6), i.e.

E sup 0≤t≤T (Z (2) j,t ) 2 ≤ Č2 T 0 R * 1 {y≥-1/2} g 2 ς j,s y dsΠ j (dy) ≤ Č2 T Π j |g(y)|1 {y≥-1/2} < ∞ .
Hence Proposition 5.2.

Proposition 5.3. For any 0 ≤ t ≤ T and x > 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 that X * t ≤ C E t (V * ) for some C > 0. Therefore, Doob's martingale inequality (A.3) and Proposition 5.1 imply

E t,x sup t≤s≤T (X * s ) 2 ≤ C E sup t≤s≤T E 2 t (V * ) ≤ 4 C EE 2 T (V * ) < ∞
and we get the bound (5.11). Moreover, using again the representation (3.17) we obtain through Proposition 5.2 the lower bound (5.12).

Now we need to study properties of the discrete investment strategy (4.4). Proof. First note, that the stock price can be represented as

S j (t) = S 0 exp t 0 µ j (u) du E t ( L j ) (5.13)
where

E t ( L j ) is defined in (2.11) with V t = L j and V t = m l=1 t 0 σ 2 j,l (
s) ds. Note now, that for any fixed 0 ≤ u ≤ T the process E u,t ( L j ) = E t ( L j )/E u ( L j ) is the Doléan exponential for u ≤ t ≤ T which is independent from F u and, therefore, Proposition 5.1 yields

max 1≤j≤m sup 0≤u<t≤T E E 2 u,t ( L j ) = max 1≤j≤m sup 0≤u<t≤T E E 2 u,t ( L j )|F u < ∞ .
Taking into account that the functions µ j are bounded we can get the same property for the process S j (t, u) = S j (t)/S j (u), i.e.

U * = max 1≤j≤m sup 0≤u<t≤T E S 2 j (t, u) = max 1≤j≤m sup 0≤u<t≤T E S 2 j (t, u)|F u < ∞ .
Moreover, note that from the definition (4.2) we obtain that for any

1 ≤ l ≤ m E T 0 (α (n) j (t)) 2 S 2 j (t)σ 2 jl (t) dt ≤ E n k=1 (X * t k-1 ) 2 t k t k-1 E S 2 j (t, t k-1 )|F t k-1 σ 2 jl (t) dt ≤ U * E sup 0≤t≤T (X * t ) 2 T 0 σ 2 jl (t) dt
and through the upper bound (5.11) we get, that max 1≤j,l≤m

sup n≥1 sup 0<t 1 <...<t n =T E T 0 (α (n) j (t)) 2 S 2 j (t)σ 2 jl (t) dt < ∞ .
In the same way we can show, that

max 1≤j≤m sup n≥1 sup 0<t 1 <...<t n =T E T 0 α (n) j (t)S j (t)|µ j (t)| dt < ∞ .
Therefore, using the definition of the stock price in (2.1) we obtain that

E T 0 α (n) j (t) dS j (t) ≤ E T 0 α (n) j (t)S j (t)|µ j (t)| dt +   E   T 0 α (n) j (t)S j (t-) d L i (t)   2    1/2 .
Taking into account here, that

E   T 0 α (n) j (t)S j (t-) d L i (t)   2 = m l=1 E T 0 (α (n) j (t)) 2 S 2 j (t)σ 2 jl (t) dt + m l=1 Π l (y 2 ) E T 0 (α (n) j (t)) 2 S 2 j (t)ς 2 jl (t) dt ,
we obtain through the conditions (2.4) the first inequality of this lemma. As to the last inequality, note that

T 0 β (n) t dB t ≤ n k=1 X * t k-1 t k t k-1 r t B t B t k-1 dt = n k=1 X * t k-1 e t k t k-1 r u du -1 ≤ U * 1 sup 0≤t≤T X * t n k=1 t k t k-1 |r u | du = U * 1 sup 0≤t≤T X * t T 0 |r t | dt ,
where

U * 1 = sup |x|≤x * e x -1 x and x * = T 0 |r t |dt .
Using again the upper bound (5.11) we get the last inequality of this proposition.

Numerical example

For Monte Carlo simulations we consider the market (2.1) for m = 1 on the interval [0, T ] for T = 3 in which B t = e r t with r = 0, 055, σ = 0, 05

S t = S 0 exp µ - σ 2 2 t + σ W t + L t s≤t (1 + ∆L s ) e -∆L s .
Here µ = 0.06, S 0 = 50 , W t is standard Brownian motion, L t is a Compound Poisson process defined by L t = Nt j=1 Y j , where N t is Poisson process with the constant intensity λ = 17 and the sizes of jumps (Y j ) j≥1 are i.i.d. random variables with the density function

p(y) = e -y 2 2 η 2 ∞ -1/2 e -z 2 2 η 2 dz -1
1 {y>-1/2} and η = 0, 01. Note, that in this case the Lévy measure Π(dy) = λ p Y (y) dy.

In this example we consider the utility functions U 1 (x, v) = (x c) γ and U 2 (x) = x γ with γ = 0.3, where the control variables κ = (v , c) ∈ [0, 1] × R + . We calculate the strategies (4.2) -(4.4) with the cumulative consumption defined as

C (n) t = t 0 ζ (n) u du.
The graphs in Figure 1 show the market behavior: b) is the log returns of the risky asset, c)e) are trading strategies with and without transaction costs, f ) is investor's wealth and him consumption. Here the portfolios are revised n = 100 times with the transaction costs coefficient κ = 0, 1 ln(100)/ √ 100 ≈ 0, 046. Moreover, the graphs in Figure 2 a) illustrate the convergence (4.10) for the strategy (4.4) and (4.11). To this end, we calculate the objective function J(x, ψ (n) ) through Monte Carlo simulations as

J(x, ψ (n) ) ≈ 1 K K l=1   T 0 ζ (n,l) u du + X (n,l) T γ +   , K = 50 , where (ζ (n,l) u ) 0≤u≤T and X (n,l)
T are the consumption and wealth processes calculated at the l-th simulations, and transaction costs coefficient κ n = 0, 5 ln(n)/ As we can see from the simulation results in Figure 1, the behavior of the optimal strategy means that when the risky asset increases one needs to increase the consumption intensity and decrease the investments in the both assets risky and risky free. It should be noted that such an investor's policy allows better control over the behavior of the financial portfolio, so as not to make its terminal value negative. Usually, after the rise in prices, there is always a decline, and this optimal strategy permits avoiding abrupt changes in the portfolio dynamics. As we can see all the changes in portfolio dynamics generally occur because of consumption, not because of the stochastic behavior of the risky assets. The numerical simulations confirm the natural behavior of the optimal strategies, i.e. the more consumption, the less capital and, moreover, as it is seen in Figure 1, f), this strategy provides an equilibriumpoint between consumption and wealth. Finally, as we see In Figure 2, the convergence (4.10) is confirmed in these simulations.

Proofs

Proof of Theorem 3.3

Proof. Let υ ∈ V. Then by the definition of an admissible control process, the equation

dX u = a(u, X u , υ u ) du + b (X u-, υ u-) d L u , X t = x ∈ X , u ∈ [t, T ] (7.1)
has an unique P-a.s. strong solution. In view of the condition H 1 ) the HJB equation (3.7) has a solution z from C 1,2 ([0, T ] × R). Therefore, the Ito formula implies

z(u, X u ) = z(t, x) + u t z t (s, X s ) ds + u t z x (s, X s-) dX s + 1 2 u t z xx (s, X s ) s ds + t≤s≤u (∆z(s, X s ) -z x (s, X s-) ∆X s ) ,
where s = tr σ s b(s, X s , κ) b (s, X s , κ) σ s and ∆z(s, X s ) = z(s, X s ) -z(s, X s-) and ∆X s = X s -X s-. Therefore, using (7.1) we can represent this equation as

z(u, X u ) = z(t, x) + u t z t (s, Xs) + z x (s, X s ) a(s, X s , υ s ) + 1 2 z xx (s, X s ) s ds + M u + t≤s≤u (∆z(s, X s ) -z x (s, X s-) ∆X s ) , (7.2)
where

M u = u t z x (s, X s-)b (s, X s-, υ s-)d L s .
The key idea of the proof is the following, we need to compensate the jump term, add to and subtract from Ito's formula the integral u t U 1 (s, X s , υ s ) ds and take a conditional expectation given X t = x. But we cannot apply the last operation directly since there are no conditions on finiteness of expectations (in particular, for stochastic integrals). Therefore we introduce a sequence of stopping times (τ n ) n>0 , which helps to control the jumps compensator and provides square integrability of stopped stochastic integral,

τ n := inf    u ≥ t : u t b s z 2 x (s, X s ) + |z(s, X s , υ s )| ds ≥ n    ∧ T , (7.3)
where b s = b (s, X s , υ s ) σ s σ s + ς s ς s b(s, X s , υ s ) and z is given as in (3.5) for g ≡ z.

Note, that E t,x M τ n = 0 and by the properties of the random measures (see, for example, Theorem 1 on the page 153 in [START_REF] Liptser | Statistics of random process: I. General theory[END_REF]) we get

E t,x t≤s≤τ n | (∆z(s, X s ) -z x (s, X s-) ∆X s ) | = E t,x τ n t |z(s, X s , υ s )|ds ≤ n .
Therefore, setting

M * n = M τ n + t≤s≤τ n (∆z(s, X s ) -z x (s, X s-) ∆X s ) - τ n t z(s, X s , υ s ) ds ,
we get E t,x M * n = 0. Furthermore, using the definition (3.5), we can see from (7.2) that

τ n t U 1 (s, X s , υ s )ds + z n = z(t, x) + τ n t (z t (s, X s ) + H 0 (s, X s , z, υ s )) ds + M * n ,
where z n = z(τ n , X τ n ). Note here, that Condition H 1 ) and the definition (3.6) yield

0 = z t (s, X s ) + H(s, X s , z(s, X s )) ≥ z t (s, X s ) + H 0 (s, X s , z(s, X s ), υ s ) ,
i.e.

τ n t U 1 (s, X s , υ s ) ds + z n ≤ z(t, x) + M * n . Therefore, for any n ≥ 1 E t,x   τ n t U 1 (s, X s , υ s ) ds + z n   ≤ z(t, x) . (7.4)
Note here, that according to Definition 3.1 for any υ ∈ V the wealth process X t and its left limit X t-belong to X . Therefore, inf 0≤t≤T X t ∈ X and sup 0≤t≤T X t ∈ X and, in view of Lemma A.1 and the condition (3.2) the integral T 0 |z(s, X s , υ s )|ds < ∞ a.s. This implies immediately that a.s. τ n → T ,

τ n t U 1 (s, X s , υ s ) ds → T t U 1 (s, X s , υ s ) ds and z n → z(T, X T ) = U 2 (X T )
as n → ∞. Moreover, note also, that U 1 ≥ 0 and by condition H 1 ) the function z is uniformly bounded from below. Therefore, the use of the Fatou lemma for the limit transition in the expectation in (7.4) yields, that for any υ ∈ V

E t,x   T t U 1 (s, X s , υ s ) ds + U 2 (X T )  = J(t, x, υ) ≤ z(t, x) , (7.5)
i.e. sup υ∈V J(t, x, υ) =: J * (t, x) ≤ z(t, x). Moreover, it should be noted, that in view of Conditions H 2 )-H 3 ), there exists the measurable function

v 0 : [0, T ] × X → K , such that H(s, x, z) = H 0 (s, x, z, v 0 (s, x))
and the control process v 0 = (v 0 (s, X * s )) 0≤s≤T is admissible, i.e. υ * ∈ V and the wealth process is defined by the equation (3.9). Now, similarly to (7.4) we can obtain that

E t,x    τ * n t U 1 (s, X s , υ * s ) ds + z * n    = z(t, x) ,
where τ * n is the stopping time defined in (7.3) for the strategy υ * and z

* n = z(τ * n , X * τ * n ).
Now, Condition H 4 ) implies that the random variable (z * n ) n≥1 are uniformly integrable, and therefore, taking into account that a.s.

z * n → z(T, X * T ) = U 2 (X * T )
as n → ∞ we can obtain through the monotone convergence theorem that

z(t, x) = lim n→∞ E t,x    τ * n t U 1 (s, X s , υ * s ) ds + z * n    = E t,x   T t U 1 (s, X s , υ * s ) ds + U 2 (X T )   = J(t, x, υ * ) .
Therefore, from (7.5) we can conclude that J(t, x, υ * ) = J * (t, x).

Proof of Theorem 4.1

Proof. First note that if use the form (3.14) in (3.12) we obtain that the coefficient A satisfies the following ordinary differential equation

Ȧ(t) + γ r t A(t) + A(t) F * (t) + (1 -γ) A γ γ-1 (t) = 0 , A(T ) = 1 ,
where the function F * (t) = max θ∈Θ F (t, θ) and F (t, θ) is given in (3.15). One can find directly, that

A(t) =   T t Ψ t,u du + Ψ t,T   1-γ , (7.6)
where Ψ t,u is given in (3.16). Therefore, the Condition H 1 holds true. Let's check Conditions H 2 ) -H 4 ). As to Condition H 2 ) note that the optimal control function v 0 = (θ 0 , c 0 ) is given in (3.13). To check H 3 ) note that the inequality (5.12) implies that X * t-> 0 a.s. for all 0 ≤ t ≤ T i.e. the strategy υ * = (θ * t , c * t ) 0≤t≤T belongs to V. Finally, the form (3.14) and the upper bound (5.11) provide Condition H 4 ) and, therefore, Theorem 3.3 implies this theorem. Then, in view of the convergence (7.7) for δ = 1, to show (7.9) it is sufficient to prove that

P -lim n→∞ D n = 0 , (7.11)
where the total amount of transactions costs D n is defined in (4.5). To do this, using the definitions (4.2) and (4.4) and taking into account that 0 ≤ θ * j (t) ≤ 1, we obtain that for

k ≥ 2 α (n) j (t k ) -α (n) j (t k-1 ) ≤ (θ * j (t k-1 ) -θ * j (t k-2 )) X * t k-1 S j (t k-1 ) + X * t k-1 S j (t k-1 ) - X * t k-2 S j (t k-2 )
.

(7.12)

Now, for N > x + max 1≤j≤m S j (0) := N 0 we introduce the set

G N = sup 0≤t≤T X * t ≤ N S * ≥ N -1 S * ≤ N , (7.13)
where S * = min Using the representation (5.13) and taking into account, that the functions µ j are bounded on the interval [0, T ], Propositions 5.1 and 5.2 imply that 0 < S * ≤ S * < ∞ a.s. Note also, that the property (5.11) implies sup 0≤t≤T X * t < ∞ a.s. Therefore, lim N →∞ P(G c N ) = 0.

Moreover, note also, that on this set through the Hölder condition (4.9) with some constant C > 0 we can rewrite the inequality (7.12) as

α (n) j (t k ) -α (n) j (t k-1 ) ≤ X * t k-1 θ * j (t k-1 ) -θ * j (t k-2 ) S j (t k-1 ) + X * t k-1 -X * t k-2 S j (t k-1 ) + X * t k-2 S j (t k-2 ) -S j (t k-1 ) S j (t k-1 ) S j (t k-2 ) ≤ N 3 θ * j (t k-1 ) -θ * j (t k-2 ) + X * t k-1 -X * t k-2 + S j (t k-2 ) -S j (t k-1 ) .
Therefore, on the set G N , using the condition (4.8), we get, that for some C > 0

D n ≤ C N 4 mκ n (1 + T n ) + N 4 κ n n k=2 X * t k-1 -X * t k-2 + N 4 κ n m j=1 n k=2 S j (t k-1 ) -S j (t k-2 ) , (7.14)
where T n = n k=1 t k -t k-1 . Now note here, that using the boundedness of the functions r t and µ t we can obtain that on the set G N for some C > 0 and for any 0 < u < t ≤ T S j (t) -S j (u)

2 ≤ C N (t -u) 2 + 2 t u Šj (v-)d L j (v) 2 and X * t -X * u 2 ≤ C N (t -u) 2 + 2 t u X * v-(θ * v ) d L v 2 ,
where Št = min(S t , N ) and X * t = min(X * t , N ). Moreover, the Cauchy -Schwarz -Bunyakovsky inequality yields

t u Šj (v-)d L j (v) 2 ≤ 2m m l=1 t u Šj (v)σ jl (v)dW l (v) 2 + 2m m l=1 t u Šj (v-)ς jl (v)dL l (v) 2 and t u X * v-(θ * v ) d L v 2 ≤ 2m m j,l=1 t u X * v θ * j (v)σ jl (v)dW l (v) 2 + 2m m j,l=1 t u X * v-θ * j (v)ς jl (v)dL l (v) 2 .
Using here the boundedness of the functions σ ij and ς ij and the inequality (A.2) for p = 2, we obtain that

E t u Šj (v-)d L j (v) 2 ≤ CN 2 (t -u)
and

E t u X * v-(θ * v ) d L v 2 ≤ CN 2 (t -u) .
Therefore, setting

η jn = n k=1 | t k t k-1 Šj (v-)d L j (v)| + | t u X * v-θ * j (v)ς jl (v)dL l (v)| T n ,
we obtain that for some constant C > 0 the maximum η * n = max 1≤j≤m η jn can be estimated as

sup k≥1 E η * n ≤ m j=1 E η jn ≤ CN .
Therefore, from (7.14) it follows that on the set G N for some C N > 0

D n ≤ C N κ n (1 + η * n )(1 + T n ) . (7.15)
From here we obtain that for any ε > 0,

P (D n ≥ ε) ≤ P C N κ n (1 + η * n )(1 + T n ) ≥ ε + P(G c N ) ≤ C N κ n (1 + T n ) ε + P(G c N ) . (7.16)
Taking into account that t j = jT /n and, that κ n = o(n -1/2 ), we obtain, that

κ n T n = √ T κ n √ n → 0 as n → ∞ ,
i.e. we get that for any N > 1 lim sup

n→∞ P (D n ≥ ε) ≤ P(G c N ) .
Since P(G c N ) → 0 as N → ∞, we get the limit property (7.11). Now, we show the uniform integrability of the sequence (X (7.17)

Since the term

T 0 ζ (n) u du + D n ≥ 0, we obtain that E x (X (n) T ) + ≤ x + E n k=1 β * t k-1 (B t k -B t k-1 ) + m j=1 E n k=1 α * j (t k-1 )(S j (t k ) -S j (t k-1 )) .
Proposition 5.4 implies upper bound (7.17). Therefore,

lim n→∞ E x (X (n) T ) γ + = E x (X * T ) γ + = E x (X * T ) γ
and we obtain Theorem 4.3.

Proof of Theorem 4.4

First of all, note that the time moments (4.11) are such that 0 < t 1 < . . . < t n = T and in view of the condition (4.12)

t k -t k-1 = qT u k u k-1 x q-1 dx ≤ qT n → 0 as n → ∞ .
Therefore, the convergences (7.7), (7.8) and (7.10) hold true and, moreover, Proposition 5.4 implies upper bound (7.17). Therefore, to end this proof it suffices to show the convergence (7.9) for which one has to establish the limit property (7.11). To do this, note that in this case

t k -t k-1 = qT u k u k-1 x q-1 dx ≤ qT u q-1 k (u k -u k-1 ) = qT u q-1 k n .
Therefore, the term T n in (7.14) can be estimated as

κ n T n = κ n n k=1 t k -t k-1 ≤ qT κ n n-1 k=1 u (q-1)/2 k √ n + √ T κ n .
Moreover, the first term

qT κ n n-1 k=1 u (q-1)/2 k √ n = qT nκ n n-1 k=1 u k+1 u k u (q-1)/2 k dx ≤ qT nκ n n-1 k=1 u k+1 u k x (q-1)/2 dx ≤ qT nκ n 1 u 1 x (q-1)/2 dx ≤ 2 √ T κ n √ n √ q q + 1 .
Therefore, in view of the last condition in (4.12) κ n T n → 0 as n → ∞. Using this in the bound (7.16) we obtain (7.11). Hence Theorem 4.4.

A Appendix

A.1 Properties of the mapping Υ .

Lemma A.1. Let X ⊂ R be an open convex set and g is some fixed arbitrary function from

C 1,2 ([0, T ] × X , R), for which sup 0≤t≤T sup x∈X |g(t, x)|/(1 + |x|) < ∞.
Then for any

x * ≤ x * from X there exists a constant C > 0 such that for all ∈ R max 1≤i≤m sup 0≤t≤T sup x * ≤x≤x * R * |Υ g (t, x, y)| Π i (dy) ≤ C(1 + 2 ) . (A.1)
Proof. First note that, for = 0 the function Υ g (t, x, 0) ≡ 0 and, therefore, (A.1) holds. Let = 0. In this case first note, that there exists some > 0 such that the interval

]x * -2 , x * + 2 [⊂ X . Therefore, x + v ∈ X for any |v| < ε and x ∈ [x * , x * ]. We set 1 = /| |. It is clear, that x + y ∈ X for |y| < 1 and |Υ (t, x, y)| = x+ y x   z x g xx (t, u) du   dz ≤ 2 y 2 2 max x-2ε≤u≤x+2ε 0 |g xx (t, u)| .
Therefore, since Π i (y 2 ) < ∞ we get for some C > 0

1 -1 |Υ (t, x, y)| Π i (dy) ≤ C 2 Π i (y 2 ) .
Moreover, using the condition of this Lemma we can deduce that for some C > 0

{|y|≥ 1 } |Υ (t, x, y)| Π i (dy) ≤ C {|y|≥ 1 } (1 + | y|) Π i (dy) + |g(t, x)| Π i ({|y| ≥ 1 }) + |g x (t, x)| | | {|y|≥ 1 } |y| Π i (dy) ≤ C(1 + | |)(1 + |g(t, x)| + |g x (t, x)|)    {|y|≥ 1 } (1 + |y|) Π i (dy)    .
Therefore, taking into account that

max 0≤t≤T max x-2ε≤u≤x+2ε 0 (|g(t, u)| + (|g x (t, u)| + |g xx (t, u)|) < ∞ ,
we obtain the upper bound (A.1). Hence Lemma A.1.

A.2 Technical tools

First, we recall the well-known Novikov inequality for purely discontinuous martingales. To end this we denote by P the predictable field and by B(R * ) the Borel field in R * .

Lemma A.2. ( [START_REF] Novikov | On discontinuous martingales[END_REF]) Let J be P ⊗ B(R * ) a measurable function for which |J| 2 * ν t < ∞ a.s. Then for any p ≥ 2 there exists some positive constant Čp > 0 such that for any t > 0

E sup u≤t |J * (ν -ν) u | p ≤ Čp E |J| 2 * ν t p/2 + E |J| p * ν t , (A.2)
where ν is a random jump measure with its compensator ν.

Lemma A.

F (t, θ) = f (t, θ) + I(ς t θ) , f (t, θ) = a 1 (t)θ + a 2 (t)θ 2 , (A.4)
where

a 1 (t) = γ(µ t -r t ), a 2 (t) = γ (γ -1)σ 2 t /2 ≤ 0 and I(x) = ∞ -1 ((1 + x y) γ -1 -γ x y) Π(dy) .
Note, that generally this function is defined only for 0 ≤ x ≤ 1. First we assume that -1/2 -1 1 (1 + y) 2-γ Π(dy) < ∞ .

(A.5)

Then, taking into account here, that Π(y 2 ) < ∞, we can show through the dominated convergence theorem, that the function I is two times continuously differentiable and for where İ and Ï denote the derivatives of the first and second order respectively. Now, we extend the function I on R, as follows Therefore, this implies the condition A 2 ). Consider now the case, when the condition (A.5) does not hold, i.e. Similarly, we get, that I δ is two times continuously differentiable and for 0 ≤ x ≤ 1 İδ (x) = γ Let now θ * δ (t) be the argmax of F δ (t, θ), i.e. max 0≤θ≤1 F δ (t, θ) = F δ (t, θ * δ (t)). Taking into account here, that the function F δ goes to F as δ → 0 uniformly over [0, T ] × [0, 1], we get that θ * δ (t) → θ * t for any t ∈ [0, T ]. Moreover, we introduce the function I 1,δ and F 1,δ by replacing in (A.6) and (A.7) the function I with I δ . Therefore, similarly to the precedent case, the function θ * δ (t) = V (θ 0,δ (t)), where the function θ 0,δ (t) is the root of the partial derivative of F 1,δ , i.e.

0 ≤ x ≤ 1 İ(x) = γ
I 1 (x) =        Ï(0)x 2 /
U δ (t, θ 0,δ (t)) = 0 and U δ (t, θ) = a 1 (t) + a 2 (t) θ + ς t İ1,δ (θ ς t ) . (A.10) Note now, that from (A.9) for θ 0,δ (t)ς t > 1 we get, that U δ (t, θ 0,δ (t)) = a 1 (t) + a 2 (t) θ 0,δ (t) + ς t İδ (1) + Ïδ (1)(ς 2 t θ 0,δ (t) -ς t ) = 0 , i.e. t > 0 and | Ïδ (1)| → +∞ as δ → 0, we get from (A.10), that lim sup δ→0 max 0≤t≤T |θ 0,δ (t)| 1 {θ 0,δ (t)ς t >1} < ∞. Moreover, again from (A.9) for θ 0,δ (t)ς t < 0 we get, that U δ (t, θ 0,δ (t)) = a 1 (t) + a 2 (t) θ 0,δ (t) + Ïδ (0)ς 2 t θ 0,δ (t) = 0 , i.e θ 0,δ (t) = a 1 (t) Therefore, for any 0 ≤ s, t ≤ T we deduce that

|θ * t -θ * s | |t -s| = lim δ→0 |θ * δ (t) -θ * δ (s)| |t -s| ≤ M * .
This implies the condition A 2 ). Hence Proposition 4.5.

H 1 )

 1 There exists solution z ∈ C 1,2 ([0, T ] × X , R) of the equation (3.7) such that,

Proposition 5 . 4 .t

 54 If the conditions (2.2), (2.4) and (2.6) hold true, then max dB t < ∞ .

√nFig. 1 Fig. 2

 12 Fig. 1 Results of simulation of the strategies (4.2) -(4.4)

7. 3 9 )

 39 Proof of Theorem 4.3 Proof. First note, that the process (ζ * t ) 0≤t≤T defined in (4.2) is càdlàg on the interval [0, T ] and through the definition (4.4) one can deduce that for any δ > the property (5.11) and the dominated convergence theorem we get lim In view of the definition in (4.4) and taking into account that the processes (α * t ) 0≤t≤T and (β * t ) 0≤t≤T have càdlàg trajectories on the interval [0, T ] we get that in probability lim

1≤j≤m inf 0≤t≤T S

 0≤t≤T j (t) and S * = max 1≤j≤m sup 0≤t≤T S j (t) .

.

  To do this in view of the Vallee-Poussin criteria it suffices to check thatsup n≥1 Ex (X (n) T ) + < ∞ .

∞ - 1 y( 1 + 1 γ

 111 xy) 1-γ -y Π(dy) and Ï(x) = ∞ -(γ -1)y 2 (1 + xy) 2-γ Π(dy) ,

  sup x,y∈R |V (x) -V (y)| |x -y| ≤ 1 .

  y) 2-γ Π(dy) = +∞ . Let now 0 < δ < 1/2. Then we set

F

  δ (t, θ) = f (t, θ) + I δ (ς t θ) and I δ (x) = ∞ -1+δ((1 + x y) γ -1 -γ x y) Π(dy) .

( 1 +

 1 xy) 1-γ -y Π(dy) and Ïδ (x) = ∞ -1+δ γ(γ -1)y 2 (1 + xy) 2-γ Π(dy) .

(A. 9 )( 1 +

 91 Note here, that for any 0 ≤ x ≤ 1 and for some positive constantC | İδ (x)| ≤ xy) 1-γ Π(dy) + C ≤ ι δ (x) + C , xy) -2+γ Π(dy) ≥ Π(-1 + δ, -1/2) → Π(-1, -1/2) > 0 as δ → 0. Note also, that | Ïδ (x)| > γ(1 -γ)ι δ (x)/4. Therefore, we get, that for any x * ≥

  Notations g t , g x , g xx mean corresponding derivatives of function g(t, x) . It should be emphasized that one needs to introduce the indicator 1 {x+v∈X } in the term Υ (t, x, v) since there are no assumptions on measures Π i (•) or function g to keep the sum x + v in the set X , for v from R * . As it is shown in appendix A.1 the function g is bounded and its Lebesgue integral will be used as jumps compensator for the process g(t, X t ).

	Remark 3.2. Now, for any x ∈ X and 0 ≤ t ≤ T , we set the Hamilton function as
	H(t, x, g) := sup

v∈K

H 0 (t, x, g, v) .

  Theorem 4.3. Assume that Conditions A 1 ) -A 2 ) hold true and the transaction coefficient is a function of n, i.e. κ = κ n such that κ n = o n -1/2 , as n → ∞. Then the strategy (4.4) -(4.6) with the revision moments (t j = jT /n) 1≤j≤n is asymptotically optimal, i.e.

	sup 0≤s,t≤T	|θ * t -θ * s | |t -s|	< ∞ .	(4.9)

.8)

Note that A 2 ) holds true if the optimal strategy (3.13) is 1/2-Holder function, i.e.

  First note, that the process V t = t 0 θ s d L s is square integrated martingale. Therefore, taking into account that dE t

Theorem 4.4. Assume that the conditions A 1 ) -A 2 ) hold true and the transaction coefficient is a function of n, i.e. κ = κ n such that κ n = o(1) as n → ∞. 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 Condition A 2 ). Proposition 4.5. Assume that, in the model (2.1) m = 1, the functions r t , µ t , σ t and ς t are continuously differentiable and ς t > 0 for all 0 ≤ t ≤ T . Then Conditions A 1 ) -A 2 ) hold true.

Remark 4.6. It should be noted that Condition A 2 ) hold for any m ≥ 1 for the homogeneous market (2.1), i.e. for the constant parameters r t , µ t , σ t , ς t .

5 Properties of the optimal strategies (3.16) -

(3.17)

In this section we study the Doéan exponentials E t (V ) defined in (2.11). Proposition 5.1. For any non random measurable [0, T ] → Θ function θ = (θ t ) 0≤t≤T the Doléan exponential (2.11) is square integrated martingale.

Proof.

  3 (Martingales inequalities). (See subsections 3.2 in [18]) If the process (X t ) t≥0 is a non negative sub martingale with EX p t < ∞ for some p > 1, then Since the functions r t , µ t , σ t and ς t in the model (2.1) are continuously differentiable, they are bounded. Hence A 1 ). For m = 1 , the function (3.15) has the following form

	P X * t > λ ≤	EX p t λ p	and E ( sup 0≤u≤t	X u ) p ≤	p p -1	p	E X p t .	(A.3)
	A.3 Proof of Proposition 4.5

  2 for x ≤ 0 ; (1) + İ(1)(x -1) + Ï(1)(x -1) 2 /2 for x ≥ 1 .It is clear, that this function is two times continuously differentiable and Ï1 (x) < 0 for any x ∈ R. Note now, thatmax 0≤θ≤1 F (t, θ) = max 0≤θ≤1 F 1 (t, θ) and F 1 (t, θ) = f (t, θ) + I 1 (ς t θ) . = a 1 (t) + 2a 2 (t)θ + ς t İ1 (θ ς t )and, moreover, since ς t > 0, then= 2a 2 (t) + ς 2 t Ï1 (θ ς t ) < 0 .Additionally, it is easy to check that U (t, θ) → +∞ as θ → -∞ and U (t, θ) → -∞ as θ → +∞, therefore, for all 0 ≤ t ≤ T there exist unique root θ 0 (t) of the equation U (t, θ 0 (t)) = 0 and by the implicit function theorem we calculate its derivativeθ0 (t) = ȧ1 (t) + 2 ȧ2 (t)θ 0 (t) + ςt İ1 (θ 0 (t)ς t ) + θ 0 (t)ς t ςt Ï1 (θ 0 (t)ς t ) 2|a 2 (t)| + ς 2 t | Ï1 (θ 0 (t)ς t )|This derivative is continuous on the interval [0, T ], i.e. bounded. The solution of the problem (A.7) can be represented as θ * t = V (θ 0 (t)) and V (x) = min(0, max(x, 1)) is Lipschitz function, i.e.

				(A.7)
	We can calculate directly
	U (t, θ) =	∂F 1 (t, θ) ∂ θ
	∂U (t, θ) ∂ θ	=	∂ 2 F 1 (t, θ) ∂ θ 2
			.	(A.8)
				(A.6)

I(x)

for 0 ≤ x ≤ 1 ; I

  θ 0,δ (t) = a 1 (t) + ς t İδ (1) -Ïδ (1)ς t |a 2 (t)| + ς 2 t | Ïδ (1)| Therefore, taking into account that min 0≤t≤T ς 2

  |a 2 (t)| + ς 2 t | Ïδ (0)| Finally, we get, that lim sup δ→0 max 0≤t≤T |θ 0,δ (t)| < ∞. Using this bound together with (A.10) in (A.8) we obtain the same property for its derivative, i.e. This implies, that θ * δ (t) satisfies the Lipschitz condition uniformly over δ, i.e.

	lim sup	max	| θ0,δ (t)| < ∞ .
	δ→0		0≤t≤T
	M * = lim sup δ→0	sup 0≤s,t≤T	|θ * δ (t) -θ * δ (s)| |t -s|	< ∞ .

.

Taking into account, that | Ïδ (0)| → γ(1 -γ)Π(y 2 ) > 0 as δ → 0, we get, that lim sup δ→0 max 0≤t≤T |θ 0,δ (t)| 1 {θ 0,δ (t)ς t <0} < ∞ .

Acknowledgments

The last author was partially supported by the Russian Federal Professor program (project no. 1.472.2016/1.4, the Ministry of Education and Science of the Russian Federation).

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


F Metze
email: metze@ira.uka.de

P Gieselmann

H Holzapfel

T Kluge

I Rogina

A Waibel

M Wölfel

J Crowley

P Reignier

D Vaufreydaz

F Bérard

B Cohen

J Coutaz

S Rouillard

V Arranz

M Bertrán

H Rodriguez

The "FAME" Interactive Space

This paper describes the "FAME" multi-modal demonstrator, which integrates multiple communication modes -vision, speech and object manipulation -by combining the physical and virtual worlds to provide support for multi-cultural or multi-lingual communication and problem solving. The major challenges are automatic perception of human actions and understanding of dialogs between people from different cultural or linguistic backgrounds.

The system acts as an information butler, which demonstrates context awareness using computer vision, speech and dialog modeling. The integrated computerenhanced human-to-human communication has been publicly demonstrated at the FORUM2004 in Barcelona and at IST2004 in The Hague. Specifically, the "Interactive Space" described features an "Augmented Table "  for multi-cultural interaction, which allows several users at the same time to perform multi-modal, cross-lingual document retrieval of audio-visual documents previously recorded by an "Intelligent Cameraman" during a week-long seminar.

Introduction

Current advances in language and vision technology as well as user interface design are making possible new tools for human-human communication. Integration of speech, vision, dialog, and object manipulation offers the possibility of a new class of tools to aid communication between people from different cultures using different languages.

The "FAME" project (EU FP5 IST-2000-28323) develops a new vision for computer interfaces, which replaces and extends conventional human-computer interaction by computer-enhanced human-to-human (CEHH) interaction. The crucial difference lies in the role that the computer plays and the demands it makes on the human user's attention in a living and working environment.

Like an invisible information butler, such systems observe and learn their users' ways and preferences, and "understand" enough about the context and purpose of their activity, to be able to provide helpful and supportive services that are informed by, and appropriate for that context. A broad range of applications can profit from CEHH interfaces, including office work, communication, entertainment and many more. What is common in most of these settings is that the goal and preoccupation of visitors is to interact with other humans and not with machines.

The work presented here demonstrates an interactive space using intelligent agents to facilitate communication among researchers from different linguistic backgrounds, who discuss several scientific topics, which have been presented at a seminar series. The agents provide three services:

1. Provide information relevant to context, i.e. make users aware of existing information in an audio-visual database, then retrieve and present it appropriately 2. Facilitate human to human communication through multi-modal interaction as well as presentation of speech and text in multiple languages 3. Make possible the production and manipulation of information, blending both electronic and physical representations

The agents therefore do not explicitely intervene, but remain in the background to provide appropriate support (this mode is known as implicit interaction), unless explicitely called for to provide a particular service, such as playing a video.

The remainder of Section 1 will formulate the concepts behind "FAME" and introduce the functions our "intelligent room" [START_REF] Gieselmann | Towards multimodal interaction within an intelligent room[END_REF] can perform. Section 2 will present the individual components and evaluate their performance individually, while Section 3 will present an overall evaluation of the integrated system from the user's point of view.

The Functions of the "FAME" Demonstrator

The functions of the "FAME -Facilitating Agents for Multi-Cultural Exchange" multimodal demonstrator are split into an off-line and on-line part, as shown in Fig. 1: Off-line Part. To provide audio-visual documents, an "Intelligent Cameraman" [START_REF] Crowley | Dynamic composition of process federations for context aware perception of human activity[END_REF] recorded a four day seminar on Language Technology and Language, Cognition, and Evolution [START_REF]Tecnologies de la llengua: darrers avenc ¸os" and "Llenguatge, cognició i evolució[END_REF], which were held on the premises of the FORUM2004 [START_REF]Universal Forum of Cultures[END_REF] in Barcelona. The cameraman runs fully automatically and does not interfere with the lecture or its visitors.

The resulting videos, which contain audio from the presenter, the translator, and the audience, are then segmented, automatically transcribed and processed by a topic detection system, which assigns one topic to every segment. The videos can then be retrieved during the on-line use of the system in different ways.

On-line Part. The goal of the "Augmented Table" is to aid and support multi-cultural interaction and multi-modal document retrieval [START_REF] Lachenal | A reference framework for multi-surface interaction[END_REF]. Students and researchers can come to the table and discuss with each other, retrieve information about the seminar as well as see recordings or automatically generated transcriptions of the lectures themselves, or see or give "testimonies". The Database. The database is the interface between the off-line and on-line parts. In addition to the videos, their transcription and topic segmentation, it contains speakers' CVs, a picture, and contact information. For every lecture, we also added information on time and place as well as course material (slides) and other meta information.

The "Interactive Space"

At a conference, attendees usually use the physical conference program organized as a time-schedule and they ask other attendees what's interesting to attend. Therefore, the design of our showcase must maintain these two forms of interaction (browsing a time-schedule and chatting with colleagues for reporting their experience and asking for advice) in a seamless and complementary way.

Preliminary Analysis. In a user-centered approach, domain concepts that users manipulate are classified to provide guidelines for the design of the user interface. In particular, 1 st -class concepts should be observable at all time, and 2 nd -class concepts should be browsable at all time, i.e. they are not necessarily observable but accessible through additional user's actions such as scrolling, selecting a navigation button such as "next-previous-top", "show details". After analysis of 15 CHI (Computer Human Interaction) conference programs, our demonstrator ranks 1 st -class domain concepts as follows: Lecture (in terms of: Title, Place, Date, BeginHour, EndHour), Speakers and Topics; while we use the 2 nd -class domain concepts Documents (papers, slides, video, audio) and Testimonies (recorded audio-visual "opinions" about lectures).

Design Solution. Given the current state-of-the-art in automatic speech recognition without close-talking microphones [START_REF] Metze | Automatically Transcribing Meetings Using Distant Microphones[END_REF], our design is based on the hypothesis that there should be two types of users: one manager and up to five users: Users are researchers knowledgeable in the task domain, i.e. they are skilled at manipulating the notions of session, lecture, topics; they are familiar with the lectures' topic, but not with augmented reality. To ensure a reliable functioning of the system, the users do not issue spoken commands to the system, but the topic spotter component of the system recognizes their speech and acts upon it. Visitors to the demonstration can be "users". A manager is skilled both in the task domain and in the interaction techniques. His spoken requests are recognized by the system, so there is no need for extra explicit interaction with the system. He can conduct a rich dialog with the system using speech and direct manipulation, he acts as a "moderator".

The manager and several users can interact with the system at the same time, i.e. they can speak between themselves, move tokens or select items etc. As shown in Fig. 2, the overall setting includes:

-A shared horizontal surface (the "Augmented Table ") for exploring information about the conference and recording testimonies -A microphone in the middle for acquiring user's utterances and detecting information of interests (i.e. topics and/ or speakers' names) as well as a loud-speaker, so that the system can talk back to users -A shared vertical surface (the "Wall") for peripheral activities, such as a presentation of the detected topics, lecture slides, or other text information from the database -A second vertical surface for video projections from the database -A camera to record testimonies Compared to the "un-augmented" setting (i.e. browsing a paper brochure or asking colleagues in the hallway), the added value of FAME comes from (see Fig. 2):

-The interaction itself (manipulating tokens and chatting at the same time), -The nature of the information that can be retrieved (lecture videos, presentation slides, testimonies, biographical or contact information, etc.), -The capacity to produce feedback (e.g. reporting a testimony, which is added to the database), which improves the feeling of engagement in the event.

Components of the FAME Demonstrator

The components of the FAME demonstrator were developed in parallel by the FAME partners, each using their own software and tools. Integration of the individual components into one system was achieved during two integration workshops held at different sites using different hardware to provide backup and to ensure that the demonstrator would not depend on specifics of either location, so as to avoid problems during several days of public exhibits at the FORUM2004 in Barcelona and IST2004 in The Hague. Communication between components is assured through Open Agent Architecture (OAA) [START_REF][END_REF], which allows agents running on several machines with different operating systems to communicate and also access the database. In total, the on-line part of the FAME demonstrator requires six PCs in a mixed configuration of Linux and Windows, which supported three speech recognizers, speech synthesis, video processing for the "Augmented Table ", projection of the table itself and the walls, video player and recorder, room control, and database/ communication server. Fig. 2. The overall setting of the "interactive space": table for focused search activity and walls for peripheral activities and information. The Feedback loop made possible by the FAME room improves engagement within communities

Cameraman and Speech Activity Detection

The automatic cameraman is a context aware system designed to record lectures or meetings. It creates movies by dynamically selecting camera views based on context analysis using a speech activity detection system running on several microphones.

The Context Aware Framework. This is based on the concept of entities, roles, relations, situations and a graph of situations [START_REF] Rey | Perceptual components for context aware computing[END_REF]. An entity is a group of observable properties detected by perceptual components. For instance, a person (an entity with (x, y) properties) can be detected by a video tracker. A role is an acceptance test which selects entities based on their properties. For instance, a long and thin object can play the role of a pointer. A relation is a predicate over entities: person 1 is near person 2. A situation is defined as a set of roles and relations between entities.

Context is described as a graph. Nodes are situations, arcs are temporal constraints between the situations. They are decorated using Allen's temporal operators [START_REF] Allen | Towards a general theory of action and time[END_REF]. For each Allen temporal operator we composed a corresponding Petri Net pattern. By applying those patterns, the context graph can be transformed in an equivalent synchronized Petri Net. The Petri Net is then compiled in a set of forward production rules (Jess) that will check the situation activation sequence.

The Cameraman. The room is equipped with four cameras: a wide angle for a global view of the scene, one for the audience, one for the lecturer and one filming the slides. The lecturer has a lapel microphone and there are ambient microphones for the audience questions. The perceptual components are a video tracker [START_REF] Caporossi | Robust visual tracking from dynamic control of processing[END_REF] for the lecturer, a "new slide" detector based on image difference, and a speech activity detector. We model for situations, each with an associated camera: lecturer speaks, audience asks a question, new slide, and unspecified.

Speech Activity Detection (SAD).

The SAD system is composed of several subsystems: an energy detector, a basic classifier and a neural net trained to recognize voiced segments, e.g. vowels. The energy detector uses pseudo energy (sum of absolute sample values to determine if a signal is speech or not. The basic classifier works in frequency bands and tags specific sound classes: fricatives, low frequency sounds like computer or air conditioning fans, and other sounds. The neural net is a multi-layer perceptron with 2 hidden layers. Input consists of band crossing, energy and 16 LPC coefficients. This module is the only sub-system that needs to be trained. Training was done on 1 hour of speech extracted from the BREF [START_REF] Lamel | BREF, a large vocabulary spoken corpus for French[END_REF] corpus. A rule based automaton determines the final result using the three sub-system's decision as input.

There are two evaluation conditions, one dealing with close-talking data from the lecturer's lapel microphone, the other dealing with far-field microphone audio. Evaluation results in terms of Mismatch Rate (MR), Speech Detection Error Rate (SDER), and Non-speech Detection Error Rate (NDER) are as follows [START_REF] Surcin | Evaluation packages for the first chil evaluation campaign[END_REF]: MR SDER NDER Close-Talking 10.8% 5.8% 27.9% Far-Field 13.4% 12.9% 15.4%

These results and field experiments suggest that the SAD system accuracy is satisfying for the purpose of the context analysis.

Information Retrieval and Topic Detection

Information Retrieval (IR). Conventional IR allows a user to type a query, the system performs a search on a previously indexed single collection of information sources and then provides the results to the user. FAME needs to go beyond this scheme, because it not only deals with "conventional" IR, but also with on-line access to relevant documents from a set of different collections triggered by explicit or implicit interaction [START_REF] Bertran | FameIr, multimedia information retrieval shell[END_REF].

Indexing is done using textual features (words, stems, lemmas, multiword terms, phrases, etc.). These can also include morphological, syntactic or semantic information. Textual features can be automatically extracted from multimedia documents or manually attached to them.

Querying can be done using text or speech. The system allows cross-lingual and conceptual-based query expansion using EuroWordNet [START_REF]The Global WordNet Association: EuroWordNet[END_REF] as lexical resource. Crosslingual IR is useful, because a speaker's ability to understand a document in a foreign language is usually higher than his active command of that language.

Topic Detection (TD). Detection of topics consists of assigning topics to lecture segments, or detecting that no in-domain topic is currently being discussed.

In order to perform (on-line) TD, topics first need to be described during a preparation phase. We refer to this off-line and highly time-consuming task as Topic Characterization (TC). The most widely used form of performing TC consists in attaching a collection of units to each topic. The problems which need to be addressed here are selecting the appropriate type of unit, choosing the criteria for selecting the appropriate units to describe each topic, and choosing a representation schema for the collection of units describing the set of topics.

Topic Signatures (TS) are used to represent topics. TS are lists of weighted units or terms. Building a TS for a collection of topics was done semi-automatically. Different TS for a topic collection depend on each other.

With the set of "topics" pre-defined and each topic being described by its TS, the input stream is examined sequentially and its features are extracted and compared with the features (TS) of the different topics. If there is enough evidence, the presence of a new topic (a change of topic) is detected and communicated [START_REF] Arranz | Which is the current topic? what is relevant to it? a topic detection retrieval and presentation system[END_REF].

The Augmented Table and Token Tracker

The token tracker component of the "Augmented Table " analyzes in real time the video stream coming from the camera located above the table. The positions of all tokens (red disks) are extracted from the video stream and converted into a stream of APPEAR, MOTION and DISAPPEAR events with associated positions (x, y). The FAME token tracker will be evaluated in terms of its resolution and latency.

Resolution. The smallest motion that will be detected and reported by the tracker depends on the video stream, which is processed at 388 x 284 pixels in order to maintain full video frame rate and low latency. Setting a 2 pixel motion threshold for static stability before reporting a motion event because of the instability of the video signal, the actual resolution on the surface depends on the geometry of the setup. Typically, an accuracy of 0.5 cm or 5 pixels for the projection, is reached, which is sufficient for the task.

Latency. The time lag between the moment a user moves a token and the moment the associated feedback is moved accordingly has a strong effect on human performance with interactive systems. [START_REF] Ware | Reaching for objects in vr displays: Lag and frame rate[END_REF] recommends that devices should not have more than 50 ms latency. As shown in [START_REF] Watson | The effects of variation of system responsiveness on user performance in virtual environments[END_REF], it is not necessary for latency to be very stable: no effect on human performance is detected with standard deviation of latency at or below 82 ms.

Following the approach reported in [START_REF] Liang | On temporal-spatial realism in the virtual reality environment[END_REF], we designed a latency measuring device for the token tracker: on a 1.4 Ghz PowerPC G4 machine, the resulting latency value is distributed between 61 ms and 141 ms, with an average at 81 ms, which is tolerable given a standard variation on latency of 16 ms in our system, which means FAME is well within the 82 ms requirement expressed in [START_REF] Watson | The effects of variation of system responsiveness on user performance in virtual environments[END_REF]. We mainly attribute the latency variation to the difference in camera delay (25 Hz frame rate) and projector refresh rate (60 Hz).

Dialog Manager

The dialog manager is the central on-line component that mediates and executes messages of the various components, maintaining a shared multi-modal context with the "Augmented Table ". In this function, the dialog manager interprets speech input from the manager (explicit interaction), input from the topic spotter using distant speech recognition (implicit interaction) and events stemming from users' actions on the table (explicit interaction). The dialog manager will call requested services as dictated by the context, for example IR queries, highlighting information on the table, playing video segments, showing slides or pieces of information on the information wall, controlling testimony recording, controlling room devices (lamps) and generating spoken output.

If the dialog manager detects clarification needs, or information is missing to execute a command, it requests further information from the speaker or calls on other services, such as IR, to obtain the information required to execute the command.

For dialog management and multi-modal integration we use the TAPAS dialog management framework, which is based on ARIADNE dialog algorithms [START_REF] Denecke | Rapid prototyping for spoken dialogue systems[END_REF] with extensions to support multilingual input and output [START_REF] Holzapfel | Towards development of multilingual spoken dialogue systems[END_REF]. The dialog manager can restrict the search space of the close-talking speech recognizers to the given context. Performance numbers for these recognizers and the dialog manager are summarized here:

Speech Recognition

Apart from the close-talking recognizers for the managers, the augmented table uses two other speech recognition systems:

Lecture Transcription. To automatically segment and transcribe the lectures given at the seminar and make them accessible to indexing, transcriptions were generated by ITC-irst [START_REF] Cettolo | Advances in the automatic transcription of lectures[END_REF]; here we describe a smaller backup system developed by UKA, which was run during pauses between demonstrations. Development of lecture transcription components used the TED [START_REF] Lamel | The translanguage english database (ted)[END_REF] corpus, because of its similarity to the Barcelona seminar presentations. The acoustic model has been trained on 180 h Broadcast News (BN) and close-talking "Meeting" data [START_REF] Metze | Automatically Transcribing Meetings Using Distant Microphones[END_REF], summing up to a total of 300 h of training material. Models were first trained on Meeting and BN, then adapted to TED by MAP and MLLR.

To generate language models (LMs), we used corpora consisting of BN (160 M words), proceedings (17 M words) of conferences such as ICSLP, Eurospeech, ICASSP or ASRU and talks (60 k words) by the TED adaptation speakers. The baseline LM is based on BN. Our final LM was generated by interpolating a 3-gram LM based on BN and proceedings, a class based 5-gram LM based on BN and proceedings and a 3-gram LM based on the talks. The overall out-of-vocabulary rate is 0.3% on a 25 k vocabulary including multi-words and pronunciation variants.

The TED decoding and perplexity results using the acoustic and language models described above are shown below. The first run did not use TED model adaptation, while the last run used MLLR model adaptation on the hypothesis of the "TED-Adapted" run. Comparing the first runs between the native vs. the non-native test data sets we see a big gap in performance which is due to the fact that the acoustic model was trained on mostly native speech. The gain of acoustic adaptation was higher for the non-native test set as for the native test set, which may be explained by the fact that the amount of adaptation material was four times smaller for the native speakers than for the nonnative speakers. Word error rates of transcriptions of the Barcelona lectures are about the same as those of the TED development set.

Performance on

Topic Spotting using Room Microphones. A real-time speech recognition system derived from the ISL "Meeting" system [START_REF] Metze | Automatically Transcribing Meetings Using Distant Microphones[END_REF], running on a single microphone placed in the middle of the "Augmented Table" is used to detect topics from the conversation between visitors to FAME.

The initial system using a BN LM interpolated with data collected from the Internet had a topic error rate of 41% at a speech recognition accuracy of 17%. Re-weighting the keywords for topic spotting and taking confidence scores into account reduced the topic spotting error rate to 38%.

Component Integration

We have conducted an evaluation to measure response time over a distributed OAA system including different components. OAA offers two kinds of method calls: asynchronous calls (messages without response) and synchronous calls (response is returned to the caller). We have evaluated both message types with sequential calls and parallel calls for maximum throughput.

We found that the first call to OAA of an agent plus registration time was quite large with an average of 667 ms. All following calls are much faster. The average delay in our distributed system for a single synchronous method call (including request and reply) ranges between 20 ms and 25 ms. The numbers have been evaluated for an agent calling himself (routed over the OAA facilitator), chained calls (a call stack of depth 10), and a single agent calling another agent. The same setting has been used to create parallel tests, where 10 agents send a message concurrently. The average response time was 157 ms. The delay from sending the first request and receiving the last response was 207 ms. Divided by 10, this corresponds to an average processing time of 21 ms per message. We thus assume the total message throughput to be 47 messages (including responses) per second. The results of the sequential tests and sequential chained calls correspond to the final setting of the demonstrator and provide realistic numbers of the expected delay of messages. In the demonstration setup, the delays of the messages are small enough. However, the tested setup can not be applied to a larger system or one that needs a significantly higher message throughput.

The FAME Demonstrator and User Study

Over 150 persons used FAME during the Barcelona demonstration. We randomly selected 5 groups of 3 users among them for a user study [START_REF] Coutaz | Evaluation of the fame interaction techniques and lessons learned[END_REF]. After a brief introduction to the system, users were asked to perform predefined tasks, such as "Can you retrieve Mr. Harold Somer's lecture?" or "What topics are being addressed in the lecture about Machine Translation?", etc. We observed their behavior while interacting within the FAME interactive space and finally, the users answered some questions about their impression of the system.

We observed two general kinds of behavior: actors who wanted to experiment with the system, and spectators who enjoyed observing the scene. People felt confident with the system in less then one minute. They were more interested in playing with it than learning from the content: unfortunately, the social context did not favor a "learning" experience, although most visitors had a "science" background. We summarize the results in two categories: General Impression. The function and the manipulation of tokens are easy to learn and to understand. The existence of multiple tokens brings a playful note to the system. We observed users dropping multiple tokens randomly on the table to see what happens. The multiplicity of tokens supports simultaneous actions but does not necessarily favor collaboration. In turn, this lack of collaboration between users may lead to confusion, because, as a result of the quasi-simultaneity of different output modalities, users did not know whether the replies of the system corresponded to their own or someone else's request.

Most users would have preferred to be able to ask IR requests directly, instead of asking the manager to do it for them. Most users did not pay attention to the fountain of words (on the wall) that correspond to the topics recognized by the topic spotter as users were talking freely around the table. From the social perspective, the testimony function was a big success. At first, some users were shy at recording themselves. Others enjoyed it for the possibility to watch themselves through the replay function. User Ratings For Different Parts of the System. In general, users' opinions are quite enthusiastic, but overall, the system is viewed as a prototype. Some users judged the quantity of information insufficient to really test the system. Some videos were perceived as too short but the content was supposed to be relevant. The icons and numbers denoting video clips in the flower menus should be replaced by a representative image of the video content and a textual title. Multi-surface and multi-user interaction were considered useful and fun but sometimes confusing. In addition, users would like to navigate within a video clip. Some users complained about the speed of the system reactions. In summary, the overall design and technical integration of the FAME interactive space were very well perceived, fun to use, and provided a special experience. However, at the detail level, there is room for both design and technical improvements, such as speeding up the response time, allowing easier navigation within videos and slides, and better spatial arrangements for projection walls and augmented table, etc.

Conclusions

From the interaction perspective, the system is easy to learn and provides people with an enjoyable experience based on the tangibility of the tokens, the table, the walls and spoken interaction. The novelty of the interaction style and the fact that it is fun, draw users' attention away from the content itself. The "testimony" feedback service provides a new social experience, although, in our experiment, users exploited this facility more to look at themselves and not to produce information for others.

We have not been able to demonstrate the benefit of implicit interaction supported by the topic spotter. We believe that implicit interaction cannot be tested in sessions of short duration, but can only be appreciated over a long period of use, which was not possible in our setting.

The experiment also showed the necessity for systems to automatically adapt to the physical environment: spatial relationships between surfaces, distance from the actuators, background color of surfaces, etc. For example, one third of the users were wearing the same color as that of the tokens, which could be a problem in bad lightning conditions. Our continuing work on context modeling and adaptation is an attempt to solve this problem.

Fig. 1 .

 1 Fig.1. Components of the "FAME" demonstrator: the database connects the off-line "Intelligent Cameraman" and indexing to the left, the and on-line multi-modal document retrieval using the "Augmented Table" on the right


Hui Liu

Xiaolong Liu

Jindong Tan

A Novel Modular Vascular Anastomosis Device for Minimally Invasive Heart Surgery

Introduction

The shift of surgical paradigm from open surgery to minimally invasive surgery (MIS) allows surgeons to perform complex operations inside the human body cavity with a few small incisions, which reduces trauma, scarring, and hospitalization. Though many surgical tasks can be operated by MIS, open surgery is still the state-ofthe-art for vascular graft in heart surgery due to the lack of effective tools. In this work, we present an innovative fully insertable vascular anastomosis device for MIS which significantly reduces the operation time and improves surgical efficiency.

Hui Liu, Xiaolong Liu, Zhe Su, and Jindong Tan Department of Mechanical, Aerospace, and Biomedical Engineering, University of Tennessee, Knoxville, TN 37996

Results and Discussion

The prototype of our vascular anastomosis module is demonstrated in Fig. 1. The SMA wires are buried in clamping wings to actuate the module by dragging rotatable pads against micro springs. Integrating a micro thermistor to detect the wire temperature, a closed-loop PID temperature control system was designed to control

Materials and Methods

This design features a miniature actuation structure for installing vessels onto couplers, stretching the vessel ends and clamping vessels with couplers by a shape memory alloy (SMA) actuator. As illustrated in Figure .1

Conclusion

This work proposed a novel modular device for vascular anastomosis in minimally invasive heart surgery. We demonstrated the prototype design that provides reliable vessel installation onto the coupler and sufficient strength for anastomosis. The module is designed into a tiny size according to the force analysis and can be mounted onto the distal of laparoscopic tools. The module will be tested by using porcine vessels for further evaluation. When SMA wire is heated, there will be two steps to install the vessel onto the couplers, as shown in Fig. 3(b). Firstly, the pads push the pins to penetrate vessel walls.

Then the pads rotate the coupler wings to stretch the vessel end. The modular actuator is designed small enough so that it can be mounted onto the flexible distal end of laparoscopic tool (Fig. 3(c)) and inserted through a trocar. The laparoscopic tool is modified into a motor control model, which can be further developed into autonomous anastomosis system.

Fig. 1

 1 Fig.1 Conceptual illustration of the vascular anastomosis module

Fig. 3 .

 3 Fig.3. (a) Force analysis of the design; (b) Install vessel onto the coupler; (c) Motor controlled laparoscopic tool with flexible distal end

  The size of the module is determined by detailed force analysis, as shown in Fig.3(a).

Fig. 2

 2 Fig.2 Design of couplers