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...nd term are common in robust statistic theory. In contact to optimal B-robustness (see Section 2), here we develop the approach, known in robust statistics as optimal V -robustness, see Hampel et al. =-=[12]-=-. Note that our approach allows incorporating current information on the underlying model, and hence is adaptive. Namely, passing from time value t to t + τ, τ > 0, when more information about market ...

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...länder and Schweizer [33], (RSch hereafter), Pham et al. [31], Gourieroux et al. [11] (GLP hereafter), Laurent and Pham [18]). A stochastic volatility model, proposed by Hull and White [13] and Scott =-=[39]-=-, where the stock price volatility is an random process, is a popular model of incomplete market, where the mean-variance hedging approach can be used (see, e.g., Laurent and Pham [18], Biagini et al....

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... t(n) < · · · < ] → 0, as n → 0, calculate the realization of yield dXs, and then calculate the sum Xs ∑n−1 Sn(t) = j=0 |R t (n) j+1 − R (n)| t j 2 . It is well-known (see, e.g., Lipster and Shiryaev =-=[23]-=-) that Sn(t) P → ∫t 0 σ 2 s ds as n → ∞. Since σ 2 t (ω) = f(Yt) is a continuous process we get where F(t, ω) = ∫t 0 σ 2 F(t + ∆, ω) − F(t, ω) t (ω) = lim , ∆↓0 ∆ σ 2 s(ω)ds. Hence, the realization (y...

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...ion rate in contrast to our consideration, where the model misspecification is due to the volatility parameter. The consideration of misspecified asset price models was initiated by Avellaneda et al. =-=[1]-=-, Avellaneda and Paras [2]. Various authors in different settings attacked the robustness problem. The method used in Section 3 was suggested by Toronjadze [41] for asset price process modelled by the...

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...eveloped in Hampel et al. [12] and in Rieder [34]; the theory of the trend parameter estimates for diffusion process with small noise is developed in Kutoyants [17]; the book of Musiela and Rutkowsky =-=[29]-=- is devoted to the mathematical theory of finance and finally, the general theory of martingales can be found in Jacod and Shiryaev [15].OPTIMAL ROBUST MEAN-VARIANCE HEDGING 9 2. Optimal B-Robust Est...

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...-measurable square-integrable random variable (r.v.), x is an initial investment, Θ is a class of admissible trading strategies, T is an investment horizon. The mean-variance formulation by Markowitz =-=[26]-=-, provides a foundation for a single period portfolio selection (see, also Merton [27]). In recent paper of Li 1991 Mathematics Subject Classification. 60G22, 62F35, 91B28, 62F35, 62M05, 62M09. Key wo...

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... a class of admissible trading strategies, T is an investment horizon. The mean-variance formulation by Markowitz [26], provides a foundation for a single period portfolio selection (see, also Merton =-=[27]-=-). In recent paper of Li 1991 Mathematics Subject Classification. 60G22, 62F35, 91B28, 62F35, 62M05, 62M09. Key words and phrases. Stochastic volatility, small diffusion, robust parameter estimate, op...

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... is a vector of unknown parameters, and ε, 0 < ε ≪ 1, is a small number. Assume that the system (1.2) has an unique strong solution. This model is analogous to the model proposed by Renault and Touzi =-=[32]-=- (RT hereafter). The principal difference is the presence of small parameter ε in our model, which due to the assumption that the volatility of randomly fluctuated volatility process is small (see, al...

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...n. Therefore, the mean-variance hedging is s powerful approach for both above mentioned major problems. The problem (1.1) was intensively investigated in last decade (see, e.g., Dufiie and Richardson =-=[9]-=-, Schwezer [36], [37], [38], Delbaen et al. [8], Monat and Striker [28], Rheinländer and Schweizer [33], (RSch hereafter), Pham et al. [31], Gourieroux et al. [11] (GLP hereafter), Laurent and Pham [1...

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... developed in Kutoyants [17]; the book of Musiela and Rutkowsky [29] is devoted to the mathematical theory of finance and finally, the general theory of martingales can be found in Jacod and Shiryaev =-=[15]-=-.OPTIMAL ROBUST MEAN-VARIANCE HEDGING 9 2. Optimal B-Robust Estimates 2.1. Basic model. CULAN estimates. The basic model of observations is described by the SDE dYs = a(s, Y ; α) ds + ε dws, Y0 = 0, ...

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...iance hedging is s powerful approach for both above mentioned major problems. The problem (1.1) was intensively investigated in last decade (see, e.g., Dufiie and Richardson [9], Schwezer [36], [37], =-=[38]-=-, Delbaen et al. [8], Monat and Striker [28], Rheinländer and Schweizer [33], (RSch hereafter), Pham et al. [31], Gourieroux et al. [11] (GLP hereafter), Laurent and Pham [18]). A stochastic volatilit...

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...2F35, 91B28, 62F35, 62M05, 62M09. Key words and phrases. Stochastic volatility, small diffusion, robust parameter estimate, optimal mean-variance robust hedging. 12 N. LAZRIEVA, T. TORONJADZE and Ng =-=[22]-=- the concept of Markowitz’s mean-variance formulation for finding the optimal portfolio policy and determining the efficient frontier in analytical form has been extended to multiperiod portfolio sele...

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...ade (see, e.g., Dufiie and Richardson [9], Schwezer [36], [37], [38], Delbaen et al. [8], Monat and Striker [28], Rheinländer and Schweizer [33], (RSch hereafter), Pham et al. [31], Gourieroux et al. =-=[11]-=- (GLP hereafter), Laurent and Pham [18]). A stochastic volatility model, proposed by Hull and White [13] and Scott [39], where the stock price volatility is an random process, is a popular model of in...

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...an-variance hedging is s powerful approach for both above mentioned major problems. The problem (1.1) was intensively investigated in last decade (see, e.g., Dufiie and Richardson [9], Schwezer [36], =-=[37]-=-, [38], Delbaen et al. [8], Monat and Striker [28], Rheinländer and Schweizer [33], (RSch hereafter), Pham et al. [31], Gourieroux et al. [11] (GLP hereafter), Laurent and Pham [18]). A stochastic vol...

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...the mean-variance hedging is s powerful approach for both above mentioned major problems. The problem (1.1) was intensively investigated in last decade (see, e.g., Dufiie and Richardson [9], Schwezer =-=[36]-=-, [37], [38], Delbaen et al. [8], Monat and Striker [28], Rheinländer and Schweizer [33], (RSch hereafter), Pham et al. [31], Gourieroux et al. [11] (GLP hereafter), Laurent and Pham [18]). A stochast...

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...r consideration, where the model misspecification is due to the volatility parameter. The consideration of misspecified asset price models was initiated by Avellaneda et al. [1], Avellaneda and Paras =-=[2]-=-. Various authors in different settings attacked the robustness problem. The method used in Section 3 was suggested by Toronjadze [41] for asset price process modelled by the one-dimensional process. ...

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...4]; the theory of robust statistics is developed in Hampel et al. [12] and in Rieder [34]; the theory of the trend parameter estimates for diffusion process with small noise is developed in Kutoyants =-=[17]-=-; the book of Musiela and Rutkowsky [29] is devoted to the mathematical theory of finance and finally, the general theory of martingales can be found in Jacod and Shiryaev [15].OPTIMAL ROBUST MEAN-VA...

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...investigated in last decade (see, e.g., Dufiie and Richardson [9], Schwezer [36], [37], [38], Delbaen et al. [8], Monat and Striker [28], Rheinländer and Schweizer [33], (RSch hereafter), Pham et al. =-=[31]-=-, Gourieroux et al. [11] (GLP hereafter), Laurent and Pham [18]). A stochastic volatility model, proposed by Hull and White [13] and Scott [39], where the stock price volatility is an random process, ...

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...9], Schwezer [36], [37], [38], Delbaen et al. [8], Monat and Striker [28], Rheinländer and Schweizer [33], (RSch hereafter), Pham et al. [31], Gourieroux et al. [11] (GLP hereafter), Laurent and Pham =-=[18]-=-). A stochastic volatility model, proposed by Hull and White [13] and Scott [39], where the stock price volatility is an random process, is a popular model of incomplete market, where the mean-varianc...

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...werful approach for both above mentioned major problems. The problem (1.1) was intensively investigated in last decade (see, e.g., Dufiie and Richardson [9], Schwezer [36], [37], [38], Delbaen et al. =-=[8]-=-, Monat and Striker [28], Rheinländer and Schweizer [33], (RSch hereafter), Pham et al. [31], Gourieroux et al. [11] (GLP hereafter), Laurent and Pham [18]). A stochastic volatility model, proposed by...

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...h above mentioned major problems. The problem (1.1) was intensively investigated in last decade (see, e.g., Dufiie and Richardson [9], Schwezer [36], [37], [38], Delbaen et al. [8], Monat and Striker =-=[28]-=-, Rheinländer and Schweizer [33], (RSch hereafter), Pham et al. [31], Gourieroux et al. [11] (GLP hereafter), Laurent and Pham [18]). A stochastic volatility model, proposed by Hull and White [13] and...

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...semimartingale X, the mean-variance approach suggests to use the quadratic criterion to measure the hedging error, i.e. to solve the mean-variance hedging problem introduced by Föllmer and Sondermann =-=[10]-=-: minimize E ( H − x − ∫T θtdXt ) 2 over all θ ∈ Θ, (1.1) 0 where contingent claim H is a FT-measurable square-integrable random variable (r.v.), x is an initial investment, Θ is a class of admissible...

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...pinion this “double robust” strategy should be more attractive to protect the hedger against the possible errors. The general asymptotic theory of estimation can be found in Ibragimov and Khas’miskii =-=[14]-=-; the theory of robust statistics is developed in Hampel et al. [12] and in Rieder [34]; the theory of the trend parameter estimates for diffusion process with small noise is developed in Kutoyants [1...

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...ipal difference is the presence of small parameter ε in our model, which due to the assumption that the volatility of randomly fluctuated volatility process is small (see, also Sircar and Papanicolau =-=[40]-=-). Thus assumption enables us to use the prices of trading options with short, nearest to the current time value maturities for volatility process filtration and parameter estimation purposes (see bel...

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...ere ϕ = (ϕt)0≤t≤T is a R d -valued, F-predictable process with ∫T 0 ϕ ′ t d〈M〉tϕt < ∞. Let τ be F-stopping time. Denote 〈k ′ · M〉Tτ = 〈k ′ · M〉T − 〈k ′ · M〉τ. Proposition 3.1 (see also Biagini et al. =-=[3]-=-, LLaurent and Pham [18]). 1. Equation (3.13) is equivalent to equation ET(ϕ ′ · M ∗ ) ET(L) = ce 〈k′ ·M〉T , (3.15) where the R d -valued process M ∗ = (M ∗ t )0≤t≤T is given by the relation dM ∗ t = ...

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...latility is an random process, is a popular model of incomplete market, where the mean-variance hedging approach can be used (see, e.g., Laurent and Pham [18], Biagini et al. [13], Mania and Tevzadze =-=[24]-=-, Pham et al. [31]). Consider the stochastic volatility model described by the following system of SDE dXt = Xt dRt, X0 > 0, dRt = µt(Rt, Yt) dt + σ.dw R t , R0 = 0, σ 2 t = f(Yt), dYt = a(t, Yt; α) d...

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... decomposition of r.v. H (see, e.g., Pham et al. [31]) putting ξ H t = ∂v(t, Xt, Yt) ∂x , L H T ∫ = ε 0 T ∂v ∂y (t, Xt, Yt) dw σ t , and calculate ψ H t , LT and V ∗ t using (4.13) and (4.14) of RSch =-=[33]-=-. Thus one get the explicit solution of the mean-variance hedging problem. Finally, here is the short summary of approach: a) Incorporate the robust procedure in statistical analysis of volatility pro...

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...nformation about market prices are available,OPTIMAL ROBUST MEAN-VARIANCE HEDGING 7 becomes smaller, and hence the estimation procedure becomes more precise. In the paper of Runggaldier and Zaccaria =-=[35]-=- the adaptive approach to risk management under general uncertainty (restricted information) was developed. As it is mentioned in this paper there exist a series of investigations dealt with various t...

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... This region shrinks to the estimate α ∗,0 t , as ε → 0. Now if the coefficient a(t, y; α) in (1.2) is such that the solution Y ε t (α) of SDE (1.2) is continuous w.r.t parameter α (see, e.g., Krylov =-=[16]-=-), then the confidence region of parameter α is mapped to the confidence interval for 0 = Y ), Further, by the function f, the latter Y ε ∗ t (α), which shrinks to Yt t (α∗,0 t interval is mapped to t...

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...s(ω)ds. Hence, the realization (yt)0≤t≤T of the process Y can be found by the formula ), 0 ≤ t ≤ T. yt = f −1 (σ2 t More sofisticated methods using the same idea can be found, e.g., in Chesney et al. =-=[5]-=-, Pastorello [30]. We can use the reconstructed sample path (yt), 0 ≤ t ≤ T. to estimate the unknown parameter α in the drift coefficient of diffusion process Y . The second, market price adjusted pro...

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...) process: Yt ε j+1 − Yt ε j = a(tε j, Yt ε j ; α)(tε j+1 − t ε j) + ε(w σ t ε j+1 − wσ t ε j ). Using the data {yt ε j } one can construct the MLE αε t of parameter α, see, e.g., Chitashvili et al. =-=[25]-=-, [26], Lazrieva and Toronjadze [19]. Remember the scheme of construction of MLE. Rewrite the previous AR(1) process, using obvious simple notation, in form Then ∂ ∂y P {Yj+1 ≤ y | Yj} = Yj+1 − Yj = a...

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... equation L ψ,ε t (X; α) = 0, where L ψ,ε t (X; α) is defined by (2.14), ψ ∈ Ψ0. The asymptotic theory of M-estimates for general statistical models with filtration is developed in Chitashvili et al. =-=[7]-=-. Namely, the problem of existence and global behaviour of solutions is studied. In particular, the conditions of regularity and ergodicity type are established, under which M-estimates have a CULAN p...

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...j, Yt ε j ; α)(tε j+1 − t ε j) + ε(w σ t ε j+1 − wσ t ε j ). Using the data {yt ε j } one can construct the MLE αε t of parameter α, see, e.g., Chitashvili et al. [25], [26], Lazrieva and Toronjadze =-=[19]-=-. Remember the scheme of construction of MLE. Rewrite the previous AR(1) process, using obvious simple notation, in form Then ∂ ∂y P {Yj+1 ≤ y | Yj} = Yj+1 − Yj = a(tj, Yj; α)∆ + ε∆w σ j . ( 1 √ exp −...

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...resents the Huber gross error model (as it explain in Remark 2.2). The model of type (1.4) of contamination of measures for statistical models with filtration was suggested by Lazrieva and Toronjadze =-=[20]-=-, [21]. In Section 2 we study the problem of construction of robust estimates for contamination model (1.4). In subsection 2.1 we give a description of the basic model and definition of consistent uni...

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...s the Huber gross error model (as it explain in Remark 2.2). The model of type (1.4) of contamination of measures for statistical models with filtration was suggested by Lazrieva and Toronjadze [20], =-=[21]-=-. In Section 2 we study the problem of construction of robust estimates for contamination model (1.4). In subsection 2.1 we give a description of the basic model and definition of consistent uniformly...

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...he realization (yt)0≤t≤T of the process Y can be found by the formula ), 0 ≤ t ≤ T. yt = f −1 (σ2 t More sofisticated methods using the same idea can be found, e.g., in Chesney et al. [5], Pastorello =-=[30]-=-. We can use the reconstructed sample path (yt), 0 ≤ t ≤ T. to estimate the unknown parameter α in the drift coefficient of diffusion process Y . The second, market price adjusted procedure of reconst...

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...odels was initiated by Avellaneda et al. [1], Avellaneda and Paras [2]. Various authors in different settings attacked the robustness problem. The method used in Section 3 was suggested by Toronjadze =-=[41]-=- for asset price process modelled by the one-dimensional process. As it will be shown in Remark 3.2 below, in simplest case when asset price process is a martingale w.r.t initial measure P, and it is ...