Nyassoke Titi

Gaston Clément

Sadefo Kamdem

Louis Aimé Fono

Dynamic Optimal Hedge Ratio Design when Price and Production are stochastic with Jump

Keywords: Jump-diffusion process, futures, stochastic dynamic programming, Lévy measure, risk management

In this paper, we focus on the farmer's risk income, by using commodity futures, when price and output processes are correlated random represented by jump-diffusion models. We evaluate the expected utility of the farmer's wealth and we determine, at each instant of time, the optimal consumption rate and hedge position at given the time to harvest and state variables.

We find a c losed f orm o ptimal p osition o f c onsumption a nd p osition r ate i n c ase o f CARA utility investor. This result (see table 1.5) is a generalization of Ho (1984) result who consider the particular case where price and output are diffusion models.

Introduction

Agricultural production is a risky business. Because agriculture is often carried out in the open air and always involves the management of living plants, it is particularly exposed to risk. Production risks are often the result of poor predictability in the medium or long term, weather conditions, and uncertainty about crop performance due to the impact of pests, disease or many other unpredictable factors. For several years, the countries of the world have intervened to help farmers cope with the risks associated with the production or the variation of the prices on the markets of agricultural commodities. Some national or international developments have led some governments to reorient their agricultural policies towards deregulation, and a more market-oriented approach. Much of the protection of some farmers against the uncertain future volatility of the markets has been removed. In addition, issues as risky as food security and environmental impacts are receiving increasing attention. Thus, future risk analysis and risk management in agriculture can be expected to receive increased attention (see [START_REF] Stulz | Optimal hedging policies[END_REF]- [START_REF] Stulz | Rethinking risk management[END_REF] and some references therein).

In addition, prices of agricultural inputs and outputs are often overlooked when a farmer has to decide how much input to use, or how much produce to produce, so that income risk due to market volatility or production is diminished. In addition to the price risk, it takes into account the exchange rate risk in the context of the import of inputs or the export of agricultural production. Note that, governments can be another source of institutional risk (political risk, sovereign risk, etc.) for farmers. Changes in the rules that affect agricultural production would have, as a corollary, implications for profitability. For example, changes in the income tax provisions or the availability of various incentive payments. Horticultural producers may be severely affected by new restrictions on the use of pesticides, just as owners of intensive live animal farms may be affected by the introduction of restrictions on the use of drugs for the prevention and control of pesticides. the treatment of diseases. One can also cite the contractual risks, that is to say the risks inherent in the relations between the trading partners. For example, the unexpected breakdown of agreements among participants in supply chains is a significant source of risk in modern agribusiness.

Many people or organizations who need to concern themselves with risk in agriculture include: farmers, farm advisers, commercial firms selling to or buying from farmers, agricultural research workers, and policy makers and planners. Policymakers and planners of public or private agricultural policies also need to consider risks and farmers' responses to risks. Thus, models that include risk could provide better predictions of farmer behavior than those that do not. It should be noted that agricultural policy development and planning are themselves risky activities. It is also necessary for farmers to plan the risks they face. The well-being of the farmer, as well as the survival of the farm business, may depend on how agricultural risks are managed. Occurrence of extreme risks could jeopardize the survival of an agricultural activity. The ambition of many farmers to return the farm to a flourishing condition in the coming years may be frustrated if risk management is neglected. Farmers must also recognize that risk and risk aversion influence their production decisions or more generally management. Mention should also be made of the risks associated with innovation. Farmers must not neglect the risks inherent in moving from an existing production system to another system that is supposed to be "superior". It should be mentioned that the adoption of untested "improved" technology may involve a high degree of risk to the farmer, especially if the adoption of the technology requires a substantial capital investment. This risk may be higher if the farmer has no direct experience of the new method.

Similarly, agricultural traders who buy from farmers may well consider the farmers' willingness to reduce market risks. Thus, some of them will be willing to accept a lower price for their production, if the buyer is willing to offer a futures contract at a guaranteed price. The buyer and the seller may benefit from such an arrangement. The purchase or sale of derivatives may be useful in reducing price risk for both future and future products. The most important examples are the futures market hedging of commodities or the purchase or sale of call or put options (see [START_REF] Sakong | Hedging production risk with options[END_REF]), depending on the farmer's need for hedging (see [START_REF] Anderson | Hedging and joint production: Theory and illustrations[END_REF] and [START_REF] Anderson | The time pattern of hedging and the volatility of futures prices[END_REF]). The hedging on the futures market is quite similar to the forward contract sale but with a number of differences. One important difference is that futures contracts are standardized and largely traded contracts, so that prices are determined more competitively than for a specific contract between a single farmer and a single trader. Thus, the farmer can get a better deal by hedging on the futures market than by contract sales. In fact, the seller of a futures contract undertakes to deliver a quantity fixed in the contract, of a defined category of the product traded at a given date. The buyer undertakes to receive the delivery of this quantity of product on this date at the price fixed in the contract. In practice, the delivery of the goods often does not take place. For example, a farmer may hedge to reduce future price risk by selling a futures contract for a product that is as close as possible to the product the farmer expects to have. For more details concerning Futures Hedging under price or production risks see [START_REF] Moschini | Uncertainty, risk aversion, and risk management for agricultural producers[END_REF], Harvey [START_REF] Harvey Lapan | Production, hedging, and speculative decisions with options and futures markets[END_REF], [START_REF] Lapan | Futures hedging under price, basis, and production risk[END_REF] and Lioui and[START_REF] Lioui | Optimal hedging in a dynamic futures market with a nonnegativity constraint on wealth[END_REF][START_REF] Lioui | General equilibrium pricing of nonredundant forward contracts[END_REF]. The cocoa producer plans to sell the cocoa in September. In May the producer decides to hedge on the futures market and so sells a futures contract for an amount of cocoa approximately equal to total production. For convenience, assuming that the October futures price at that moment is 500 euros/ton. By selling such a contract the farmer is agreeing to supply the specified quantity of cocoa of the specified quality in October at the contract price.

It is also important to take into account the risk aversion of the farmers (see [START_REF] Morgenstern | Theory of games and economic behavior[END_REF] and [START_REF] Hardaker | Whole-farm planning under uncertainty: impacts of subsidy scheme and utility function on portfolio choice in norwegian agriculture[END_REF] for more details). The function of absolute risk aversion to risk can be categorized according to its evolution in relation to the increase in wealth, such as the increase in absolute risk aversion (IARA), the absolute risk aversion ( CARA) or decreasing absolute risk aversion (DARA). As a result, relative risk aversion can be categorized as increasing, constant or decreasing with wealth (IRRA, CRRA and DRRA, respectively). Constant risk aversion (CARA) means that preferences are unchanged if a constant sum is added or subtracted from all payments. Constant relative risk aversion (CRRA) means that preferences among at-risk perspectives are unchanged if all gains are multiplied by a positive constant. A more general and flexible utility function that characterize risk aversion is proposed in [START_REF] Aït-Sahalia | Portfolio choice with jumps: A closedform solution[END_REF]. In this paper, we consider as in [START_REF] Ho | Intertemporal commodity futures hedging and the production decision[END_REF], that the farmer has an aversion to the constant risk of CARA type.

Risk hedging can be thought of as the process of simultaneously selecting the futures position and the underlying asset positions to build a portfolio of assets. The futures markets on commodities are these organized markets with a clearing which eliminates counterparty risk, liquidity risk and allows hedge against adverse exchange rate movements. To do this the agent must establish a strategy for risk management. What strategy should he choose ? Considering the approach of [START_REF] Morgenstern | Theory of games and economic behavior[END_REF] who demonstrated in that the maximization of expected utility is a criterion for rational decision. The optimum consumption and portfolio rules in a continuous-time model has been introduced in [START_REF] Merton | Optimum consumption and portfolio-rules in a continuous-time framework[END_REF]. In the same spirit, [START_REF] Ho | Intertemporal commodity futures hedging and the production decision[END_REF] maximizes the consumption and wealth utility function of a farmer in the context of agricultural commodities . The latter paper use a model of consumption and investment in continuous time to determine the optimal behavior to have in the markets by a US wheat producer subject to the risks of prices and quantities through the monthly data for the period 1977 -1980. [START_REF] Anderson | The time pattern of hedging and the volatility of futures prices[END_REF], [START_REF] Marcus | Futures markets and production decisions[END_REF], [START_REF] Ho | Intertemporal commodity futures hedging and the production decision[END_REF] and [START_REF] Hey | The dynamic competitive firm under spot price uncertainty[END_REF] first develop dynamic Hedging models which assumes that producers can revise their hedge position during the growing season. [START_REF] Ho | Intertemporal commodity futures hedging and the production decision[END_REF] allowed for hedge positions to be continuously adjusted over time. [START_REF] Karp | Dynamic hedging with uncertain production[END_REF] extended the Anderson and Danthine's models by adding stochastic production. [START_REF] Karp | Methods for selecting the optimal dynamic hedge when production is stochastic[END_REF] has developed a continuous model similar to [START_REF] Ho | Intertemporal commodity futures hedging and the production decision[END_REF]. Unfortunately, paper [START_REF] Ho | Intertemporal commodity futures hedging and the production decision[END_REF] assumes that the evolution of commodity prices follows a Brownian-type continuous diffusion process. Indeed, the Brownian-type processes do not really take into account, the occurrence of jumps or strong turbulence in the evolution of the price of convenience. It should be noted that jumps diffusion process are among the stochastic processes, that are most often used in financial econometrics, to model the dynamics of asset prices. Discontinuities in these random processes are modeled with jumps. In addition, asset prices sometimes have breaks in their evolution, especially during periods of turbulence, such as the occurrence of bubbles or financial crises. In our paper extends Ho's model by assuming that the evolution of price follows a stochastic process with Levy jump, and that the logarithmic variation of the quantity produced by the farmer follows a normal distribution. Fore more details on jumps diffusion processes (see for instance [START_REF] Aït-Sahalia | Portfolio choice with jumps: A closedform solution[END_REF], [START_REF] Cont | Financial modelling with jump processes, chapman & hall/crc financ[END_REF], [START_REF] Hanson | Applied stochastic processes and control for Jump-diffusions: modeling, analysis, and computation[END_REF] and some references therein).

In this paper, we study a normative model of optimal hedging strategy of the farmer coupled with its consumption behavior in a continuous time jumps-diffusion framework. Some papers such as [START_REF] Rolfo | Optimal hedging under price and quantity uncertainty: The case of a cocoa producer[END_REF] has analyzed the farmer's hedging strategies in a one-period context. However, as mentioned in [START_REF] Ho | Intertemporal commodity futures hedging and the production decision[END_REF], in a one-period model some rather restrictive conditions are imposed on the individual's behavior. The one-period model assumes that the farmer ignores all past information. As a result, the farmer does not adjust over time its consumption and its hedging strategy against the risk of price and production through future contracts. These conditions can be relaxed in a continuous-time jump-diffusion framework. In an intertemporal framework, the farmer's hedging problem may be based on the assumptions that follows. We assume that the farm income (at harvest) depends on both the spot price and the output (in bushels), the farmer is subject to both price and output uncertainties during the production period. The farmer can hedge in commodity futures market, but he cannot eliminate both the price and production uncertainties by any hedging strategy. Other papers have also looked at intertemporal setups. In a discrete-time framework, [START_REF] Neuberger | Hedging long-term exposures with multiple short-term futures contracts[END_REF] examines hedging long-term commodity supply commitment with multiple short term futures contracts. [START_REF] Stanton | Pricing continuously resettled contingent claims[END_REF] price continuously resettled contingent claims, which bear some similarities with continuoustime instantaneous forward contracts, as the current market value of such claims is always zero. [START_REF] Smith | The determinants of firms' hedging policies[END_REF] classify rationales for hedging into two categories: costs and risk aversion. They devote entire section IV in their paper to hedging motivated by risk aversion of managers. Mathematically, manager maximizes an objective function (expected utility) which is already concave. In that case individual preferences result directly in hedging behavior.

Our contribution is to revisit the work of Thomas Ho in the same context, taking into account jumps in price trends. Under our new assumptions regarding the stochastic evolution of price and quantity, added to that of [START_REF] Ho | Intertemporal commodity futures hedging and the production decision[END_REF], we obtain a coverage ratio (see the last line of Table 1.5), which is quite different from that found in Ho's paper. This article is divided into four sections. This article is divided into four sections. The first section entitled "Modeling" study the optimal portfolio and determine the design pf its optimal decisions, and the last section presents the final conclusion and remarks and possible some possible orientations of Farm Risk management. The first section of the appendix entitled "jump-diffusion process" introduces the preliminaries concepts of stochastic calculus for jumpdiffusion model, and the section of the appendix entitled "Stochastic Dynamic Programming" provides tools for dynamic programming process jump-diffusion.

Modeling

Assumptions

A-1

• The farmer's optimal behavior is determined in a continuous-time, finite horizon framework. The harvest time T is assumed to be known.

• At the beginning of the period (the planting season), the farmer makes his production decision on the quantity of crops, Q bushels, to be produced. At the end of the period (the harvest time), he sells all the crops at the prevailing spot price P (per bushel), and hence, his farm income is given by PQ.

A-2 • During the production period (time between the planting and harvest seasons) the farm income is subject to two sources of uncertainty:

1. the price at which the crop will be sold (Price risk), P 2. the quantity of the crop which will be sold (output risk)Q.

• At each time t, the farmer forms expectations of the spot price P t and output Q t at harvest, and then at the next instant, with more information, the farmer revises the expectations:

• P t and Q t are Itô-Lévy processes.

dQ t = σ Q Q t dZ t (1)
dP t P t - = σ P dω t + (e J -1)dN t -λ E(e J -1)dt (2)
dZ t dω t = ρdt (3)
σ Q and σ P are the constant instantaneous standard deviation of the anticipated output and anticipated spot price, respectively, ρ is the instantaneous correlation coefficient, dZ t and dω t are standardized Wiener processes and dN t is a Poisson process with arbitrary distributed jump amplitude J. The symbol E in front of (e J -1) stands for the expectation under physical measure and the constant variable λ is the jump intensity of the Poisson process.The two stochastic processes N t , ω t and J are mutually independent. For more details see [START_REF] Zhang | Equilibrium asset and option pricing under jump diffusion[END_REF][START_REF] Ho | Intertemporal commodity futures hedging and the production decision[END_REF].

A-3 There are frictions in the real sector.

i) During the production period, the farmer cannot affect the output level by committing further investment ( buy more acreage) or disinvestment (land abandonment or selling off portions of the business).

ii) Significant agency costs prohibit the farmer from shifting the uncertainty of farm income to the market through issuance of shares.

A-4

• The futures market is assumed to be perfect: * mark-to-market settlements occur continuously through time, so that the net value of the contract always remains zero.

*
• Each contract calls for a delivery of one bushel.Those holding long positions promise to take delivery of the underlying crop and make payment at the futures price of the contract; those holding short positions promise to make delivery of the underlying com-modity and receive payment at maturity at the futures price of the contract.

• Let F t denote the equilibrium settlement price on the contract at time t. F t is stochastic, and it is characterized by F t = P t for more details see [START_REF] Mahul | Hedging crop risk with yield insurance futures and options[END_REF]. Thus dF t = dP t .

A-5 The farmer has a cash account:W t Wt that may be positive (negative) on which a constant rate of interest r is received (paid).Any net cash flow resulting from either position in futures contracts or from consumption is deposited or withdrawn from this cash account. Therefore, the borrowing rate equals the lending rate.

A-6 During the period, at each instant t, the farmer has to determine optimal consumption rate c * t : optimal consumption rate and x * t the futures position : such that the expected additive utility of consumption is maximized

max c,x E[ T 0 U(c,t)dt + B(Y T , T )] with Y T = P T Q T +W T
where ∀t ≥ 0 Y t the total wealth at time t, P t Q t is the crop at time t, W t : the cash account value at time t, B(., T ) is the terminal utility of wealth and is assumed to be a concave function and U(c,t) an instantaneous utility function for consumption such that :

   ∂U ∂ c > 0 ∂ 2 U ∂ c 2 < 0.

Objective function and the Optimal Decisions

• Let x t denote the number of contracts held by the farmer at time t in a short position

• Let dF t : variation due to an increase in the settlement price of the futures • x t dF t : denote the amount, in cash, to pay at the clearing corporation by the farmer.

• the change in the farmers cash account is the sum of three cash flows:

-The interest earned from the cash account: rW t dt -The consumption : -c t dt

the mark-to-market settlement of his futures position

• This is summarized by the following budget constraint equation

dW t = (rW t -c t )dt -x t dF t (4)
• To derive optimal decisions, x * t and c * t , we utilize the Bellman stochastic dynamic programming technique. The objective function

J T (W, F, Q,t) = max c t ,x t ,t<T E t [ T t U(c, s)ds + B(Y T , T )] (5)
where E t is an expectation operator, conditional on W (t) = W, F(t) = F, Q(t) = Q and c s > 0.

• Substituting the minimization operator by maximization one into (75) J T is the solution of Hamilton-Jacobi-Bellmann equation

J T t + max c,x [H (y,t)] = 0 (6)
this can also be written as follows

max c,x [dJ T +U(c,t)dt] = 0 (7)
satisfying the boundary condition

J T (W, F, Q, T ) = B(F T Q T +W T , T ). (8)
• The boundary condition impose that the optimized derived utility equal to, (at time T), the terminal utility of wealth, since the consumption equal to zero when the wealth equal to zero to. This is the initial value to our dynamic programming. Thus, this problem is a time backwardation problem. See [START_REF] Beckmann | Non-linearities in the relationship of agricultural futures prices[END_REF]

dQ t = σ Q Q t dZ t (10) dZdω = ρdt (11)
Substituting dF t into (4), we obtain

dW t = (rW t -c t )dt -x t dF t (12) = (rW t -c t )dt -x t F t -σ F dω t + x t F t -[λ E(e J -1)dt -(e J -1)dN t ] (13)
Thus

dW t = [rW t -c t + x t λ F t -E(e J -1)]dt -x t F t -σ F dω t -x t F t -(e J -1)dN t (14)
Let us evaluate a dynamic dY t of the wealth farmer Y t . Since

Y t = P t Q t +W t = F t Q t +W t , (15)
then

dY t = dF t Q t + dW t (16)
Applying ( 57) to dF t Q t , such that ( 16) takes the form

dY t = F t dQ t + Q t dF t + d[F t , Q t ] + dW t (17) Applying (62) to d[F t , Q t ],
such that (17) takes the form

dY t = F t dQ t + Q t dF t + d < F t , Q t > c +∆F s ∆Q s + dW t (18) Since d < F t , Q t > c = ρσ F σ Q F t-Q t dt
and ∆Q s = 0, then (17) takes the form:

dY t = [(rW t -c t ) -λ (Q t -x t )E(e J -1)F t-+ ρσ F σ Q F t-Q t ]dt + σ F (Q t -x t )F t-dω t + σ Q F t-Q t dZ t + (Q t -x t )F t-(e J -1)dN t (19)
Let (ω 1 , ω 2 ) be two independent standard Wiener Processes satisfying : ω t = ω 1 t and Z t = ρω 1 t + 1ρ 2 ω 2 t . Thus (19) takes the form:

dY t = [(rW t -c t ) -λ (Q t -x t )E(e J -1)F t + (ρσ F σ Q F t Q t )] α 1 dt + [(Q t -x t )F t σ F + ρσ Q F t Q t ] α 2 dω 1 t + σ Q F t Q t 1 -ρ 2 α 3 dω 2 t + (Q t -x t )F t (e J -1)dN t (20) Let us evaluate dJ T . Applying (71) to J T (Y t ,t) = max c t ,x t E t [ T t U(c, s)ds + B(Y T , T )],
we obtain:

dJ T = J T t dt + J T Y α 1 dt + 1 2 J T Y 2 (α 2 2 dt + α 2 3 dt) + λ dt [J T (Y t + (Q t -x t )F t (e J -1)z,t) -J T (Y t ,t)]ν(dz) (21) as E t (dω i t ) = 0 for i = 1; 2.
End this paragraph by the dynamic programming equation. Substituting (21) into max c,x [dJ T +U(c,t)dt] = 0, we obtain this following result:

max c t ,x t J T t + J T Y [(rW t -c t ) -λ (Q t -x t )E(e J -1)F t + (ρσ F σ QF t Q t )] + 1 2 J T Y 2 t F 2 t [(Q t -x t ) 2 σ 2 F + 2ρ(Q t -x t )σ F σ Q Q t + σ 2 Q Q 2 t ] + λ [J T (Y t + (Q t -x t )F t (e J -1)z,t) -J T (Y t ,t)]ν(dz) +U(c t ) = 0. ( 22
)
As equation ( 22) exist, the optimal decisions (c * , x * ) satisfy the first order conditions.

Evaluation of c * t and x * t

Let us evaluate c * t . The first order condition is obtained by deriving the terms into maximization operator with respect to c. We obtain:

-J T Y +U c (c,t) = 0 (23)
thus, t The optimal consumption rate is determined such that the marginal utility of consumption equates the marginal derived utility in wealth. we obtain:

U c (c * ,t) = J T Y (Y,t) (24)
. This result was obtained by [START_REF] Ho | Intertemporal commodity futures hedging and the production decision[END_REF] in case of no jump into dynamic processes. Hence c * is determined independently of the hedging decisions:

c * = [ ∂U ∂ c ] -1 [ ∂ J T (Y,t) ∂Y ]. (25)
Let us evaluate x * t .

Let us write the second condition of first order and derive the optimal position x * t . To do this, we utilize an utility functions defined by: U(c,t) = e -βt V (c) and G(Y,t) = e -βt L(Y ) where V is the utility function of consumption, L is the utility function of wealth. The choice of U and G (as product of two functions with separable variables) is motivated by the fact that the consumption and wealth processes are Markov processes. More, we suppose that these two utility functions are exponential utility function type: V (c) = -1 q exp(-qc) and L(y) = -K q exp(-rqy) where q > 0 and K a positive constant. Let us evaluate c * t in this case.

since ∂ L(y) ∂ y = rK exp(-rqy) = -rqL(y) and ∂ 2 L(y) ∂ y 2 = r 2 q 2 L(y), thus (25) takes the form:

c * = rY - 1 q log(rK). ( 26
)
Substituting these two functions into (22), we obtain:

0 = max c t ,x t V (c t ) -β L(Y t ) -rqL(Y t )[(rW t -c t ) + ρσ F σ Q F t Q t -λ (Q t -x t )E(e J -1)F t ] + 1 2 r 2 q 2 L(Y t )[(Q t -x t ) 2 σ 2 F F 2 t + 2ρ(Q t -x t )σ F σ Q F 2 t Q t + σ 2 Q F 2 t Q 2 t ] + λ [e -rq[(Q t -x t )F t (e J -1)z] L(Y t ) -L(Y t )]ν(dz) . ( 27
)
We divide this term by rqL(Y t ). since qL(Y t ) < 0, thus the max is replaced by the min ( 27) takes the form:

0 = min c t ,x t V (c t ) rqL(Y t ) - β rq -[(rW t -c t ) + ρσ F σ Q F t Q t -λ (Q t -x t )E(e J -1)F t ] + 1 2 rq[(Q t -x t ) 2 σ 2 F F 2 t + 2ρ(Q t -x t )σ F σ Q F 2 t Q t + σ 2 Q F 2 t Q 2 t ] + λ rq [e -rq[(Q t -x t )F t (e J -1)z] -1]ν(dz) . ( 28
)
Let us evaluate K : We deduce its value by replacing (c t , x t ) by (c * t , x * t ) into ( 28). So we obtain:

K = 1 r exp 1 - β r -q[(rW t -rqY ) + ρσ F σ Q F t Q t -λ (Q t -x t )E(e J -1)F t ] + 1 2 rq 2 [(Q t -x t ) 2 σ 2 F F 2 t + 2ρ(Q t -x t )σ F σ Q F 2 t Q t + σ 2 Q F 2 t Q 2 t ] + λ r [e -rq[(Q t -x t )F t (e J -1)z] -1]ν(dz) . ( 29
)
To evaluate x * t , we distinguish two case: λ = 0 and λ = 0.

First case: λ = 0, i.e, the dynamic of price is pure diffusive process . The condition of first order, i.e, the derivative of second term of equation ( 28) with respect to x is zero, applying it to (28) we obtain:

rq[(Q -x)σ 2 F F 2 + ρσ Q σ F F 2 Q] = 0 (30) ainsi x * t,P = (1 + ρ σ Q σ F )Q. (31)
Remark: 3.1. The optimal hedge ratio

x * t,P Q = (1 + ρ σ Q σ F
) depend to the correlation ρ between the two uncertainties, the price P and the quantity Q. If the correlation is negative, the farmer'revenue is less uncertain, thus the farmer would not hedge his entire position in futures market.

Second case: λ = 0. Like [START_REF] Aït-Sahalia | Portfolio choice with jumps: A closedform solution[END_REF], we choose a Levy measure to obtain a closed form. Consider the Levy measure defined by: ν(dz) = β e -ηz 1I {z≥0} dz where β and η are strictly positive constant. This measure satisfies IR min(1, |z|)ν(dz) < ∞ The calculation of integral term into (28) gives:

λ rq [e -rq(Q t -x t )F t (e J -1)z -1]ν(dz) = λ rq +∞ 0 [e -rq(Q t -x t )F t (e J -1)z -1]ce -ηz dz (32) = λ β rq +∞ 0 [e -[rq(Q t -x t )F t (e J -1)+η]z -e -ηz ]dz (33) = λ β rq 1 rq(Q t -x t )F t (e J -1) + η - 1 η . (34)
The second condition of first order applying to (28) give us:

-λ E(e J -1)F t -rq(Q t -x t )σ 2 F F 2 t -rqρσ F σ Q F 2 t Q t + λ β F t (e J -1) (rq(Q t -x t )F t (e J -1) + η) 2 = 0. (35) (35) it is an implicit form of cubic equation in (Q t -x t ) contains x t . Otherwise x t is solution of (35). Setting S = (Q t -x t ), (35) takes the form aS 3 + bS 2 + cS + d = 0 and setting X = S + b/a
3 , the equation takes the form X 3 + pX + q = 0. The discriminant is ∆ = p 3 27 + q 2 4 . We solve in case where ∆ < 0 (this implies that p < 0). The equation has three real solutions. Since we minimize in the objective function , we consider the smallest solution defined by:

(Q t -x * t ) = - b/a 3 + -4p 3 cos( 1 3 arccos(-q -27 4p 3 - 2π 3 )) (36)
By replacing p and q by its values we obtain:

x * t = (1 + 1 3 ρ σ Q σ F )Q + 2η 3rqF + λ E(e J -1) rqσ 2 F F (e J -1) + + 4 9 ρ 2 σ 2 Q Q 2 σ 2 F + 8 3 2 3 (e J -1)-1 ηλ E(e J -1) r 2 q 2 σ 2 F F 2 - 8ηρσ Q Q 3rqσ F F + 16 3 - 4 (e J -1) η 2 3r 2 q 2 F 2 + 16ρσ Q Qη 9σ F rqF + 8λ ρσ Q QE(e J -1) 9rqσ 3 F F (e J -1)+ 4λ 2 E 2 (e J -1) 9r 2 q 2 σ 4 F F 2 (e J -1) 3 1 2 cos 1 3 arccos - 2 27 ρ 3 σ 3 F Q 3 σ 3 F - 16η 3 27r 3 q 3 F 3 - 2λ 3 E 3 (e J -1) 27r 3 q 3 σ 3 F F 3 (e J -1) 3 - λ E(e J -1) r 3 q 3 (e J -1)σ 2 F F 3 + λ c r 2 qσ 2 F F 3 - η 2 ρσ Q Q r 2 q 2 σ 2 F F 2 (e J -1) - 2λ E(e J -1)(e J -1) 9rqσ 2 F F ρ 2 σ 2 Q Q 2 σ 2 F + 4η 2 r 2 q 2 F 2 - 2ρσ Q Q 9σ F λ 2 E 2 (e J -1)(e J -1) 2 r 2 q 2 σ 4 F F 2 + 4η 2 r 2 q 2 F 2 - 4η 9rqF
λ 2 E 2 (e J -1)(e J -1) 2 r 2 q 2 σ 4 F F 2 + ρ 2 σ 2 Q Q 2 σ 2 F - 2λ 2 E 2 (e J -1)(e J -1) 2 9r 2 q 2 σ 4 F F 2 ρσ Q Q σ F + 2η rqF - 8λ ηE(e J -1)(e J -1)ρσ Q Q r 2 q 2 σ 3 F F 2 + 4ηρσ Q Q 3rqσ F F + 4ηE(e J -1)λ 3r 2 q 2 σ 2 F F 2 + 2η 2 3r 2 q 2 (e J -1)F 2 × 3 √ 3 2 + 1 3 ρ 2 σ 2 Q Q 2 σ 2 F +2 2 3 (e J -1)-1 ηλ E(e J -1) r 2 q 2 σ 2 F F 2 - 2ηρσ Q Q rqσ F F + 4 3 - 1 (e J -1) η 2 r 2 q 2 F 2 + 4ρσ Q Qη 3σ F rqF + 2λ ρσ Q QE(e J -1) 3rqσ 3 F F (e J -1) + λ 2 E 2 (e J -1) 3r 2 q 2 σ 4 F F 2 (e J -1) 3 -3 2 - 2π 3 . (37)

Comparative tabular

c * = [ ∂U ∂ c ] -1 [ ∂ J T (Y,t) ∂Y ] c * = [ ∂U ∂ c ] -1 [ ∂ J T (Y,t) ∂Y ]
Optimal position:

x * P = (1 + ρ σ Q σ F )Q x * t = (1 + 1 3 ρ σ Q σ F )Q + 2η 3rqF + λ E(e J
-1)

rqσ 2 F F (e J - 1) + + 4 9 ρ 2 σ 2 Q Q 2 σ 2 F + 8 3 2 3 (e J - 1) - 1 ηλ E(e J -1) r 2 q 2 σ 2 F F 2 - 8ηρσ Q Q 3rqσ F F + 16 3 - 4 (e J
-1)

η 2 3r 2 q 2 F 2 + 16ρσ Q Qη 9σ F rqF + 8λ ρσ Q QE(e J
-1)

9rqσ 3 F F (e J - 1) + 4λ 2 E 2 (e J -1) 9r 2 q 2 σ 4 F F 2 (e J - 1) 3 1 2 cos( 1 3 arccos(-q -27 4p - 2π
3 ))

Conclusion

This work proposes an investment-consumption model, in continuous-time, which determine the optimal instantaneous consumption and optimal hedging position using futures by the farmer in managing income risk (price risk and production uncertainty). In our model we assume that the price process is a jump-diffusion process and it generalizes [START_REF] Ho | Intertemporal commodity futures hedging and the production decision[END_REF] who assumed that this price process is a diffusion process. We justify that the optimal instantaneous consumption is unchanged in both models. Assuming the farmer preferences for consumption and wealth are represented by exponential utility functions, determine a closed formula of the optimal hedge ratio. Our principal result is the last line of the precede table 1.5 (see also the remark 1.1)

An open question is the implementation of the theoretical results with real data from Cameroonian cocoa market, to provide a hedging policy to stakeholder.

A. Jump-diffusion process

A.1. Counting measure of jumps

Definition A.1. Let be t ≥ 0, A ∈ B(IR d \ {0}) and the random quantity

N(t, A) = #{0 ≤ s ≤ t; ∆X s ∈ A} = ∑ 0≤s≤t 1I A (∆X s ). (38)
For all ω, the function of sets A → N(t, A)(ω) is a counting measure on B(

IR d \ {0}). So A → E[N(t, A)] (39)
is a Borel measure on B(IR d \ {0}). 0ne denote µ(.) = E[N(1, .)] is called intensity measure of process X .

For all t ≥ 0 and A lower bounded , we define a compensated random jump measure Ñ by :

Ñ(t, A) = N(t, A) -tµ(A). (40)
Remark: A.1. N(t, A) is a shorthand measure notation for the measure set N([0,t] × A).

A.2. Space-time Poisson process

Also called general compound Poisson processes, marked Poisson point processes. The spacetime Poisson process is a generalization of Poisson process denoted by

N (dt, dq) = N (]t,t + dt], ]q, q + dq]) (41)
Poisson mark space Q :

Q N (dt, dq) = dN(t; Q) (42) then the quadratic covariation [X 1 , X 2 ] is equal to [X 1 , X 2 ] t = t 0 φ 1 s φ 2 s ds + t 0 IR d ψ 1 (s, y)ψ 2 (s, y)N(ds, dy). (59) or [X 1 , X 2 ] t = t 0 φ 1 s φ 2 s ds + ∑ 0≤s≤t ∆X 1 s ∆X 2 s . (60)
Under differential form :

d[X 1 , X 2 ] t = φ 1 t φ 2 t dt + IR d ψ 1 (t, y)ψ 2 (t, y)N(dt, dy). ( 61
) or d[X 1 , X 2 ] t = φ 1 t φ 2 t dt + ∆X 1 t ∆X 2 t . (62)

B. Stochastic Dynamic Programming

In this section the essential is provided by [START_REF] Hanson | Applied stochastic processes and control for Jump-diffusions: modeling, analysis, and computation[END_REF]

B.1. Model and principle of optimality

The general jump-diffusion SDE ( 45) is reformulated with an additional process, the control process U(t) on U , thus the dynamic'state is defined by : dX(t) = f (X(t),U(t),t)dt + g(X(t),U(t),t)dω(t)

+ Q h(X(t),U(t),t, q)N (dt, dq; X(t),U(t),t) (63)
where ω(t) is a wiener process or diffusion process, N(t; Q, X(t),U(t),t) is a Poisson process or jump process with jump amplitude mark random variable Q, and N (dt, dq; X(t),U(t),t) is the Poisson random measure.

The objective function has a control formulation which is the combination of a final cost at time t f and cumulative instantaneous costs, given the initial data (t, x(t)). That is

V [X,U,t f ](x,t) = t f t C(X(s),U(s), s)ds + S(X(t f ),t f ). (64)
It is a functional of the processes X(t) and U(t), where C(x, u,t) is the scalar instantaneous or running cost function on the time horizon ]t,t f ] given the state at t and S(x,t) is the final cost function; both are assumed continuous. This is the Bolza form of the objective.

Let us end this section with the Principle of optimality

The expected cost for (64) is:

E (ω,N)]t,t f ] t f t C(X(s),U(s), s)ds + S(X(t f ),t f ) (65)
Hence, the optimal expected cost for ( 64) is :

min u E (ω,N)]t,t f ] t f t C(X(s),U(s), s)ds + S(X(t f ),t f ) (66)
the optimal expected cost (66) can be decomposed as follows:

Proposition B.1. Under the hypothesis of decomposition rule and properties of jump-difffusion processes,

v * (x,t) = min U]t,t+δt] E (ω,N)]t,t+δt] t+δt t C(X(s),U(s), s)ds + v * (X(t + δt),t + δt) X(t) = x,U(t) = u (67)
Thus, we have formally derived the fundamental recursive formula of stochastic dynamic programming : Proof:

v * (x,t) = min U]t,t+δt] E (ω,N)]t,t+δt]

Programming

Using the Principle of Optimality (67) and by taking the limit of small δt, replacing δt by dt, we can systematically derive the partial differential equation of stochastic dynamic programming, also called the stochastic Hamilton-Jacobi-Bellman (HJB) equation, for the general, multi-dimensional Markov dynamics case. From the increment form of the state differential dX(t) = X(t + dt) -X(t),

we consider the expansion of the state argument X(t + dt) = X(t) + dX(t) (70) about X(t) for small dX(t) and about the explicit time argument t + dt about t in the limit of small time increments dt, using an extension of Taylor approximations extended to include discontinuous (i. v * (x + ĥ(x, u,t, q),t)v * (x,t)N (dt, dq, ; x, u,t)

Recall that dN(t; x, u,t) = Q N (dt, dq; x, u,t) where the first t argument of dN is the time implicit to the Poisson process, while the second t argument is an explicit time corresponding to the implicit state and control parametric dependence and ĥ(x, u,t, q) is jump amplitude with a corresponding multiplicative factoring of the Poisson random measure. The next step is to take a conditional expectation over the isolated differential Wiener and Poisson processes by considering these expectations: (dt, dq; x, u,t)] = λ (t; x, u,t)dtΦ Q (dq; x, u,t) = λ (t; x, u,t)Φ Q (q; x, u,t)dqdt. We note that the v * (x,t) value on both sides of the equation cancel and then the remaining common multiplicative factors of dt also cancel. The Hamiltonian (technically, a pseudo-Hamiltonian) functional is given by: H (x, u,t) ≡C(x, u,t) + ∂ x v * (x,t). f (x, u,t) + 1 2 g 2 (x, u,t)∂ xx v * (x,t) + λ (t; x, u,t) Q v * x + ĥ(x, u,t, q),tv * (x,t) .φ Q (q; x, u,t)dq,

     E[dω(t)] = 0 E[dω(t)dω(t)] = dt E[N

  Participants can trade costlessly and continuously * The futures contracts are perfectly divisible. * The contracts have the settlement date at the time of harvest.

  s),U(s), s)ds+ S(X(t f ),t f )|X(t + δt),U(t + δt) X(t) = x,U(t) = u s),U(s), s)ds + v * (X(t + δt),t + δt) X(t) = x,U(t) = u .

  ), (71) is transformed as follows v * (x,t) =v

for further details 3. Solution 3.1. Evaluation of dynamics dW

  t , dY t and dJ T

	Let us evaluate dW t . since   dF t = dP t	then
	 F t = P t ,
		dF t F t -	= σ F dω t + (e J -1)dN t -λ E(e J -1)dt	(9)

  e, Poisson) v * (x,t) = dt min u E (dω,dN)(t) C(x, u,t)dt + v * (x,t) + v * t (x,t)dt + ∂ x v * (x,t). f (x, u,t)dt + g(x, u,t)dω(t)

	+	1 2	∂ xx v * (x,t).dω(t)g(x, u,t) g(x, u,t)dω(t)

Q

.

Example: if Q = {1}, the number of jumps in ]t,t + dt] is deduced from:

Q N (dt, dq) = N (dt, {1}) = N(dt) = dN(t; 1) = dN(t) = dN t (44)

A.3. Jump-diffusion and conditional infinitesimal moments

The Poisson and Wiener processes provide stochastic differential equations (SDE) in continuous time for simple jump-diffusion state process X(t),

Where X(0) = x 0 , with a set of continuous function { f , g, h} taken as coefficients and eventually non linear.

N, a random Poisson measure with intensity dt dµ on IR + × (IR d \ {0}).

Ito processes are jump-diffusion processes. Levy processes are essentially jump-diffusion processes, but extended to processes with infinite rate of jump. Jump-diffusion processes with constants coefficients are Levy processes.

Assuming Poisson and Wiener processes are independents. The conditional moments are determined as follows:

and

The jump in state process is given by

where

The infinitesimal moment and jump properties are very useful for modeling approximations of real applications, by providing a basis for estimating the coefficient functions f , g, and h, as well as some of the process parameters, at least in the first approximation, through comparison to the empirical values of the basic probability corresponding of the stochastic integral equation.

A.4. Stochastic Jump-Diffusion Chain Rule

The state process is decomposed into continuous changes,

and discontinuous or jump changes,

such that

Thus, the change of a composite function of the state process X(t), dF(X(t),t), can be decomposed into the sum of continuous and discontinuous changes. The function F(x,t) is assumed to be at least twice continuously differentiable in x and once in t. This leads to

For continuous composite and to

for jump composite.

Combining the continuous and discontinues changes, we derive:

A.5. Quadratic covariation

where [X,Y ] is the quadratic covariation.

If N is a random Poisson measure on [0, T ] × IR and (ω t ) t∈[0,T ] , a Wiener process, both are independent. If

Also, with sufficiently small dt, U]t,t + dt] has been replaced by the conditioned control vector u at t. Theorem B.1. Hamilton-Jacobi-Bellman Equation (HJBE) for Stochastic Dynamic Programming (SDP) If v * (x,t) is twice differentiable in x and once differentiable in t, while the operator decomposition rules are valid, then

The optimal control, if it exists, is given by

subject to any control constraints.

(75) is not an ordinary PDE.

The equation ( 75) is the Hamilton-Jacobi-Bellman equation, is also called simply the Bellman equation, or the stochastic dynamic programming equation or the PDE of stochastic dynamic programming, or in particular, the PIDE of stochastic dynamic programming where PIDE denotes a partial integral differential equation.


Global Carbon Cycle

The European Union of Geosciences held its 9 th biannual meeting in Strasbourg, March 23-27, 1997. During this meeting, Symposium N8 18, Global carbon Cycle, was held under the sponsorship of the IGCP 1 n8404 on the «Terrestrial Carbon in the past 125 Ka», the INQUA 2 Carbon Commission and the ESCOBA-Biosphere 3 project of the EC Environment and Climate Programme. The «Global Carbon Cycle» Symposium attracted 28 oral and poster presentations and about one hundred participants, reflecting the interest of the Earth Sciences community in the global carbon cycle.

The aim of the Symposium was to bring together inputs from diverse sources and disciplines, and from different temporal perspectives in a hope that they can contribute to validation of models of carbon cycle on geological time scales. For present-day carbon cycle, the relative role of the ocean Õersus land vegetation and soils as the possible «missing sink» is still debated. On geological time scales, it is essential that we understand the volcanic and tectonic input fluxes of carbon from the mantle and the lithosphere and the opposing output fluxes due to prolonged storage in carbonates, hydrocarbonates, and disseminated organic matter in sediments. The ice core records show that, during the Quaternary, the concentration of CO and CH in the contempo- important to understand how and why the global carbon cycle has changed over time, altering the composition of the ancient atmosphere, and how it may change in the future as human activities continue to add ever-greater quantities of these gases into the atmosphere.

A major aim of this Symposium was to contribute to our understanding of dynamics among reservoirs and fluxes of carbon. The presentations focussed mainly on terrestrial system including ecosystem carbon storage, rock weathering, riverine transport, groundwater and lake bed carbon, soil organic matter and carbonates. This special issue of Chemical Geology groups together 15 papers, with 12 selected from Symposium presentations and 3 submitted subsequently by their authors. The 7 first papers are devoted to the CO uptake by continental erosion and riverine car- The subsequent two papers are devoted to methane production from freshwater sediments in Lake Bled Ž . Slovenia and Sulistrowiczki and Nowa Cerekiew Ž . Poland , using the carbon isotope tracing. The twelfth paper estimates the amount of terrestrial organic carbon stored in sediments of the Amazon fan during the Last Glacial, utilizing carbon isotopes of the organic matter, organic carbon content and age models based on oxygen isotopes, faunal data and magnetic excursions. The last but one paper in this special issue models the role of the coastal zone for the coupled C-N-P-S biogeochemical cycles during the past three centuries using the TOTEM 6 approach. The final paper utilizes a modified version of an earlier model to sketch the evolution of the carbon cycle from the Archaean to the planetary future.

The guest editors thank the authors for the time and effort invested into their contributions and the reviewers for their helpful comments on the manuscripts. We express our thanks also to Celine Ž . Gavet-Durr CGSrCNRS Strasbourg, France , to the Ž Editor in Chief, Tim Drever University of Wyoming, . USA and his Editorial assistant, Patricia Massar, for their assistance and hard work ensuring the prompt publication of this special issue.

Jean-Luc Probst

Hugues Faure Jan Veizer 4 Global Erosion Model 5 Carbon Assimilation in the Biosphere 6 Terrestrial-Ocean-aTmosphere-Ecosystem Model

2

  bon transport. The first paper addresses silicate weathering in large river basins, the second focusses on the Ganges-Bramaputra river basin and the subsequent four papers present new results, based on Ž 13 12 . carbon isotopes Cr C , for carbon cycle in large Ž river systems Ottawa and St Lawrence in Canada, . Ž Rhone in France and in a small catchment Streng-ˆ.bach in the Vosges, France . The seventh paper riverine carbon transport into the oceans can be applied to the Last Glacial Maximum. The next section contains two papers dealing with the global reconstruction of carbon storage in the terrestrial biosphere during the Holocene and the Last GlacialrInterglacial Cycle, one using CARAIB 5 model and the other the paleogeographic distribution of vegetation in northern Eurasia. The paper N8 10 quantifies the CO fluxes of degassing 2 from the groundwater aquifer in central Italy, probably one of the largest deep-CO production systems. 2


1

Random telegraph signals in nanoscale vertical junctionless transistor with gate-all-around

Simon Grall 1 , Abhishek Kumar 2 , Laurent Jalabert 1 , Soo-Hyeon Kim 1 , Guilhem Larrieu 1,[START_REF] Timp | International Electron Devices Meeting[END_REF] and Nicolas Clement 1a

1 LIMMS/CNRS, Institute of Industrial Science, University of Tokyo, Tokyo, 153-8505, Japan 2 LAAS-CNRS, Université de Toulouse, Toulouse, 31031, France

The role of a single defect on the electrical performance of transistors is unclear for transistors at nanometric dimensions. This role must be understood in order to improve the design and fabrication process of nanoscale transistors. Herein, we detected a single defect by a noise analysis approach on 18-nm-long gate junctionless (JL) vertical gate-all-around nanowire (VGAANW) transistors. Random telegraph signals (RTS) are reported to have a relative amplitude as high as 60%. We studied these RTS to assess the capacitive network in such nanoscale devices, providing routes to avoid large noise in the further downscaling of VGAANW and to exploit the high charge sensitivity for sensing applications. Defect density comparisons between JL and classic accumulation-mode transistors offered the opportunity to discuss a noise model developed when transistors were of several micrometers in size, questioning the significance of a defect density of less than one defect per device at the nanoscale.

a Author to whom correspondence should be addressed. E-mail: nclement@iis.u-tokyo.ac.jp.

As we are reaching scales where one-dimensional transport and ballistic behaviors can be observed [1][START_REF] Timp | International Electron Devices Meeting[END_REF][START_REF] Javey | [END_REF][4][5][6][7] , the transport mechanisms at the nanoscale need to be better understood for their applications in single-electron sensors [8][9][10][11] , quantum computing 12,13 , and the next generations of transistors 14,[START_REF] Bae | IEEE International Electron Devices Meeting (IEDM)[END_REF] . At the nanoscale, the continuous concepts of dopant concentrations and trap densities are challenged by their discrete occurrence, increasing variability from device to device, and uncertain transport mechanism at this scale. Whereas the study of noise in nanoscale devices gives access to a lot more information than just direct current (DC) measurements, there are several candidate models for noise in nanodevices. One model was developed at a time when device dimensions were on the order of micrometers and considers a correlated number and mobility fluctuation (known as unified number fluctuation), corresponding to the trapping/detrapping of charges from the channel in interface defects, affecting the overall mobility of charge [START_REF] Ghibaudo | [END_REF][17][18] . Another model has been recently proposed 19,[START_REF] Clement | 2015 International Conference on Noise and Fluctuations (ICNF)[END_REF] from the analysis of Coulomb repulsion between neighboring traps in nanoscale devices, an effect that does not involve mobility. This model is based on the same number fluctuation postulate but with an effective charge (q * ) of the involved traps that depends on the carrier's density in the channel. Very interestingly, both models adapted here for the gate-all-around architecture are very similar mathematically (Equations ( 1) and (2), respectively).

𝑆 𝑔 = 𝑞 𝑁 𝐶 2𝑅𝐿 1 - 𝛼 𝜇 𝑞 𝐶 𝑉 -𝑉 1 𝑓 (1)
𝑆 𝑔 = 𝑞 𝑁 𝐶 2𝑅𝐿 2𝜖 𝜖 + 𝜖 1 - 𝑞𝑡 2𝑘 𝑇𝜖 𝜖 + 𝜖 𝐶 𝑉 -𝑉 1 𝑓 ( 2
)
where SI is the current power spectrum density, gm is the transconductance, q is the elementary charge, Not is the density of traps, Cg is the gate capacitance, R is the half-width of the transistor (or radius for nanowire architectures), αsc is the scattering parameter, Vg is the gate voltage, Vth is the threshold voltage, Lg is the length of the gate, 𝜖 and 𝜖 are the permittivities of SiO2 and Si, respectively, tacc is the thickness of the accumulation layer, T is the temperature, and kB is the Boltzmann constant. See Figure S1 for the comparison of both models with standard parameters.

Junctionless (JL) transistors consist of uniformly and highly doped nanowires, which are sufficiently narrow to be entirely depleted by a gate. In practice, it has been demonstrated that these gated resistors have the full functionality of complementary metal-oxide semiconductor transistors [START_REF] Colinge | [END_REF][START_REF] Colinge | Semiconductor-On-Insulator Materials for Nanoelectronics Applications[END_REF] . Several reports have been published on the JL configuration that effectively address short channel issues and the difficulty of fabricating highly doped p-n junctions at the nanoscale 21- 27 . This design can be efficiently combined with a gate-all-around nanowire architecture, which has been actively explored with promising perspectives on integration and downscaling, especially for the vertical approach 14,[START_REF] Bae | IEEE International Electron Devices Meeting (IEDM)[END_REF][START_REF] Colinge | [END_REF][START_REF] Colinge | Semiconductor-On-Insulator Materials for Nanoelectronics Applications[END_REF]24,[27][28][29][30] . Noise studies widely acknowledge their superiority compared to classic accumulation-mode transistors, mostly thanks to their core-conduction mechanism [START_REF] Colinge | [END_REF][START_REF] Colinge | Semiconductor-On-Insulator Materials for Nanoelectronics Applications[END_REF][START_REF] Nazarov | [END_REF][25][26][27]29 .

In this paper, we use transistors with 18 nm-long gates in an all-around configuration, allowing fine tuning of the aperture of the channel. We show how the observation and analysis of a random telegraph signal (RTS) resulting from trapping/detrapping of a single charge and low frequency noise can bring insights to noise models. In addition, we compare JL vertical gate allaround nanowire transistors (VGAANWs) with equivalent low-doping-level devices (commonly called Schottky barrier accumulation-mode transistors (SB)) made with the same fabrication process to discuss the models and their implications on the density of the traps and the charge transport mechanism.

The devices were fabricated following the main process steps described in ref 14 . However, in this new generation, the fabrication of the symmetrical source and drain silicide contacts have been specifically optimized by introducing a novel lift-off-free approach, leading to an improvement of the access resistance 31 . The schematics of the device shown in Figure 1a. Briefly, the devices consist of a monocrystalline array with ~138-nm-long Si nanowires vertically arranged with top and bottom contact with PtSi, a 60-nm SiOx layer at the spacers on both sides of a 18-19nm-long Cr gate, and a 4-5-nm-thick SiO2 gate dioxide layer around the nanowires. The silicon for the JL devices was heavily p-doped (Np = 3×10 19 at/cm 3 ), and the silicon for the SB devices was undoped (Np = 1.8×10 15 at/cm 3 ). The number of nanowires in parallel (nNWs) varied from 1 to 36, and their diameter varied from 16 nm to 43 nm.

The measurement setup is described in the supplementary information (Figure S2). The drain current (Ids) and transconductance (gm) were measured for all devices versus Vg for a drainsource voltage (Vds) of -0.1 V or -1 V. It was also measured versus Vds, with Vg in the subthreshold and saturation regimes. The characteristics of four devices are shown in Figure 1, with a summary of the measured ION/IOFF values for all measured devices shown in Figure S3 (8 JL, 7 SB). The maximum current ranged from 1 nA to 1µA per nanowire for these devices. Wider JL devices showed a larger IOFF current as Vds increased, which was not the case for a smaller radius or SB devices (Figure S4). By default, the JL devices had a fully opened channel that the gate must deplete entirely to turn off the device. This becomes more difficult as the diameter increases 32 . As for smaller radii, they showed a lower ION, though more dispersity was observed, which was expected with scaling down. The mobility was extracted using Equation (3) in the saturation region for JL devices and Equation (4) in the subthreshold regime for SB devices, with 1 < 𝜇 , < 50 cm 2 /V and 1 < 𝜇 , < 75 cm 2 /V, respectively.

𝜇 , = 𝐼𝐿 𝑞𝑁 𝜋𝑅 𝑉 (3)
𝜇 , = 𝑔 𝐿 2𝑅𝑛 𝐶 𝑉 ( 4
)
where L is the length of the nanowire and Np is the dopant concentration.

For 16-nm-diameter JL devices, Ids tended to saturate and gm decreased sharply after reaching saturation, as expected for extremely scaled transistor devices for which access resistances are not negligible. This behavior is representative of 16-nm-diameter JL devices (4/4 measured devices of this diameter) and was observed for one SB device (43-nm diameter, nNWs = 1). Overall, the performances were improved compared to a previous study on similar devices, in which smaller currents per nanowire were measured, attributed to the access to the contacts 29 .

Two JL devices with diameters of 16 nm (nNWs = 4 and 25, respectively) showed a clear RTS, thus providing a good opportunity to assess information on the capacitive network and electron transport in these nanoscale devices. A time trace of the 4-nanowire one and its amplitude evolution versus Vds and Vg are shown in Figure 2. The relative amplitude was as high as 15% of the overall current (Ids vs. Vg in the inset of Figure 3a), i.e., as high as 60% of the current flowing in the nanowire holding the defect. This RTS amplitude was extremely large when compared to the RTS usually obtained in nanotransistors [START_REF] Asenov | International Electron Devices Meeting 2000. Technical Digest. IEDM (Cat. No.00CH37138[END_REF][START_REF] Kutovyi | [END_REF] . The evolution of the time spent in the up state (up) or down state (down) showed no correlation with Vg (Figure S5) but demonstrated a clear Fermi distribution with varying Vds (Figure 2b). This finding suggests that a defect is loosely coupled to the gate, i.e., located in the spacers. The amplitude of IRTS varied in both cases but could be well explained versus Vg by the relative series resistance of the spacers and opened channel = Rs/Rtot, as shown in Figure 2c, with Rtot = Vds/Ids and Rs = max(Rtot) (measured at the saturation) obtained from DC characteristics.

Considering the Fermi distribution behavior observed from the evolution of up and down

versus Vds, we proposed a thermally activated trap occupancy with the Fermi level being mainly tuned by Vds:

𝛥𝐼 = 𝜂 𝑞 * 𝑔 𝐶 * 1 1 + exp 𝑞 𝑉 -𝛽𝑉 𝑘 𝑇 (5)
where q * is the effective charge of the trap, gm is the transconductance, Cg * is the effective capacitance felt by the trap, q×Vtrap is the activation energy of the trap,  is a parameter that takes into account the position of the trap relative to the drain contact, kB is the Boltzmann constant, and T is the temperature. Being in the saturation regime, we considered ≈ 1 and constant (i.e., Ids vs.

Vds is linear). On a first approximation, q * could be assumed to be equal to the elementary charge, and gm was estimated using a linear regression of Ids vs. Vg data (𝑔 (𝑉 ) ≈ -𝑉 × 1.42𝜇𝑆). Of note, the parameter  was found to be very close to 1 for JL devices in which RTSs were observed, which corresponds to traps located close to the drain side of the transistor. It is reasonable to assume that this will often be the case as a sweep of Vds barely impacts the energy level of the trap if it is located on the source side, whereas a sweep of Vds scans a much larger amplitude of energy levels if the trap is located closer to the drain.

If we considered Cg * = Cg in Equation ( 5), as usually assumed for larger devices, it led to a variation of the current from the trap corresponding to more than 40 elementary charges (Figure 2b). On the other hand, taking Cg * ≈ Cg/15 fit well with the RTS measurement. This, in addition to  = 1 and  ≈ 1, suggests that the trap is located in the spacer, with a small coupling to the gate capacitance. This analysis on Cg is consistent with previous estimations of the capacitive network on similar structures 14 . This work emphasizes the importance of the capacitive network to explain the large relative RTS observed. The fact that previous studies did not report such large RTSs due to a trap in the oxide layer of JL devices could be due to Schottky resistances that are too high, the use of another implementation than VGAANWs (leading to higher Cg * ), larger nanowire diameters, or the use of large numbers of nanowires in parallel. 14,[START_REF] Nazarov | [END_REF][24][25][26][27][START_REF] Cheung | IEEE International Conference on IC Design & Technology[END_REF] Other studies already have demonstrated a very high RTS amplitude in nanowires due to the quantum confinement effect for smaller devices 28 , random dopant fluctuations, and traps positioned along the corresponding percolation path, but the effects were observed at a low temperature or for a RTS occurring in the subthreshold region 6,[START_REF] Asenov | International Electron Devices Meeting 2000. Technical Digest. IEDM (Cat. No.00CH37138[END_REF][START_REF] Cheung | IEEE International Conference on IC Design & Technology[END_REF][START_REF] Fantini | [END_REF][37] . Considering that the RTS was observed in the saturation region and that the experiments were performed at room temperature, the percolation path explanation seems unlikely here.

The fact that a RTS was observed for JL devices only with small diameters can be explained by the larger relative resistance of the spacers compared to that of devices with larger diameters, as it scales directly with the section of the nanowires (Equation S4). Another way to look at it is to estimate the apparent radius of the trap (rtrap). Adapting work by Ohata et al. 38 to our geometries gave an apparent radius of rtrap ≈ 2 nm (Figure 2c). This could explain why large RTSs are observed only for JL devices with a small diameter, as the trap occupies a relatively larger portion of the whole conductive section compared to that of larger JL devices. In the following section, we analyze the current power spectra.

Drain current power spectrum densities (SI) were measured versus Vds acquired at Vg in the subthreshold and the saturation regions as well as versus Vg at Vds = -0.1 V and Vds = -1 V (examples in Figure S6). An example of SI measured at different Vg values (Vds = -0.1 V) for a JL device (nNWs = 4, 16 nm in diameter) is shown in Figure 3a along with the corresponding Ids and gm measurements. The contribution of RTS to SI is visible by the emergence of a shoulder at around 20 Hz. The spectra were fit using a 1/f α background contribution superimposed on Machlup's mathematical description of RTS 39 . The Hooge constants (H) were estimated in the saturation regime, where the channel is fully opened and where the whole length of the transistors was considered for JL devices. Values of H were obtained (2. 10 < 𝛼 < 0.4) in the range of expectations using Equation ( 6) 40 .

where 𝑁 = 𝐿𝜋𝑅 𝐶 𝑛 𝑁 for JL and 𝑁 = 𝐿 𝜋𝑅 𝐶 𝑛 (𝑉 -𝑉 )/𝑞 for SB.

Previous work on vertical nanowires has demonstrated an exponential increase of SI/I 2 with Vds for Schottky barrier contact noise 41 . For all devices measured herein, the current-normalized noise SI/I 2 measured versus Vds remained constant (see Figure S7), which confirms the technological improvements achieved on the contacts of these devices 31 . This finding suggests that the Schottky

𝛼 = 𝑆 𝑓𝑁 𝐼 (6)
contact is no longer the limiting factor in noise measurements, unlike those previously reported for similar structures and carbon nanowires 4,[START_REF] Mukherjee | 2017 47th European Solid-State Device Research Conference (ESSDERC)[END_REF] . The improvement of the fabrication of such nanoscale devices offers an opportunity to assess the intrinsic carrier transport properties in a nanotransistor.

The two types of devices with different doping levels but manufactured with the same technology allowed a fair comparative analysis of the defect densities. To this end, the defect density Not was measured near Vth, where SI/gm 2 is nearly constant (Figure 3b), a marker of number fluctuation-induced noise 27 , and where the effective capacitance can still reasonably be considered equal to Cg. This regime corresponds to a fluctuation of the threshold voltage rather than a fluctuation of the mobility. Note that the possibility of a dielectric polarization noise 8 (that leads to a thermal 1/f noise in the absence of defects) is excluded because the estimated dielectric loss would be orders of magnitude too large. Therefore, a simplified version of Equation ( 1) was considered, Equation (7) [START_REF] Ghibaudo | [END_REF]18 , to estimate the Not at f = 10 Hz. The results are summarized in Figure 3c-d.

𝑁 = 𝑆 𝑔 𝑓𝐶 2𝜋𝑅𝐿 * 𝑛 𝑞 (7)
The Not values were approximately 2 × 10 11 cm -2 for JL devices and 2 × 10 10 cm -2 for SB devices. Although a non-negligible device-to-device fluctuation was observed on these values, both types of devices were fabricated using the same equipment and techniques and should thus have very similar defect densities. Considering the number of defects per nanowire, there was less than one defect per nanowire (Figure 3d). However, considering an effective charge q * (Equation ( 2)), Not became more similar for JL and SB devices, and the number of defects per nanowire became greater than one. Therefore, these results suggest that Equation ( 2) is more appropriate than Equation (1) for nanotransistors.

In conclusion, we observed an unprecedentedly high RTS on JL VGAANWs, which was allowed thanks to low Schottky access resistances. The study of this RTS revealed the importance of the capacitive network, Cg * , for defects that are not located under the gate. Indeed, if traps are uniformly distributed along the length of the nanowire, as the gate tends to be much shorter than the total length of the device, traps will statistically have a greater chance to be controlled by Cg * than by Cg. Importantly, keeping a relatively large Cg * to decrease IRTS is contradictory with the requirements for the general demands of the higher switching speed and power efficiency obtained by increasing Cg/Cg * in transistors. If we consider the expression of q * , the use of high-K materials for the gate could have a favorable impact on the RTS amplitude, even with the expected increase in the number of defects. With the general trend in downscaling and the defect densities currently obtained, the tradeoff still makes the high permittivity route promising to decrease the impact of traps in nanoscale devices. A recent publication using such a configuration has reported low noise and no RTSs for similar devices 27 . The comparison of defect densities obtained on JL and SB devices suggest that the effective charge (q * ) model may be the most appropriate to describe the defects in nanoscale devices. Several publications have continued to use the unified number fluctuation model for nanoscale devices, but one can wonder about the physical meaning of such a model when the dimensions of devices get so small that they sometimes have less than one defect per device, on average (e.g., ~0.04/device 25 , 0.03-2/device 42 , or 0.002/device 27,43 ). We believe that this study of RTS noise on JL VGAANW devices provides critical information toward a better understanding of nanoscale charge transport and opens new perspectives for high-sensitivity sensors [START_REF] Clement | 2015 International Conference on Noise and Fluctuations (ICNF)[END_REF][START_REF] Kano | [END_REF]45 as the charge noise and charge sensitivity are correlated like in quantum computing, where 1/f noise is an important cause of decoherence 12,13 . q (red) or the effective charge of the traps q * (blue) for the estimation. The diameters and nNWs of the devices included in these results are available in Table S1. For JL devices, q * is estimated with tacc = 0; and for SB devices, q * is estimated with tacc = 1 nm.

Figure 1 .

 1 Figure 1. (a) Schematics of the JL and SB structures. (b) Ids vs. Vg and gm vs. Vg curves of two JL

Figure 2 (

 2 Figure 2 (a) RTS vs. time trace of the corresponding device at Vg = -1.39 V and Vds = -0.1 V. q * /q

Figure 3 .

 3 Figure 3. (a) Power spectrum densities of a JL device (16 nm, nNWs=16) measured at different Vg

Acknowledgement: This work has been supported by the ANR SIBI and ANR LEGO (ANR-18-CE24-0005-01) and the LAAS-CNRS micro and nanotechnology platform, a member of the Renatech French national network.