Ahmed Belhadjayed

Grégoire Loeper

Sofiene El Aoud

Frédéric Abergel

Ahmed Bel Hadj Ayed

Performance analysis of the optimal strategy under partial information

come

Introduction

Optimal investment was introduced by Merton in 1969 (see [START_REF] Merton | Lifetime portfolio selection under uncertainty: the continuoustime case[END_REF] for details). He assumed that the risky asset follows a geometric Brownian motion and derived the optimal investment rules for an investor maximizing his expected utility function. Several generalisations of this problem are possible. One of them is to consider a stochastic unobservable trend, which leads to a system with partial information. This hypothesis seems to be realistic since only the historical prices of the risky asset are available to the public. For example, Karatzas and Zhao (see [START_REF] Karatzas | Bayesian adaptive portfolio optimization[END_REF]) study the case of an unobservable constant trend, Lakner (see [START_REF] Lakner | Optimal trading strategy for an investor: the case of partial information[END_REF]) and Brendle (see [START_REF] Brendle | Portfolio selection under incomplete information[END_REF]) consider a stochastic asset price model where the trend is an unobservable Ornstein Uhlenbeck process, and Sass and Hausmann (see [START_REF] Sass | Optimizing the terminal wealth under partial information: The drift process as a continuous time markov chain[END_REF]) suppose that the trend is given by an unobserved continuous time, finite state Markov chain.

In this paper, we consider a stochastic asset price model where the trend is an unobservable Ornstein Uhlenbeck process and we focus on the optimal strategy with a logarithmic utility function under partial or complete information.

The purpose of this work is to characterize the performance of these strategies as functions of the signal-to-noise ratio and of the trend mean reversion speed and to quantify the loss of performance due to partial information. The loss of utility due to incomplete information was already studied by Karatzas and Zhao (see [START_REF] Karatzas | Bayesian adaptive portfolio optimization[END_REF]), by Brendle (see [START_REF] Brendle | Portfolio selection under incomplete information[END_REF]) and by Rieder and Bäuerle (see [START_REF] Rieder | Portfolio optimization with unobservable markovmodulated drift process[END_REF]). Here, the trading strategy performance is measured with the asymptotic Sharpe ratio (see [START_REF] Sharpe | Mutual fund performance[END_REF] for details).

The paper is organized as follows: the first section presents the model and recalls some results from filtering theory.

In the second section, the optimal strategy with complete information is investigated. This portfolio is built by an agent who is able to observe the trend and aims to maximize his expected logarithmic utility. We provide, in closed form, the expectation and variance of the logarithmic return as functions of the signal-to-noise ratio. We also show that the asymptotic Sharpe ratio of the optimal strategy with complete information is an increasing function of the signal-to-noise ratio.

In the third section, we consider the optimal strategy under partial information. This corresponds to an unobservable trend process and to an agent who aims to maximize his expected logarithmic utility. In this case, we provide, in closed form, the expectation and variance of the logarithmic return as functions of the signal-to-noise ratio and of the trend mean reversion speed. Then, we derive the asymptotic Sharpe ratio and we show that this is an increasing function of the signal-tonoise ratio and an unimodal (increasing then decreasing) function of the trend mean reversion speed. After that, we introduce the partial information factor which is the ratio between the asymptotic Sharpe ratio of the optimal strategy with partial information and the asymptotic Sharpe ratio of the optimal strategy with full information. This factor measures the loss of performance due to partial information. We show that this factor is bounded by a threshold equal to 2 3 3/2 . In the fourth section, numerical examples illustrate the analytical results of the previous sections. The simulations show that, even with a high signal-to-noise ratio, a high trend mean reversion speed leads to a negligible performance of the optimal strategy under partial information compared to the case with complete information.

Setup

This section begins by presenting the model, which corresponds to an unobserved mean-reverting diffusion. After that, we reformulate this model in a completely observable environment (see [START_REF] Liptser | Statistics of random processes I[END_REF] for details). This setting introduces the conditional expectation of the trend, knowing the past observations. Then, we recall the asymptotic continuous time limit of the Kalman filter.

1.1. The model. Consider a financial market living on a stochastic basis (Ω, F, F, P), where F = {F t , t 0} is the natural filtration associated to a two-dimensional (uncorrelated) Wiener process (W S , W µ ), and P is the objective probability measure. The dynamics of the risky asset S is given by

dS t S t = µ t dt + σ S dW S t , (1)
dµ t = -λµ t dt + σ µ dW µ t , (2)
with µ 0 = 0. We also assume that (λ,

σ µ , σ S ) ∈ R * + × R * + × R * + .
The parameter λ is called the trend mean reversion speed. Indeed, λ can be seen as the "force" that pulls the trend back to zero. Denote by F S = F S t be the natural filtration associated to the price process S. An important point is that only F S -adapted processes are observable, which implies that agents in this market do not observe the trend µ.

1.2. The observable framework. As stated above, the agents can only observe the stock price process S. Since, the trend µ is not F Smeasurable, the agents do not observe it directly. Indeed, the model ( 1)-( 2) corresponds to a system with partial information. The following proposition gives a representation of the model ( 1)-( 2) in an observable framework (see [START_REF] Liptser | Statistics of random processes I[END_REF] for details or Appendix A for a proof).

Proposition 1. The dynamics of the risky asset S is also given by

dS t S t = E µ t |F S t dt + σ S dN t , (3)
where N is a P, F S Wiener process.

Remark 1.1. In the filtering theory (see [START_REF] Liptser | Statistics of random processes I[END_REF] for details), the process N is called the innovation process. To understand this name, note that:

dN t = 1 σ S dS t S t -E µ t |F S t dt .
Then, dN t represents the difference between the current observation and what we expect knowing the past observations. 1.3. Optimal trend estimator. The system (1)-(2) corresponds to a Linear Gaussian Space State model (see [START_REF] Brockwell | Introduction to time series and forecasting[END_REF] for details). In this case, the Kalman filter gives the optimal estimator, which corresponds to the conditional expectation 2) is a controllable and observable time invariant system. In this case, it is well known that the estimation error variance converges to an unique constant value (see [START_REF] Kalmar | Fundamental study of adaptive control systems[END_REF] for details). This corresponds to the steady-state Kalman filter. The following proposition (see [START_REF] Bel Hadj Ayed | Forecasting trends with asset prices[END_REF] for a proof) gives a first continuous representation of the steadystate Kalman filter: Proposition 2. The steady-state Kalman filter has a continuous time limit depending on the asset returns:

E µ t |F S t . Since (λ, σ µ , σ S ) ∈ R * + × R * + × R * + , the model (1)-(
d µ t = -λβ µ t dt + λ (β -1) dS t S t , (4)
where

β = 1 + σ 2 µ λ 2 σ 2 S 1 2 . ( 5
)
The following proposition gives a second representation of the steadystate trend estimator μ: Proposition 3. Based on Equation (4), it follows that:

dμ t = -λμ t dt + λσ S (β -1) dN t . (6)
Proof. Replacing dSt St in Equations ( 4) by the expression of Equation (3), we find Equation ( 6).

Remark 1.2. It is well known that the Kalman estimator is a Gaussian process. Here, we find that the steady-state trend estimator μ is an Ornstein Uhlenbeck process. In practice, the parameters (λ, σ µ , σ S ) are unknown and must be estimated (see [START_REF] Bel Hadj Ayed | Forecasting trends with asset prices[END_REF] where the authors assess the feasibility of forecasting trends modeled by an unobserved meanreverting diffusion). In this paper, we assume that the parameters are known.

Optimal strategy under complete information

In this section, the optimal strategy under full information is investigated. This strategy is built by an agent who is able to observe the trend µ. Formally, it corresponds to the case F S = F. Given this framework, we consider the optimal strategy with a logarithmic utility function. We provide, in closed form, the asymptotic expectation and variance of the logarithmic return, and the asymptotic Sharpe ratio of this strategy as functions of the signal-to-noise ratio.

2.1. Context. Consider the financial market defined in the first section with a risk free rate and without transaction costs. Let P o be a self financing portfolio given by: where ω o t is the fraction of wealth invested in the risky asset (also named the control variable). The agent aims to maximize his expected logarithmic utility on an admissible domain A o for the allocation process. In this section, we assume that the agent is able to observe the trend µ. Formally, it means that A o represents all the F-progressive and measurable processes and the solution of this problem is given by:

dP o t P o t = ω o t dS t S t , P o 0 = x,
ω * = arg sup ω∈A o E [ln (P o t ) |P o 0 = x] .
As is well known (see [START_REF] Lakner | Optimal trading strategy for an investor: the case of partial information[END_REF] or [START_REF] Bjork | Optimal investment under partial information[END_REF] for examples), the solution of this problem is given by:

dP o t P o t = µ t σ 2 S dS t S t , (7)
P o 0 = x. (8)
2.2. Performance analysis of the optimal strategy under complete information. The following proposition gives the stochastic differential equation of the portfolio P o : Proposition 4. Consider the portfolio P o given by Equation [START_REF] Kalmar | Fundamental study of adaptive control systems[END_REF]. In this case,

d ln(P o t ) = µ 2 t 2σ 2 S dt + µ t σ S dW S t . (9)
Proof. Using Equation ( 7) and Itô's lemma on the process ln(P o t ), the result follows.

The asymptotic expected logarithmic return is the first indicator to assess the potential of a trading strategy. The second one can be the variance of the logarithmic return. This indicator can be useful as a measure of risk. Moreover, let SR T be the annualized Sharpe-ratio at time T of a portfolio (P T ) defined by:

SR T = E ln P T P 0 T Var ln P T P 0 . ( 10
)
This indicator measures the expected logarithmic return per unit of risk. The Sharpe ratio is a prime metric for an investment.

Remark 2.1. This definition of the Sharpe ratio is different from the original one (see [START_REF] Sharpe | Mutual fund performance[END_REF]). Here, this indicator is computed on logarithmic returns.

The following theorem gives the asymptotic expectation, variance and Sharpe ratio of the logarithmic return: Theorem 2.2. Consider the portfolio given by Equation [START_REF] Kalmar | Fundamental study of adaptive control systems[END_REF]. In this case:

lim T →∞ E ln P o T P o 0 T = SNR 2 , (11) lim
T →∞ Var ln

P o T P o 0 T = SNR, (12)
SR o ∞ = √ SNR 2 . ( 13
)
where SNR is the signal-to-noise-ratio:

SNR = σ 2 µ 2λσ 2 S . (14)
Proof. Integrating the expression of Proposition 4 from 0 to T and taking the expectation, it gives:

E ln P o T P o 0 = 1 2σ 2 S T 0 E µ 2 t dt + 0.
Since µ is an Ornstein-Uhlenbeck process:

E [µ t ] = 0, Var [µ t ] = σ 2 µ 1 -e -2λt 2λ .
Then, tending T to ∞, Equation ( 11) follows. Since the processes W S and µ are supposed to be independent:

Var ln P o T P o 0 = 1 4σ 4 S Var T 0 µ 2 t dt + 1 σ 2 S Var T 0 µ t dW S t .
Since the process

T 0 µ t dW S t T ≥0
is a martingale:

Var T 0 µ t dW S t = T 0 E µ 2 t dt = σ 2 µ 2λ T + 1 -e -2λT 2λ ,
Moreover, Isserlis' theorem (see [START_REF] Isserlis | On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables[END_REF] for details) gives:

Var T 0 µ 2 t dt = 2 T 0 T 0 (E [µ s µ t ]) 2 dsdt.
Since µ is an Ornstein Uhlenbeck:

Var T 0 µ 2 t dt = σ 4 µ e -4λT e 4λT (4λT -5) + e 2λT (8λT + 4) + 1 8λ 4 .
Equation ( 12) follows. Finally, using the definition of the Sharpe ratio (see Equation [START_REF] Liptser | Statistics of random processes I[END_REF]) and the results of Equations ( 11) and ( 12), Equation (13) follows.

Theorem 2.2 shows that the asymptotic expectation and the asymptotic variance logarithmic return are linear functions of the signal-tonoise ratio and that the asymptotic Sharpe ratio is a linear function of the ratio between the asymptotic trend standard deviation and the volatility.

Optimal strategy under partial information

In this section, the Merton's problem under partial information is investigated. We consider the case of a logarithmic utility function. We provide, in closed form, the asymptotic expectation and variance of the logarithmic return, and the asymptotic Sharpe ratio of this strategy as functions of the signal-to-noise ratio and of the trend mean reversion speed. After that, we introduce the partial information factor which is the ratio between the asymptotic Sharpe ratio of the optimal strategy with partial information and the asymptotic Sharpe ratio of the optimal strategy with complete information. We close this section by showing that this factor is bounded by a threshold equal to 2 3 3/2 . 3.1. Context. Consider the financial market defined in the first section with a risk free rate and without transaction costs. Let P be a self financing portfolio given by:

dP t P t = ω t dS t S t , P 0 = x,
where ω t is the fraction of wealth invested in the risky asset. The agent aims to maximize his expected logarithmic utility on an admissible domain A for the allocation process. In this section, we assume that the agent is not able to observe the trend µ. Formally, A represents all the F S -progressive and measurable processes and the problem is:

ω * = arg sup ω∈A E [ln (P t ) |P 0 = x] .
The solution of this problem is well known and easy to compute (see [START_REF] Lakner | Optimal trading strategy for an investor: the case of partial information[END_REF] for example). Indeed, it has the following form:

ω * t = E µ t |F S t σ 2 S .
Using the steady-state Kalman filter, the optimal portfolio is given by:

dP t P t = µ t σ 2 S dS t S t , (15)
P 0 = x, ( 16
)
where µ is given by Equation (4).

3.2.

Performance analysis of the optimal strategy under partial information. The following proposition gives the stochastic differential equation of the portfolio:

Proposition 5. The optimal portfolio process of Equation (15) follows the dynamics:

d ln(P t ) = 1 2σ 2 S λ (β -1) d µ 2 t + µ 2 t σ 2 S β (β -1) - 1 2 - 1 2 λ (β -1) dt,
where β is given by Equation (5).

Proof. Equation ( 15) is equivalent to (by Itô's lemma):

d ln(P t ) = µ t σ 2 S dS t S t - 1 2
µ 2 t σ 2 S dt.
Using Equation (4),

d ln(P t ) = µ t σ 2 S d µ t λ (β -1) + µ 2 t σ 2 S β (β -1) dt - 1 2
µ 2 t σ 2 S dt,
Itô's lemma on Equation (4) gives:

d µ 2 t = 2 µ t d µ t + λ 2 β σµ,λ,σ S -1 2 σ 2 S dt.
Using this equation, the dynamic of the logarithmic wealth follows.

Remark 3.1. Proposition 5 shows that the returns of the optimal strategy with partial information can be broken down into two terms. The first one represents an option on the square of the realized returns (called Option profile). The second term is called the Trading Impact. These terms are introduced and discussed in [START_REF] Bruder | Risk-Return Analysis of Dynamic Investment Strategies[END_REF]. The option profile at the time T is:

Option Profile T = 1 2σ 2 S 1 λ (β -1) ( µ 2 T -µ 2 0 ).
With the assumption of an initial trend estimate equal to 0, the Option profile is always positive. The Trading Impact is a cumulated function of the trend estimate:

Trading impact T = T 0 µ 2 t σ 2 S β (β -1) - 1 2 - 1 2 λ (β -1) dt.
When T → ∞, it becomes the preponderant term. The Trading Impact is positive on the long term T if the drift estimate µ t verifies:

1 T T 0 µ 2 t dt > λσ 2 S (β -1) 2 β β-1 -1 , (17)
Equation ( 17) can be seen as a condition for the trend following strategy to generate profits in the long term.

The following theorem gives the asymptotic expectation, variance and Sharpe ratio of the logarithmic return: Theorem 3.2. Consider the portfolio given by Equation (15). In this case:

lim T →∞ E ln P T P 0 T = λ 4 (β -1) 2 , (18) lim
T →∞

Var ln

P T P 0 T = λ 8 β 2 -1 2 , (19)
lim

T →∞ SR T = λ 2 β -1 β + 1 , ( 20
)
where β is given by Equation ( 5).

Proof. Based on Equation ( 6), μ is an Ornstein-Uhlenbeck process:

E [μ t ] = 0, Var [μ t ] = (λσ S (β -1)) 2 1 -e -2λt 2λ .
Integrating the expression of Proposition 5 from 0 to T and taking the expectation, it gives:

E ln P T P 0 = (β -1) 4 1 -e -2λT - λ(β -1) 2 T + β β -1 - 1 2 λ 2 (β -1) 2 T - λ 2 (β -1) 2 1 -e -2λT 2λ
Then, tending T to ∞, Equation (18) follows. Integrating the expression of Proposition 5 from 0 to T and taking the variance, it gives:

Var ln P T P 0 = 1 (2λ (β -1) σ 2 S ) 2 Var μ2 T + 1 σ 4 S β β -1 - 1 2 2 Var T 0 μ2 t dt + 2 β β-1 -1 2 (2λ (β -1) σ 4 S ) Cov μ2 T , T 0 μ2 t dt . Moreover Var μ2 T = Cov μ2 T , μ2 T , Var T 0 μ2 t dt = T 0 T 0 Cov μ2 s , μ2 t dsdt, Cov μ2 T , T 0 μ2 t dt = T 0 Cov μ2 s , μ2 T ds,
and the expression of Cov [μ 2 s , μ2 t ] is given in Lemma 5.1 (see Appendix B). Then

Var μ2 T = λ 2 σ 4 S (β -1) 4 2 1 -2e -2λT + e -4λT , Var T 0 μ2 t dt = λ 2 σ 4 S (β -1) 4 2 1 -e -2λT 2λ + e -2λT -e -4λT 2λ -2T e -2λT , Cov μ2 T , T 0 μ2 t dt = λσ 4 S (β -1) 4 2 T - 5 4λ + e -2λT λ + e -4λT 4λT + 2T e -2λT .
Finally, using these expressions and tending T to ∞, Equations ( 19) and ( 20) follow.

The following result is a corollary of the previous theorem. It represents the asymptotic expectation, variance and Sharpe ratio of the logarithmic return as a function of the signal-to-noise-ratio and of the trend mean reversion speed λ. Corollary 3.3. Consider the portfolio given by Equation (15). In this case:

lim T →∞ E ln P T P 0 T = 1 2 SNR + λ -λ (λ + 2SNR) , (21) lim T →∞ Var ln P T P 0 T = SNR 2 2λ , (22)
lim

T →∞ SR T = λ 2 3/2 1 + 2SNR λ -1 2 SNR , ( 23
)
where SNR is the signal-to-noise-ratio (see Equation ( 14)). Moreover:

(1) For a fixed parameter value λ, -the asymptotic expected logarithmic return is an increasing function of SNR, -the asymptotic Sharpe ratio is an increasing function of SNR.

(2) For a fixed parameter value SNR, -the asymptotic expected logarithmic return is a decreasing function of λ, -the asymptotic Sharpe ratio is a decreasing function of λ if:

SNR < 3 2 λ, (24)
and an increasing function of λ if SNR > 3 2 λ. The maximum asymptotic Sharpe ratio is attained for λ = 2 3 SNR and is equal to:

SR Max ∞ = √ SNR 3 3/2 . ( 25
)
Proof. Using Equation ( 14) and Equation ( 5), it follows that:

β = 1 + 2SNR λ .
Injecting this expression in Equation ( 18), we find: lim

T →∞ E ln P T P 0 T = L (SNR, λ) ,
where

L (SNR, λ) = 1 2 SNR + λ -λ (λ + 2SNR) . Since ∂L (SNR, λ) ∂SNR = 1 2   1 - 1 1 + 2SNR λ   ≥ 0,
the asymptotic expected logarithmic return is an increasing function of SNR. Moreover:

∂L (SNR, λ) ∂λ = 1 2 1 - λ + SNR λ (λ + 2SNR) ≤ 0,
it follows that the asymptotic expected logarithmic return is a decreasing function of λ. Moreover, using Equations ( 14), ( 5) and (19), Equation ( 22) follows. Now, with Equations ( 14), ( 5) and (20), we find: lim

T →∞ SR T = SR ∞ (SNR, λ) ,
where

SR ∞ (SNR, λ) = λ 2 3/2 1 + 2SNR λ -1 2 SNR .
Since

∂SR ∞ ∂SNR = λ 3 2 1 + 2SNR λ -1 2 2SNR 2 2 1 + 2SNR λ ≥ 0,
the asymptotic Sharpe ratio is an increasing function of SNR. Moreover:

∂SR ∞ ∂λ = 1 + 2SNR λ -1 3λ 1 -1 + 2SNR λ + 2SNR 4 √ 2λSNR 1 + 2SNR λ .
Then, the sign of ∂SR∞ ∂λ is given by the sign of:

A (SNR, λ) = 3λ 1 -1 + 2SNR λ + 2SNR . Using β = 1 + 2SNR λ , this expression can be factorised: A (SNR, λ) = λ (β -1) (β -2) .
Since β ≥ 1, A (SNR, λ) is negative if and only if β ≤ 2 (and positive if and only if β ≥ 2), which is equivalent to the condition of Equation (24). Equation ( 25) is obtained using SNR = 3 2 λ in Equation ( 23). Note that SR ∞ is always positive. Since SR ∞ is an increasing function of λ if λ < 2 3 SNR and a decreasing function after this point, the maximum value of this function is given by Equation (25).

3.3.

Impact of partial information on the optimal strategy. In order to measure the impact of the investor's inability to observe the trend on the optimal strategy performance, we introduce the partial information factor. This indicator represents the ratio between the asymptotic Sharpe ratio of the optimal strategy with partial information and the asymptotic Sharpe ratio of the optimal strategy with complete information:

PIF = SR ∞ SR o ∞ , (26)
where SR ∞ is the asymptotic Sharpe ratio of the optimal strategy with partial information, and SR o ∞ is the asymptotic Sharpe ratio of optimal strategy with full information. The following theorem gives the analytic form of this indicator.

Theorem 3.4. The partial information factor is given by:

PIF = λ SNR 3/2 1 + 2SNR λ -1 2 √ 2 , ( 27
)
where SNR is the signal-to-noise-ratio (see Equation ( 14)).

If SNR < 3 2 λ (respectively, SNR > 3 2 λ): (1) For a fixed parameter value SNR, this indicator is a decreasing function (respectively, an increasing function) of λ.

(2) For a fixed parameter value λ, this indicator is an increasing function (respectively, a decreasing function) of SNR.

Moreover:

PIF ≤ 2 3 3/2 , ( 28
)
and this bound is attained for λ = 2 3 SNR. Proof. This expression of the partial information factor is a consequence of Equations ( 13) and (23). Moreover:

∂PIF ∂SNR = √ λ 2SNR λ + 1 -1 3λ 2SNR λ + 1 -1 -2SNR 2SNR 5/2 4SNR λ + 2 .
This expression is positive if and only if SNR ≤ 3 2 λ. The dependency on the mean reversion speed λ comes from Corollary 3.3. Remark 3.5. Equation (28) shows that in the best configuration (with λ = 2 3 SNR), the asymptotic Sharpe ratio of the optimal strategy with partial information is approximatively equal to 38.49% of the asymptotic Sharpe ratio of the optimal strategy with complete information.

Moreover, the intuition tells us that a high signal-to-noise ratio and a small trend mean reversion speed λ involves a small impact of partial information on the optimal strategy performance (and then a high PIF). This is true if and only if SNR ≤ 3 2 λ.

Simulations

In this section, numerical examples are computed in order to illustrate the analytical results of the previous sections. The figure 1 represents the asymptotic Sharpe ratio of the optimal strategy with full information as a function of the signal-to-noise ratio. If the signalto-noise ratio is inferior to 1, which corresponds to a trend standard deviation inferior to the volatility of the risky asset, the asymptotic Sharpe ratio of the optimal strategy with complete information is inferior to 0.5. Now, suppose that λ ∈ [START_REF] Bel Hadj Ayed | Forecasting trends with asset prices[END_REF]252] and that the trend is an unobservable process. The figure 2 represents the asymptotic Sharpe ratio of the optimal strategy with partial information as a function of the trend mean reversion speed λ and of the signal-to-noise ratio. Since λ ∈ [START_REF] Bel Hadj Ayed | Forecasting trends with asset prices[END_REF]252] and SNR< 1, Equation ( 24) is satisfied and this Sharpe ratio is an increasing function of SNR and a decreasing function of λ. Moreover, the maximal value is inferior to 0.2. We also observe that, even with a high signal-to-noise ratio, a high mean reversion parameter λ leads to a small Sharpe ratio.

The figure 3 represents the partial information factor, which corresponds to the ratio between the asymptotic Sharpe ratios of the optimal strategy with partial and full information (see Equation ( 26)). Using Equation (28), this indicator is bound by 2 3 3/2 . Since SNR < 3 2 λ, this indicator is a decreasing function of λ and an increasing function of SNR. Even with a high signal-to-noise ratio, a high mean reversion parameter λ leads to a negligible performance of the optimal strategy with partial information compared to the case with full information.

The figures 4 and 5 represents the asymptotic Sharpe ratio of the optimal strategy with partial information and the partial information factor as functions of the signal-to-noise ratio and of λ with λ ∈ [0, 2]. Theses figures illustrate that, if SNR > 3 2 λ, these quantities are increasing functions of the trend mean reversion speed λ (and the partial information factor is also a decreasing function of the signal-to-noise ratio).

Conclusion

The present work quantifies the loss of performance in the optimal trading strategy due to partial information with a model based on an unobserved mean-reverting diffusion.

If the trend is observable, we show that the asymptotic Sharpe ratio of the optimal strategy is only an increasing function of the signal-tonoise ratio.

Under partial information, this asymptotic Sharpe ratio becomes a function of the signal-to-noise ratio and of the trend mean reversion speed. Even if the asymptotic Sharpe ratio is also an increasing function of the signal-to-noise ratio, we find that the dependency on the trend mean reversion speed is not monotonic. Indeed, this is an unimodal (increasing then decreasing) function of the trend mean reversion speed.

We also show that the ratio between the asymptotic Sharpe ratio of the optimal strategy with partial information and the asymptotic Sharpe ratio of the optimal strategy with complete information is bounded by a threshold equal to 2 3 3/2 . Given this result, we surely conclude that the impact of partial information on the optimal strategy is not negligible.

Moreover, the simulations show that even with a high signal-to-noise ratio, a high trend mean reversion speed leads to a negligible performance of the optimal strategy under partial information compared to the performance of the optimal strategy with complete information.

Cov B 2 f (s) , B 2 f (t) = E B 2 f (s) , B 2 f (t) -E B 2 f (s) E B 2 f (t) .

5 SRFigure 1 .

 51 Figure1. Asymptotic Sharpe ratio of the optimal strategy with complete information as a function of the signal-to-noise ratio.

Figure 2 .

 2 Figure 2. Asymptotic Sharpe ratio of the optimal strategy with partial information as a function of the trend mean reversion speed λ and of the signal-to-noise ratio.

Figure 3 .

 3 Figure 3. Partial information factor as a function of the trend mean reversion speed λ and of the signal-tonoise ratio.

Figure 4 .

 4 Figure 4. Asymptotic Sharpe ratio of the optimal strategy with partial information as a function of the trend mean reversion speed λ and of the signal-to-noise ratio with λ ∈ [0, 2].

Figure 5 .

 5 Figure 5. Partial information factor as a function of the trend mean reversion speed λ and of the signal-tonoise ratio with λ ∈ [0, 2].

Lemma 5 . 1 .

 51 Appendix B: Auto-covariance function of the square steady state Kalman filter the following lemma gives the auto-covariance function of the process μ2 : Consider the process μ defined in Equation (4

  ). Its autocovariance function is given by: Proof. Since μ is a centred Ornstein Uhlenbeck process, there exists a Brownian motion B such that, for all s ∈ R + : μs = e -λs λσ S (β -1) B f (s) , where f (s) = e 2λs -1 2λ is a time change. Then, for all s, t such that 0 ≤ s ≤ t, we have: (s) = 3f (s) 2 + (f (t) -f (s)) f (s) .

	Cov μ2 s , μ2 t =	λ 2 σ 4 S (β -1) 4 2	e -2λt e 2λs + e -2λs -2 ,	(29)
	with 0 ≤ s ≤ t.
	Cov μ2
	Let F B t	be the filtration generated by the process B. So:
	E B 2 f (s) , B 2 f (t)	= E B 2 f (s) E B 2 f (t) |F B s
			= E B 2 f (s) E B 2 f (s) +2
	Then
		E B 2 f (s) , B 2 f (t) = 2f (s) 2 + f (t) f (s) ,
	and Equation (29) follows using:

s , μ2 t = e -2λ(t+s) λ 4 σ 4 S (β -1) 4 Cov B 2 f (s) , B 2 f (t) . Since B is a Wiener process: E B 2 f (s) = f (s) . f (t) f (s) BudBu+f (t)-f (s) |F B s = E B 2 f (s) B 2 f (s) + f (t) -f

Chaire of quantitative finance, laboratory MAS, CentraleSupélec

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Appendix A: Proof of Proposition 1

Proof. Let K be a (P, {F t }) martingale defined by:

and the probability measure P defined by:

With the Girsanov's theorem, it follows that the process:

is a P, {F t } Wiener process. Note also that:

Now, introduce the process N , defined by:


M Krifa

N Bouhaddi

S Cogan

Estimation of Modal Damping for Structures with Localized Dissipation

Keywords: Bolted joints, loss factor, proportional damping, localized damping, energetic method

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Introduction

The study of the dissipation in the joints has been the subject of several studies [START_REF] Bograd | Modeling the dynamics of mechanical joints[END_REF]. However, the prediction of damping in structures assembled from the design phase started since a decade [START_REF] Gaul | Structural damping with friction beams[END_REF][START_REF] Ibrahim | Uncertainties and dynamic problems of bolted joints and other fasteners[END_REF]. Bolted joints are responsible for the major part of the energy dissipation in assembled structure [START_REF] Gaul | Structural damping with friction beams[END_REF]. Constitutive modeling of mechanical joints is an important part of predictive dynamic modeling of joints structures because joints are often the dominant source of energy dissipation and vibration damping in those structures. For very expensive components that are safety critical, predictive dynamic modeling is necessary. In order to predict modal damping in an assembled structures two kinds of methods exist in the literature : experimental methods [START_REF] Dion | Improvement of measurement techniques for damping induced by micro-sliding[END_REF] and calculating methods [START_REF] Caignot | Virtual testing for the prediction of damping in joints[END_REF]. Afterwards we will only be interested in the calculation methods. Mechanical engineers often use proportional damping assumption for vibration calculations, even if the structures have a configuration of localized damping. The modal assumption risks to not respect physical properties of the dynamic system. This can causes vibration amplitudes differences and modal damping errors when comparing the proportional model with the real model [START_REF] Adhikari | Rates of change of eigenvalues ans eigenvectors in damped dynamic systems[END_REF]. The purpose of this paper is to investigate the limitations of the energetic method when predicting modal damping for localized structures. Therefore, this paper is structured as follows : section 2 presents the energetic method to estimate modal damping and section 3 illustrates the prediction level of this methodology and its validity domain on academic examples.

Energetic method

To estimate the damping with localized dissipation in bolted joints of an assembled structures when the eigenmodes are well separated, we can use state space method and the energetic method. In this work we describe just the second method. But results of these two methods are taken into account to calculate modal damping of proposed academic examples.

The present study employs this approach to calculate the loss factor associated with the localized linear interfaces of a globally linear structure. In practice, the response of the system is not calculated explicitly due to the high computational burden. When assuming a linear and dissipative problem of assembled structures, the discrete form of the damped vibration problem may be governed by the equation :

M ÿ(t) + C ẏ(t) + Ky(t) = 0 (1)
where K, M and C are respectively the stiffness, mass and damping matrices, y is the vector of the external loads.

The objective of the energetic method is to determine the damping factor corresponding to each vibration mode of the structure. It is based on the concept of the dissipated energy in the interfaces for which the close form expression of the loss factor is the ratio between dissipated energy and maximal potential energy, over a cycle of periodic vibration. as shown in this relation :

ξ ν = 1 4π E diss ν E def ν ; ν = 1, 2, ...., n (2)
where E diss ν and E def ν are respectively the dissipated energy and maximal potential energy. In the following we will present the necessary formulas for calculating these energies.

The estimation of the dissipated and maximal potential energies requires an accurate characterization of the response levels of the system and the latter remain an approximation since they depends a priori on a knowledge of the different dissipation mechanisms. So, dissipated energy is calculated by the following expression :

E diss ν = T 0 ẏ(t) T f c (t)dt (3)
Where

• T = 2π ων the cycle of periodic vibration • f c (t) = C ẏ(t) : dissipated force
• ẏ(t) = Re(jω ν y(ω ν )e jων t ) the velocity of harmonic response Potential energy is calculated as follow :

E def ν = 1 2 {y(w ν )} T K {y(w ν )} (4)
where y(w ν ) is the frequency response of the system. We distinguish to cases : the proportional damping and localized damping. In the first case the frequency response of the system is equal to

y(w ν ) = n ν=1 φ ν q ν (5)
If the response is projected on a single mode of vibration then equation (5) becomes

y(ω ν ) = φ ν q ν ( 6
)
where φ ν and q ν are respectively the eigenmode and modal amplitude corresponding to eigenfrequency ω ν . The exacte frequency response is equal to :

y(ω) = (K + jωC -ω 2 M ) -1 × f (7)
In the case of proportional damping, if the eigenmodes φ ν are orthonormalized with respect to the mass matrix M, i.e. φ T k M φ l = δ kl , so from equations (4) and ( 5) the total strain energy can be derived in the form :

E def ν = 1 2 n ν=1 ω 2 ν q 2 ν (8)
If the damping is assuming to be hysteretic; i.e.

C = η ω K (9)
where η is the unknown loss factor. One can easily show that at frequency ω ν , the modal damping q ν can be expressed in fonction of ξ ν by :

q ν = -j φ ν f ω 2 ν η ν = -j φ ν f 2ω 2 ν ξ ν (10)
Where : η ν = 2 * ξ ν . Equation (10) leads to the amplitude q ν , from which one can determine the energy dissipated in each joint. Hence, the energy dissipated in all joint leads to the estimated value of the total damping factor :

ξ joints ν = 1 4π E diss E def (11)
3 Advantages and limitations of the energetic method

The energetic method proposed has been implemented on a demonstrator as a model in Matlab. Three numerical simulations have been performed in order to highlight the validity domain of energetic method. The goal of this part consists in studying the influence of the following parameters on the accuracy of this method: modal projection effect, localized damping level and modal density when predicting modal damping.

Influence of the mode projection

The system studied to demonstrate the effect of a projection on a user device is composed of two beams connected together by a localized dissipative interface (figure 1). The dissipation force is expressed in terms of the velocity by the equation:

f c (t, u) = c i × ∆ u (12)
The modal damping results calculated with three methods are illustrated on Figures 2: The reference method is, in this case, the state space method [START_REF] Geradin | Mechanical vibrations[END_REF], the methods EM1 and EM2 are the variants of the Energetic Method EM respectively with mode projection on truncated modal sub-basis using equation ( 6) and with modal projection on a single mode of [START_REF] Adhikari | Rates of change of eigenvalues ans eigenvectors in damped dynamic systems[END_REF]. The method EM1 predicts the modal damping with unacceptable errors: for mode 2 (19%), 4(55%) and 6(>80%). Whereas for the method EM2 these error levels drop significantly to reach 0.06 and 0.07 respectively for modes 2 and 4. This improvement is due, in this case, at the projection in a single mode of the forced response of the system. Results of damping errors found between the energetic method EM1 and EM2 versus the reference method (figure 3) are expected if one takes into account the contribution of neighboring terms before calculating the modal damping coefficients. The frequency response of the system (figure 1) when looking for a modal damping can be expressed by :

y(ω) = n ν=1 φ ν φ T ν f ext ω 2 ν -ω 2 + 2jωω ν ξ ν = n ν=1 φ ν × q ν (13)
where q ν is the contribution of mode ν. The coefficient ( qi qj ) presents the ratio of the contribution of mode i relative to mode j. We can thus define a theoretical coefficient of the sum of neighboring mode contribution on single mode j as follows :

γ j = 100 × i =j q i 2 q j 2 (14)
Figure 4 presents the variation of damping errors with respect to the variation of the sum of neigbouring mode contribution γ j .

When γ increases for one mode, the damping error of this mode increases too. For this case modes 2,4 and 6 have the biggest neighboring mode contribution and the biggest damping error values. In the reminder of this study, we will use the dynamic responses with projection on a single mode as this generates better results in the prediction of the modal damping coefficients.

Effects of the localized damping level

In this section, we are interested in studying the effect of the localized damping level (figure 1) on damping prediction by the Energetic Method EM2. So, six numerical simulations are performed by increasing the viscous damping (c i = 10, 20, 30, 40, 50 and 60 N s/m). When localized damping value c i increases as if the dissipated energy increases. Figure 5 illustrates the evolution of relative error obviously for modes 1 and 2, which is calculated between the energetic method EM2 and the reference method. Figure 5 represents also the curves rise for both modes 1 and 2 when localized damping value increases. For other modes we notice the same curve shape but with lower levels of error. When c i increases, for example when c i = 60N/(m/s), damping value ξ 2 goes up 10% and attains a peak of 13% (figure 6). Beyond this modal damping level, damping error increases to unacceptable values (figure 7). The reference method (state space method) may not give good results when c i exceeds a specific modal damping level.

The raison is that state space method is based on the assumption described by the following equation Where s ν is the eigensolution of ν th mode ; j 2 = -1. So to evaluate the quality of the estimated modal damping by the energetic method, and the previous assumption is valid only with low modal damping values. In the following, we compare the frequency responses functions FRFs of the four methods (energetic method EM1, energetic method EM2, state space and exact method). The exact method consists in the calculation of the FRFs by direct inversion of the full impedance matrix at each frequency (equation ( 7)). The FRFs computing by the energetic method(EM1, EM2) use a truncated basis of ten modes contained in the frequency band (0, 2 × f max ), f max = 2000Hz is the maximum frequency of interest.

s ν = -ξ ν ω ν + jω ν 1 -ξ ν 2 , ξ ν << 1 (15)
To calculate FRFs level error in (%) between two responses y 1 and y 2 in the frequency band of interest [0,2000 Hz], we use the following expression :

ε y = 100 × y 2 -y 1 y 1 (16)
For State space method, the FRF comparaison with exact method gives almost similar results (figures 8) when c i = 20N s/m. But it don't gives the same amplitude levels for high level of viscous damping, for exemple when c i = 60N s/m, (figures 9). That's way in the following we will use exact method as a reference instead of state space method, especially when dealing with high damping level. Figures 10 illustrates that the method EM2 estimates the damping factor with good accuracy for low damping level. But, Figure 11 shows that this method is not valid for high damping value. As studied in section 3.1 The method EM1 doesn't give good results when comparing with exact method for both cases (c i = 20N s/m and c i = 60N s/m), (figures 12 and 13).

c i = 60N s/m
So in the rest of this work we will just investigate the method EM2 in the aim to know it's limits for the specific condition : the influence of modal density.

Influence of modal density

In this section, the system studied to investigate the effect of high modal density or the effect of neighboring modes is the triple beam as described on figure 14. The structure was chosen for its ability to generate neighboring modes by adjusting the parameters of the beam and those of linear interfaces. The triple beam structure is composed of three linear interfaces respectively between nodes 6 and 12, 12 and 18, and between the frame and node number 18, (figure 14). Each interface consists of a stiffness k i and a linear damper c i . The simple bending excitation is applied to the node number 1. the used parameters are performed in table 2. Hasselman [START_REF] Hasselman | Modal coupling in lightly damped structures[END_REF] proposed the following condition of decoupling modes :

ω j -ω i 2 × ξ i × ω i (17)
Namely, two modes i and j are decoupled if the pulsation difference between these two modes is much greater than the product of the pulsation and the damping coefficient of one mode multiplied by two. We focus on the case of neighboring modes so eigenfrequencies (ω i = 2 * π * f i ) must respect the following inequation :

f j -f i ≤ 2 × ξ i × f i (18)
We can thus define a theoretical limit frequency neighborhood as follows :

f j lim = (1 + 2 × ξ i )f i (19)
To investigate the effect of high modal density a study campaign has been performed. This study consists at evolving the parameters of the stiffness and keeping damping parameters unchangeable. The aim of changing the stiffness of the interfaces is to get neighboring modes, so that we can study the importance of this phenomena when applying the iterative energetic method to calculate damping. In this section, the used parameters are performed in table 3.

k 1 = k 2 = k 3 = 1 10 4 ... case 2 : k 1 = k 2 = k 3 = 2 10 4 case 3 : k 1 = k 2 = k 3 = 4 10 4 ... case N : k 1 = k 2 = k 3 = 6 10 5
For the first case damping results are presented in table 4. We get two triplets of neighboring modes, the first one is composed of modes {1, 2, 3}, and the second one is composed of modes {4, 5, 6}. It is found that the error increases when the eignemodes are more close. The error can exceed 10 % when the frequency of neighbor modes becomes less than the theoretical limit frequency. For example the mode 4 depicts an error equal to 12.08% with a frequency spacing of 0.35 Hz. (f 1 = 920.59Hz, f 2 = 920.93Hz, f lim = 923.13Hz and f 2 ≺ f lim ). Figure 15 shows the evolution of the damping error of mode 4 (blue curve) and mode 5 (red curve) based on the difference between the frequencies of modes 4 and 5.

Each point of the two curves presents a test case. The frequency difference between Mode 4 and 5, presented in Table 4 and equal to 0.35 Hz, shows the first point on the x-axis graph. The first point on the y-axis graph of the blue curve shows the relative error (%) between damping calculated by the reference method and the energetic method EM2. The error is equal to 12.08%. It is the same for the red curve which represents the error for mode 5, the first point on the y-axes is an error of 5.24%. Figure 15 illustrates also that when modes are more close then the damping error increases. The method EM2 may have some limitations for closed modes. Because, in case of high modal density it seems difficult to isolate a single mode, also because in this case the generalized damping matrix presents high non-diagonal terms, these terms affect the final results.

Conclusion

In this paper, the energetic method is presented and investigated for particular conditions to linear interface joints. The method is based on the ratio between the dissipated energy in the interfaces, and potential energy of the global structure. Firstly, the authors compared the localized damping in interfaces calculated by three methods. He aims at showing the importance of the mode projection by using a double beam system. Secondly, the effects of the localized damping level when localized viscous damping increases was investigated. Finally the influence of increasing modal density when calculation modal damping by the energetic method was studied. To do so, the academic structure of triple beams system was simulated. As a result, the energetic method is not valid when the system has a high modal density.

Figure 1 :

 1 Figure 1: Double-beam with simple bending

Figure 2 :

 2 Figure 2: Comparaison of estimated modal damping

Figure 3 :

 3 Figure 3: Error in estimating modal damping

Figure 4 :

 4 Figure 4: Modal participation of energetic method EM1

ξ 1 Damping error ξ 2

 12

Figure 5 :

 5 Figure 5: Infuence of the degree of dissipated energy at the interface

Figure 6 :Figure 7 :

 67 Figure 6: Comparaison of estimated modal damping

  Method FRF : State Space Method Error (%): c i = 20 Ns/m

Figure 8 :Figure 9 :

 89 Figure 8: FRFs error: State space vs Exact method -c i = 20N s/m

Figure 10 :Figure 11 :

 1011 Figure 10: FRFs error: EM2 vs Exact Method -c i = 20N s/m

  Figure 12: FRFs error: EM1 vs Exact Method -c i = 20N s/m

Figure 14 :

 14 Figure 14: Triple-beam with simple bending

Figure 15 :

 15 Figure 15: Evolution of damping errors vs frequency difference

Table 1 :

 1 Physical and geometric properties of the structure / Inerface parameters

	E	Steel's Young modulus (GP a) 210	k i	Linear stiffness (N/m)	10 5
	ρ s Steel density (kg/m 3 )	7800	c i	Viscous damper (N/(m/s)) 20
	h	Beam thikness (m)	2 10 -2 m 1 Mass (Kg)	0.2
	b	Beam width (m)	2 10 -2 m 2 Mass (Kg)	0.2
	L 1 Beam first length (m)	0.6
	L 2 Beam second length (m)	0.5
	ν	Steel's Poisson ratio	0.3
	interest using equation

Table 2 :

 2 Physical and geometric properties of the triple beams

	E	Steel's Young modulus (GP a) 210
	ρ s Steel density (kg/m 3 )	7800
	h	Beam thikness (m)	10 10 -3
	b	Beam width (m)	50 10 -3
	L 1 Beam first length (m)	0.6
	L 2 Beam second length (m)	0.5
	L 3 Beam second length (m)	0.5
	ν	Steel's Poisson ratio	0.3

Table 3 :

 3 interfaces parametres evolution Damping Stiffness Nonidentical and fixed damping c i Identical and evolutive Stiffness k i c 1 = 40 ; c 2 = 80 ; c 3 = 40 case 1 :

Table 4 :

 4 Comparison of modal damping calculated using three methods

	M odes(Hz)	1	2	3	4	5	6
	F requenciesf i 141.26 143.09 145.25 920.58 920.93 921.01
	f i+1 -f i	1.83	2.17	775.33 0.35	0.08
	f lim		145.74 173.79		923.13 938.43
	ξ ν	Ref (%)	0.79	5.36	16.53	0.06	0.47	1.48
	ξ ν	var2 (%)	0.88	5.64	16.14	0.07	0.50	1.45
	Error(%)	11.74	5.15	2.32	12.07	5.23	2.06

Acknowledgments

Autors are grateful to the National Research Agency ANR for their financial support of this project (contract ANR 11).