LEAST-SQUARES APPROXIMATION OF RANDOM VARIABLES BY STOCHASTIC INTEGRALS

Transcription

1 LEAST-SQUARES APPROXIMATION OF RANDOM VARIABLES BY STOCHASTIC INTEGRALS CHUNLI HOU Nomura Securities International 2 World Financial Center, Building B New York, NY 1281 chou@us.nomura.com IOANNIS KARATZAS Departments of Mathematics and Statistics, 619 Math. Bldg. Columbia University New York, NY 127 ik@math.columbia.edu November 2, 24 Abstract This paper addresses the problem of approximating random variables in terms of sums consisting of a real constant and of a stochastic integral with respect to a given semimartingale X. The criterion is minimization of L 2 distance, or least-squares. This problem has a straightforward and well-known solution when X is a Brownian motion or, more generally, a square-integrable martingale, with respect to the underlying probability measure P. We address the general, semimartingale case by means of a duality approach; the adjoint variables in this duality are signed measures, absolutely continuous with respect to P, under which X behaves like a martingale. It is shown that this duality is useful, in that the value of an appropriately formulated dual problem can be computed fairly easily; that it has no gap i.e., the values of the primal and dual problems coincide; that the signed measure which is optimal for the dual problem can be easily identified whenever it exists; and that the duality is also strong, in the sense that one can then identify the optimal stochastic integral for the primal problem. In so doing, the theory presented here both simplifies and extends the extant work on the subject. It has also natural connections and interpretations in terms of the theory of variance-optimal and mean-variance efficient portfolios in Mathematical Finance, pioneered by H. Markowitz and then greatly extended by H. Föllmer, D. Sondermann and most notably M. Schweizer. Paper dedicated to Professor K. Itô on the occasion of his 88 th Birthday. This work was supported in part by the National Science Foundation under Grant NSF-DMS

2 1 INTRODUCTION Suppose we are given a square-integrable, d dimensional process X = {Xt; t T } defined on the finite time-horizon [, T ], which is a semimartingale on the filtered probability space Ω, F, P, F = {Ft} t T. How closely can we approximate in the sense of leastsquares a given, square-integrable and FT measurable random variable H, by a linear combination of the form c + T ϑ dx? Here c is a real number and ϑ a predictable d dimensional process for which the stochastic integral ϑ dx d i=1 ϑ idx i is welldefined and is itself a square-integrable semimartingale. In other words, if we denote by Θ the space of all such processes ϑ, how do we compute V c = T 2 inf E H c ϑ dx 1.1 ϑ Θ if c R is given and we have the freedom to choose ϑ over the class Θ as above? How do we find T 2 V = inf E H c ϑ dx = inf V c 1.2 c,ϑ R Θ c R when we have the freedom to select both c and ϑ? And how do we characterize, or even compute, the process ϑ c and the pair ĉ, ˆϑ that attain the infimum in 1.1 and 1.2, respectively, whenever these exist? To go one step further: How does one minimize the variance T Var H ϑ dx 1.3 over all ϑ Θ as above? Or even more interestingly, how does one minimize the variance Var H T ϑ dx [ ] T over ϑ Θ with E ϑ dx = µ 1.4 for some given µ R? Questions such as 1.3 and 1.4 can be traced back to the pioneering work of H. Markowitz 1952, 1959, and have been studied more recently by Föllmer & Sondermann 1986, Föllmer & Schweizer 1991, Duffie & Richardson 1991, Schäl 1992, in the modern context of Mathematical Finance. Most importantly, problems have received an exhaustive and magisterial treatment in a series of papers by Schweizer 1992, 1994, 1995.a,b, 1996 and his collaborators cf. Rheinländer & Schweizer 1997, [Ph.R.S.] 1998, [DMSSS] 1997, as well as Hipp 1993, [G.L.Ph.] 1996, Laurent & Pham 1999, Grandits 1999, Arai 22. In this context, the components X i, i = 1,..., d of the semimartingale X are interpreted as the discounted stock-prices in a financial market, and H as a contingent claim, or liability, that one is trying to replicate as faithfully as possible at time T, starting with initial capital c and trading in this market. Such trading is modelled by the predictable portfolio process ϑ, whose component ϑ i t represents the number of shares being held at time t in the i th stock, for i = 1,..., d. Then T ϑ dx d T i=1 ϑ isdx i s corresponds to the discounted gains from trading accrued by the terminal time T, with 2

3 which one tries to approximate the contingent claim H, and one might be interested in minimizing the variance of this approximation over all admissible portfolio choices problem of 1.3, or just over those portfolios that guarantee a given mean-rate-of-return problem of 1.4. It turns out that solving the problem of 1.1 provides the key to answering all these questions. For instance, if ϑ c attains the infimum in 1.1 and ĉ argmin c R V c, then ˆϑ ϑ ĉ is optimal for the problem of 1.3; the pair ĉ, ˆϑ is optimal for the problem of 1.2; and the process ϑ c µ with c µ = E[πH] EH + µ / E[π1] 1 is optimal for the problem of 1.4. Here π denotes the projection operator { from the Hilbert space L 2 P } T onto the orthogonal complement of its linear subspace ϑ dx / ϑ Θ. The problem of 1.1 has a very simple solution, if X is a square-integrable martingale; then every H as above has the so-called Kunita-Watanabe decomposition H = EH + T ζ H dx + L H T, where ζ H Θ and L H is a square-integrable martingale strongly orthogonal to ϑ dx for every ϑ Θ. Then the infimum in 1.1 is computed as V c = E[H] c 2 +E L H T 2 and is attained by ζ H Θ, which also attains the infimum V = E L H T 2 in 1.2. In order to deal with a general semimartingale X we develop a simple duality approach, which in a sense tries to reduce the problem to the easy martingale case just described. This approach is the main contribution of the present paper. The dual or adjoint variables in this duality are signed measures Q, absolutely continuous with respect to P and with dq/dp L 2 P, under which X behaves like a martingale Definition 2.1 and Remark 2.2. A simple observation, described in , leads to a dual maximization problem. The resulting duality is useful because, as it turns out, the dual problem is relatively straightforward to solve Proposition 3.1; its value is easily computed as E[π 2 H c] and coincides with the value V c of the original problem 1.1, so there is no duality gap ; and furthermore the duality is strong, in that one can identify the optimal integrand ϑ c of 1.1 rather easily, under suitable conditions Theorem 4.1 and Remark 4.1. Several examples are presented in Section 5. We follow closely the notation and the setting of Schweizer 1996, our great debt to which should be clear to anyone familiar with this excellent work. Indeed, the present paper can be considered as complementing and extending the results of this work, by means of our simple duality approach. 2 THE PROBLEM On a given complete probability space Ω, F, P equipped with a filtration F = {Ft ; t T } that satisfies the usual conditions, consider a process Xt = X + Mt + Bt, t T, 2.1 defined on the finite time-horizon [, T ] and belonging to the space S 2 S 2 P of squareintegrable d-dimensional semimartingales. This means that each X i is in L 2 Ω, F, P ; 3

4 that each M i belongs to the space M 2 P of square-integrable F-martingales with M i = and RCLL paths; and that we have B i = A + i A i, where A± i are increasing, right-continuous and predictable processes with A ± i = and E A ± i T 2 <, for every i = 1,, d. We denote by Θ the space of good integrands for the square-integrable semimartingale X = {Xt, t T }, namely, those F predictable processes whose stochastic integrals with respect to X are themselves square-integrable semimartingales: { Θ = ϑ LX / } G ϑ ϑ dx S 2 P. 2.2 Here LX stands for the space of all R d -valued and predictable processes, whose stochastic integrals G t ϑ = t ϑ s dxs = with respect to X are well-defined. d i=1 t ϑ i s dx i s, t T 2.3 Suppose now we are given a random variable H in the space L 2 P L 2 Ω, FT, P. The following problem will occupy us in this paper. Problem 2.1. Given H L 2 P, compute V = inf c,ϑ R Θ E H c G T ϑ and try to find a pair ĉ, ˆϑ R Θ that attains the infimum, if such a pair exists. In other words, we are looking to find the least-squares approximation of H, as the sum of a constant c R and of the stochastic integral G T ϑ, for some process ϑ Θ. This problem has a rather obvious solution, if it is known that the random variable H is of the form H = h + G T ζ H 2.5 for some h R and ζ H Θ; because then we can take ĉ h, ˆϑ ζ H, and deduce that V = in 2.4. Now it is a classical result e.g. Karatzas & Shreve 1991, pp for a proof that every H L 2 P can be written in the form 2.5, in fact with h = EH, if X is Brownian motion and if F is the augmentation of the filtration F X generated by X itself. One can then also describe the integrand ζ H in terms of the famous Clark 197 formula, under suitable conditions on the random variable H L 2 P L 2 Ω, F X T, P. Thus, in this special case, we can take ĉ = EH, ˆϑ = ζ H, and have V = in 2.4. A little more generally, suppose that X M 2 P is a square-integrable martingale i.e., X = and A in 2.1. Then again it is well-known that every H L 2 P admits the so-called Kunita-Watanabe 1967 decomposition H = h + G T ζ H + L H T 2.6 for h = EH, some ζ H Θ, and some square-integrable martingale L H M 2 P which is strongly orthogonal to G ϑ for every ϑ Θ; in particular, E [ L H T G T ϑ ] =, ϑ Θ

5 Then E H c G T ϑ 2 = h c 2 + E G T ζ H ϑ 2 + E L H T 2, and it is clear that Problem 2.1 admits again the solution ĉ = EH, ˆϑ = ζ H, but now with V = E L H T 2 in 2.4. What happens for a general, square-integrable semimartingale X S 2 P? In view of the above discussion it is tempting to try and reduce this general problem to the case where X is a martingale. This can be accomplished by absolutely continuous change of the probability measure P. We formalize this idea as in Schweizer Definition 2.1. A signed measure Q on Ω, F is called a signed Θ-martingale measure, if QΩ = 1, Q P with dq/dp L 2 P, and [ ] dq E dp G T ϑ =, ϑ Θ. 2.8 We shall denote by P s Θ the set of all such signed Θ-martingale measures, and introduce the closed, convex set D = {D L 2 P / D = dq/dp, some Q P s Θ} = {D L 2 P / ED = 1 and EDG T ϑ =, ϑ Θ}. 2.9 We shall assume from now onwards, that P s Θ equivalently, D =. 2.1 Remark 2.1 : The linear subspace G T Θ = { T ϑ s dxs / ϑ Θ} of the Hilbert space L 2 P is not necessarily closed for a general semimartingale X it is, if B in 2.1, or equivalently if X is a square-integrable martingale, since then the stochastic-integral of 2.3 is an isometry. For a semimartingale X with continuous paths, necessary and sufficient conditions for the closedness of G T Θ have been provided by Delbaen, Monat, Schachermayer, Schweizer & Stricker [DMSSS] 1997; see also Theorem 2 in Rheinländer & Schweizer 1997, as well as Corollary 4 in Pham, Rheinländer & Schweizer [Ph.R.S.] 1998 and section 5, equation 5.11 of the present paper, for sufficient conditions. As noted by W. Schachermayer see Schweizer 1996, p. 21; Lemma 4.1 in Schweizer 21, the assumption 2.1 is equivalent to the requirement that { the closure in L 2 P of G T Θ does not contain the constant 1 On the other hand, the orthogonal complement }. 2.1 G T Θ = { D L 2 P / EDG T ϑ =, ϑ Θ }

6 of G T Θ is a closed linear subspace of L 2 P, and its orthogonal complement G T Θ is the smallest closed, linear subspace of L 2 P that contains G T Θ. Clearly, G T Θ includes the set D of 2.9, and the requirement 2.1 amounts to 1 / G T Θ. 2.1 Remark 2.2 : The notion of signed Θ-martingale measure in Definition 2.1 depends on the space Θ itself, as well as on the definition of the stochastic integral G T ϑ, ϑ Θ. In many cases of interest, though, every Q P s Θ belongs also to the space P 2 sx of signed martingale measures for X, namely those signed measures Q on Ω, F with Q P, dq/dp L 2 P, QΩ = 1 and [ dq E dp Xt Xs ] Fs =, a.s for any s t T. If Q is a true probability measure, as opposed to a signed measure with QΩ = 1, then 2.12 amounts to the martingale property of X under Q. See Müller 1985, Lemma 12b in Schweizer 1996, as well as conditions and the paragraph immediately following them in the present paper. In addition to Problem 2.1, it is useful to consider also the following question, which is interesting in its own right. Problem 2.2. Given H L 2 P and c R, compute V c = 2 inf E H c G T ϑ 2.13 ϑ Θ and try to find ϑ c Θ that attains the infimum in 2.13, if such exists. Clearly, inf c R V c coincides with the quantity V of 2.4; and if this last infimum is attained by some ĉ R, then the pair ĉ, ϑ ĉ attains the infimum in 2.4. In the next Sections we shall try to solve Problems 2.2 and 2.1 using very elementary duality ideas. In this effort, the elements of the set D in 2.9 will play the role of adjoint or dual variables. For the duality methodology to work in any generality it is critical to allow, as we did in Definition 2.1, for signed measures Q with QΩ = 1, as opposed to just standard probability measures. 3 THE DUALITY. The duality approach to Problem 2.2 is simple, and is based on the elementary observation [ H x 2 + yx ] = H H y/2 2 + y H y/2 3.1 min x R = yh y 2 /4, y R. The key idea now, is to read 3.1 with x = c + G T ϑ, y = 2kD for given c R and arbitrary ϑ Θ, D D as in 2.2 and 2.9, and with arbitrary k R, to obtain H c G T ϑ 2 + 2kD c + G T ϑ 2kDH k 2 D

7 Note also that 3.2 holds as equality for some ϑ ϑ c Θ, D D c D and k k c R, if and only if c + G T ϑ c = H k c D c. 3.3 Now let us take expectations in 3.2 to obtain, in conjunction with the properties of 2.9: E H c G T ϑ 2 k 2 ED 2 + 2k [EDH c], 3.4 for every k R, D D and ϑ Θ. Clearly, ED 2 1 = VarD, D D 3.5 and the mapping k k 2 ED 2 + 2k [EDH c] attains over R its maximal value EDH c 2 /ED 2 at the point k D,c Thus, we obtain from 3.4 the inequality V c = EDH c ED = inf E H c G T ϑ 2 ϑ Θ [ sup sup k 2 ED 2 + 2k EDH c ] 3.7 D D k R EDH c 2 = sup D D ED 2 =: Ṽ c, which is the basis of our duality approach. Here V c is the value of our original primal optimization Problem 2.2, whereas Ṽ c is the value of an auxiliary dual optimization problem. This kind of duality is useful, only if the dual problem is easier to solve than the primal Problem 2.2 and if there is no duality gap i.e., equality holds in 3.7, so that by computing the value of the dual problem we also compute the value of the primal. Both these features hold for our setting, as we are about to show. Furthermore, the duality is strong, in the sense that we can identify an optimal Dc D for the dual problem, namely E D 2 c H c Ṽ c = E D 3.8 c 2 for all but a critical value of the parameter c, and then obtain from this an optimal process ϑ c for the primal problem via 3.3. In order to make headway with this program, let us start by introducing the projection operator π : L 2 P G T Θ with the property In particular, E [H πh D] =, H L 2 P, D G T Θ. 3.9 E [H 1 πh 1 πh 2 ] =, H 1, H 2 L 2 P, 3.9 and from 3.9 and 2.1 we have E[π1] = E[π 2 1] >

8 Proposition 3.1. The value of the dual problem in 3.7, namely can be computed as Ṽ c = The supremum in 3.11 is attained by sup D D EDH c 2 ED 2, 3.11 Ṽ c = E [ π 2 H c ], c R it is not attained for c = ĉ. D c = πh c E [πh c], c ĉ = E[πH] E[π1] ; 3.13 For every c ĉ, we shall call the random variable D c D of 3.13 the density of the dual-optimal signed martingale measure in P s Θ, namely Q c A = D c dp, A F Remark 3.1 : Suppose that for some h R we have A EDH = h, D D For instance, this is the case when H is of the form 2.5. Then the dual value function of 3.11 becomes h c 2 Ṽ c = inf D D ED and, for c h, the dual-optimal Dc of 3.13 coincides with D = arg min D D ED2 = π1 E[π1] = E D 2 + R D 3.17 for some R G T Θ, and E D 2 = 1/E[π1] 1, as we shall establish below. Following Schweizer 1996, we shall call D the density of the variance-optimal signed martingale measure QA = D dp, A F 3.18 A in P s Θ. This terminology should be clear from 3.5 and the definition in Proof of Proposition 3.1 : For every D D, we have EDH c = E[DH c] = E[D πh c] thanks to 3.9. Thus, from the Cauchy-Schwarz inequality, EDH c 2 = E [D πh c] 2 ED 2 E [ π 2 H c ], which implies Ṽ c E [ π 2 H c ]. 8

9 Now these last two inequalities are valid as equalities, if and only if we can find D c D of the form D c = const πh c, 3.13 where the constant has to be chosen so that E D c = 1. This is impossible to do if E[πH c] = E[πH c π1] is equal to zero, i.e., if c = ĉ as in 3.13; in other words, the supremum of 3.11 cannot be attained in this case. But for c ĉ, the normalizing constant in 3.13 can be taken as 1/E[πH c], leading to the expression of 3.13 and to 3.12 as well. It remains to show that 3.12 holds even for c = ĉ. For this, let c n = c 1/n, n N and so that E D 2 cn H c ϕ n = πh cn, ϕ = πh c, Dcn = ϕ n Eϕ n D E D 2 c n = = E D 2 cn H c n 1/n E D c 2 n E D 2 cn H c n E D 2 c n [ ] + 1/n2 E D c 2 n 2 E Dcn H c n n E D c 2 n = E [ π 2 H c n ] + 1/n2 E D 2 c n 2 n E[ πh c n ] = Eϕ 2 n + 1/n2 E D 2 c n 2 n Eϕ n E ϕ 2 = E [ π 2 H c ] as n. We have used the inequality < 1/E D2 cn 1; the facts ϕ n ϕ = π1/n a.s., ϕ n ϕ + π1 L 2 P ; the Dominated Convergence Theorem; and the observation that, for c ĉ, we have [ ] [ ] E Dc H c E Dc πh c E = D2 c E 3.19 D2 c = E [ π 2 H c ] E[πH c]2 E [πh c] E [π 2 = E[πH c] H c] from 3.9, Proof of 3.17 : For any D D, we have 1 = ED 1 2 = ED π1 2 E D 2 E [ π 2 1 ], from 3.9 and the Cauchy-Schwarz inequality. Equality holds if and only if D = D = const π1, 9

10 and the normalizing constant has to be chosen so that E D = 1, namely, equal to 1/E[π1]. We conclude that D = π1/e[π1] satisfies E D 2 1 E [π 2 1] = 1 E [π1] = E D 2, D D. 3.2 On the other hand, since 1 = π1 + η for some η G T Θ, 3.21 we have D = 1 η/e[π1] = E D 2 + R, with R = η/e[π1]. Remark 3.2 : If G T Θ is closed in L 2 P, then 3.21 becomes 1 = π1 + G T ξ 1, for some ξ 1 Θ 3.21 and 3.17, 3.2 give D = E D 2 + G T ξ, with ξ = E D2 ξ 1 Θ RESULTS We are now in a position to use the duality developed in the previous section in order to provide solutions to Problems 2.1 and 2.2. First, a lemma from Schweizer 1996, pp ; we provide the proof for completeness. Lemma 4.1. Suppose that the infimum in 2.13 is attained by some ϑ c Θ. Then this process satisfies E [H c G T ϑ c] = E DH c E D and E [H c G T ϑ c] 2 in the notation of 3.9 and Proof of 4.1 : fε = c2 2c E DH E D 2 + E [ π 2 H ], 4.2 The assumption implies that, for any given ξ Θ, the function = E [H c G T = ε 2 E [ G 2 T ξ ] 2ε E ] 2 ϑ c + εξ [H c G T ϑ c G T ξ ] + V c attains its minimum over R at ε =. This gives f =, or equivalently E [H c G T ϑ c ] G T ξ =, ξ Θ

11 Let us also notice that the mapping since we have D E DD is constant on D, 4.4 E D2 E DD = E [ D D D ] = E [ π1 D D ] /E [π1] = E D D /E [π1] = thanks to 3.9. Now denote γ = E [ H c G T ϑ c ]. If γ =, then the random variable D 1 = D + H c G T ϑ c belongs to D by virtue of 4.3, and 4.4 implies = E [ D D1 D ] [ = E D H c G T ϑ c] = E DH c, [ so that 4.1 holds. If γ, then D 2 = H c GT ϑ c ] /γ is in D, and by 4.4 once again: E D2 = E 1 DD2 = E DH c, γ and so 4.1 holds in this case too. Proof of 4.2 : From 4.3, the random variable H c G T ϑ c belongs to the closed subspace G T Θ of 2.11, so we have E [H c G T ϑ c] 2 = E [H c G T ϑ c] [H πh + πh c G T ϑ c] = E [H c G T ϑ c] [πh c ] = E [ π 2 H ] c E [πh] c E DH c E D2 thanks to 4.3, 3.9 and 4.1. The equation 4.2 now follows from E DH = E D2 E [πh]. 4.5 In order to check 4.5, recall 3.17, 3.2 and use 3.9 repeatedly, to wit: E DH E D2 = E [Hπ1] = E [πhπ1] = E [πh]

12 Theorem 4.1. i Suppose that there exists some ϑ c Θ which attains the infimum in Then this ϑ c satisfies H c G T ϑ c = πh c, 4.7 and there is no duality gap in 3.7, namely V c = Ṽ c = E [ π 2 H c ] E DH c = E D E [ π 2 H ] EπH2 E [π1]. 4.8 ii Conversely, suppose there exists some ϑ c Θ that satisfies 4.7; then this ϑ c attains the infimum in 2.13, and the equalities of 4.8 hold. Proof of 4.8, Part i : Under the assumption of i, we claim that V c = E H c G T ϑ c 2 E DH 2 c = E D + E [ π 2 H ] EπH 2 E D = E [ π 2 H c ] = Ṽ c, which clearly proves 4.8 in light of the last equality in 3.2. Indeed, the first equality in 4.9 holds by assumption, whereas the second is a consequence of 4.2, 4.5. The third equality is a consequence of 4.6, 3.2 and 4.2, thanks to the simple computation E [ π 2 H c ] = E πh c π1 2 Finally, the last equality in 4.9 is just = E [ π 2 H ] + c 2 E [ π 2 1 ] 2c E [π1πh] = E [ π 2 H ] + c2 2c E DH E D2. Proof of 4.7, Part i; c ĉ : Let us write 3.2 with ϑ ϑ c, D D c as in 3.13, and k k c = k Dc,c = E Dc H c E D2 c as in 3.6: namely, H c G T ϑ c 2 + 2kc Dc c + G T ϑ c 2 2k c Dc H k c Dc, a.s. 4.1 Taking expectations in 4.1, and recalling the optimality of ϑ c as well as Proposition 3.1, we obtain 12

13 V c = E H c G T ϑ c 2 kc 2 E D2 c + 2kc E Dc H c E Dc H 2 c = E = Ṽ c. D2 c 4.11 But from 4.8 we know that 4.11 actually holds as equality, which means that the lefthand side and the right-hand side of 4.1 have the same expectation. In other words, 4.1 must hold as equality, which we know happens only if 3.3 holds, namely only if H c G T ϑ c = k c Dc = E Dc H c E D2 c πh c E [πh c] = πh c, holds a.s., thanks to Proof of 4.7, part i; c = ĉ : with L = In this case we shall need a new kind of duality, namely { L G T Θ / } EL = 4.12 replacing the space D of 2.9; the elements of L will be the dual adjoint variables in this new duality. We begin by writing 3.1 with x = c + G T ϑ, y = 2L for arbitrary ϑ Θ, L L: H c G T ϑ 2 + 2L c + G T ϑ 2LH L 2, 4.13 with equality if and only if holds a.s. Taking expectations in 4.13, we obtain H c G T ϑ = L 4.14 E H c G T ϑ 2 E [ 2LH c L 2] = E [ 2L πh c L 2] 4.15 = E [ π 2 H c ] E L πh c 2. This suggests that we should read with ϑ ϑ c, the element of Θ that attains the infimum in 2.13 and is thus optimal for Problem 2.2, and L L = πh c, noting that E L = since c = ĉ = E [πh] /E [π1]. With these choices, the left-most member of 4.15 becomes E H c G T ϑ c 2 = V c, whilst its right-most member is E [ π 2 H c ] = Ṽ c. From 4.8 we know that these two quantities are equal, so the two sides of 4.13 have the same expectation. This means that 4.13 must holds as equality, which happens only if 4.14 is valid, namely H c G T ϑ c = πh c, a.s. 13

14 Proof of Part ii : Suppose there exists some ϑ c Θ that satisfies 4.7; then E H c G T ϑ c 2 = E [ π 2 H c ] = Ṽ c from Proposition 3.1. But we also have Ṽ c V c E H c G T ϑ c 2, thanks to 3.7 and In other words, these last two inequalities are valid as equalities, ϑ c attains the infimum in 2.13, and 4.8 holds. Remark 4.1 : The case of G T Θ closed. If G T Θ is closed in L 2 P, then the infimum in 2.13 is attained, as was assumed in Lemma 4.1 and in Theorem 4.1i. In this case we have of course G T Θ = G T Θ, and every H L 2 P admits a decomposition of the form H = πh + G T ξ H for some ξ H Θ In particular, there exists ξ 1 Θ so that 3.21 holds, and thus H c πh c = [H πh] c [1 π1] = G T ξ H c ξ 1. Comparing this expression with 4.7, we conclude that 4.7 is satisfied with the choice ϑ c = ξ H c ξ According to Theorem 4.1ii, the process ϑ c Θ of 4.17 is then optimal for Problem 2.2, and 4.8 holds. Example 4.1. Föllmer-Schweizer decomposition. The assertion at the end of Remark 4.1 remains valid even if G T Θ is not closed in L 2 P, provided that 3.21 and 4.16 still hold. Consider, for example, the case of a semimartingale X S 2 P and of a random variable H L 2 P which admits the so-called Föllmer-Schweizer decomposition ; this means that H can be written in the form H = h + G T ζ H + L H T of 2.6, for h = EH, some ζ H Θ, and some martingale L H M 2 P that satisfies the property 2.7. Suppose that 3.21 is also satisfied; then πh = hπ1 + L H T and we have H πh = h 1 π1 + G T ζ H = G T hξ 1 + ζ H, so we may take ξ H hξ 1 + ζ H in 4.16 and ϑ c h c ξ 1 + ζ H 4.17 in 4.7. The process ϑ c of 4.17 is then optimal for the Problem 2.2, i.e., attains the infimum in 2.13, which can be readily computed as V c = h c 2 E[π1] + E L H T 2. This simple result may be compared with Theorem 3 and Proposition 18 in Schweizer We are now in a position to discuss the solution of Problem 2.1 as well. 14

15 Theorem 4.2. Suppose that G T Θ is closed in L 2 P. Then the value of Problem 2.1 is given as V = V ĉ = E [ π 2 H ] E [πh] E [π1] with the notation ĉ = E [πh] E [π1] = E DH = E d Q dp H 4.19 of Furthermore, the infimum in 2.4 is attained by the pair ĉ, ˆϑ with ĉ as in 4.19 and with ˆϑ = ϑ ĉ = ξ H ĉ ξ Proof : Immediate from Theorem 4.1 and Remark 4.1, when it is observed that the number ĉ of 4.19 minimizes the expression of 4.8 over c R. Note that when the signed measure Q of 3.8 is a probability measure i.e., when P [π1 ] = 1, the quantity of 4.19 is just the expectation of the random variable H under the dual-optimal measure Q. Sufficient conditions are spelled out in the next section. Remark 4.2 : Variance Minimization. If G T Θ is closed in L 2 P, then the process ˆϑ of 4.2 also minimizes Var H G T ϑ, over all ϑ Θ This is because for any ϑ Θ, and with c ϑ = E [H GT ϑ], we have: from Theorem 4.2. Var H G T ϑ = E H c ϑ G T ϑ 2 ˆϑ 2 E H ĉ G T = Var H G T ˆϑ, More generally, for any given c R, the process ϑ c Θ of 4.17 has the meanvariance efficiency property { Var H G T ϑ c } Var H G T ϑ, for any ϑ Θ that satisfies E [H G T ϑ] = E [ H G T ϑ c ] Indeed, let µ c = E [ H GT ϑ c ] and observe that, for any ϑ Θ with E [H G T ϑ] = µ c, we have Var H G T ϑ = Var H c G T ϑ = E H c G T ϑ 2 µ c c E H c G T ϑ c E H c G T ϑ c = Var H c G T ϑ c = Var H G T ϑ c. Remark 4.3 : Mean-Variance Frontier. Suppose that G T Θ is closed in L 2 P, that we have P [ π1 Eπ1 ] > ; this implies E D 2 > 1 in 3.5, hence E[π1] < 1. For some given m R, consider the following problem: 15

16 { To minimize the variance Var H GT ϑ, over ϑ Θ with E [H G T ϑ] = m. } 4.23 In view of the property 4.22, it suffices to show that we can find c c m such that E [H G T ϑ c] = m Then the solution of the problem 4.23 will be given by ϑ c Θ as in 4.17, with c c m. Now from 4.1 we have E [ H G T ϑ c ] = c + E DH c, so that 4.24 amounts to c = c m = m E D2 E DH E D2 1 E D2 = m E [πh] 1 E [π1] thanks to 4.6 and 3.2. We take these two Remarks 4.2, 4.3 from Schweizer 1994, Remark 4.4 : Failure of Condition 2.1. If the condition 2.1 fails, that is, if we have ED = for every D G T Θ, then it can be seen easily that Problem 2.1 has the solution V = E [ π 2 H ] = E H ĉ G T ˆϑ 2 with ĉ = and ˆϑ ξ H as in 4.16, at least when G T Θ is closed., 5 A MATHEMATICAL FINANCE INTERPRETATION The Problems 2.1, 2.2 have an interesting interpretation in the context of Mathematical Finance, when one interprets the components X i of the semimartingale in 2.1 as the discounted prices of several risky assets in a financial market. In this context, ϑ i t represents the number of shares in the corresponding i th asset, held by an investor at time t [, T ]. The resulting process ϑ Θ stands then for the investor s self-financing trading strategy, and G t ϑ = t ϑ s dxs = d t i=1 ϑ is dx i s for the discounted gainsfrom-trade associated with the strategy ϑ by time t. Suppose now that the investor faces a contingent claim liability H at the end T of the time-horizon [, T ]. Starting with a given initial capital c, and using a trading strategy ϑ Θ, the investor seeks to replicate this contingent claim H as faithfully as possible, in the sense of minimizing the expected squared-error loss E H c G T ϑ 2. This leads us to Problem 2.2. When the determination of the right initial capital c is also part of the problem, one is led naturally to the formulation of Problem 2.1. Similarly, one may consider minimizing the variance of the discrepancy H c G T ϑ over all trading strategies ϑ Θ problem of 4.21, or just over those strategies that guarantee a given mean-gains-from-trade level E [G T ϑ] = EH m problem of If one decides to stick with this interpretation, it makes sense to ask whether the model for the financial assets represented by 2.1 admits arbitrage opportunities; these are trading strategies ϑ Θ with P [G T ϑ ] = 1 and P [G T ϑ > ] >. To this effect, let A be an 16

17 increasing, predictable and RCLL process with values in [, and A =, M i A for i = 1,, d, and suppose that the processes M = M 1,..., M d and B = B 1,..., B d, in the decomposition X = X+M +B of 2.1 for the semimartingale X = X 1,..., X d, satisfy B i M i 5.1 B i t = M i, M j t = t t γ i s das, t T 5.2 σ ij s das, t T 5.3 for i = 1,, d and j = 1,, d. Here γ = γ 1,, γ d and σ = {σ ij } 1 i,j d are suitable predictable processes that satisfy σtλt = γt, a.e. t [, T ] 5.4 almost surely, for some λ = λ 1,, λ d in the space L 2 M = { / ϑ : [, T ] R d predictable T T 2 } E ϑ sσsϑs das = E ϑ s dms <. 5.5 Following Schweizer 1996, we shall refer to as the structure conditions on the semimartingale X. Under these conditions, it can be shown that the semimartingale X does not admit arbitrage opportunities, and that we have equality P s Θ = P 2 sx in Remark 2.2 cf. Ansel & Stricker 1992; Schweizer 1995; and Schweizer 1996, Lemma 12. If, in addition, X has continuous paths, then it can be shown that the variance-optimal martingale measure Q of 3.18 is nonnegative, namely a probability measure: P [ D ] = 1 and QΩ = E D = in 3.17, This Q is in fact equivalent to P i.e., P [π1 > ] = 1, under the extra assumption { X is a Q local martingale under some probability measure Q P with dq/dp L 2 P }. 5.7 cf. Schweizer 1996, Theorem 13; Delbaen & Schachermayer 1996, Theorem 1.3. This condition 5.7 also implies that cf. [DMSSS], Lemma 3.5. the mapping ϑ G T ϑ is injective;

18 Always under the structure conditions on the semimartingale X of 2.1, consider now the so-called mean-variance tradeoff process t t Kt = λ s dbs = λ sσsλs das = λ dm t, t T. If this process is P -a.s. bounded, then in the notation of 2.2, 5.5 and Θ = L 2 M 5.9 G T Θ is closed in L 2 P 5.1 [Ph.R.S.], Corollary 4 and below; Schweizer 1996, Lemma 12. See also [DMSSS] where conditions both necessary and sufficient for 5.1 are presented. Remark 5.1 : In the one-dimensional case d = 1, the structure conditions are satisfied if there exists a process λ L 2 M with Xt = X + Mt + t λs d M s, t T In this case, the mean-variance tradeoff process of 5.9 takes the form Kt = t λ 2 s d M s Remark 5.2 : Suppose that X has continuous paths, and that 5.11 and 5.7 hold. If the random variable H L 2 P is of the form H = h + G T ζ H in 2.5, then H πh = G T hξ 1 + ζ H and the injectivity property 5.8 allows us to make the identifications in 4.16 and 4.17, respectively. ξ H = hξ 1 + ζ H, ϑ c = h c ξ 1 + ζ H 5.13 More generally, under the assumptions of Remark 5.2 but now for any H L 2 P, we have the Kunita-Watanabe decomposition under the variance-optimal probability measure Q of 3.18, namely Ẽ[H Ft] = h + G t ζ H + L H t, t T 5.14 cf. Ansel & Stricker 1993, or Theorem 3 in Rheinländer & Schweizer Here Ẽ denotes expectation with respect to the probability measure Q, h = ẼH = E DH, ζ H Θ, and L H S 2 P is a Q-martingale with L H = and LH, X i, i = 1,, d

19 On the other hand, since L H T belongs to the space L 2 P, we also have its decomposition L H T = G T ϑ H + π L H T 5.16 from the closedness of G T Θ, where ϑ H Θ is such that [ LH E T G T ϑ ] H G T ϑ =, ϑ Θ Back in 5.14, this gives H = h + G T ζ H + ϑ H + π L H T, so that H c πh c = h c1 π1 + G T ζ H + ϑ H = G T h cξ 1 + ζ H + ϑ H from Again, the injectivity property 5.8 allows us to make the identification in Finally, let us consider the positive Q-martingale ϑ c = h cξ 1 + ζ H + ϑ H 5.18 Dt = Ẽ[ D Ft] = E D 2 + G t ξ obtained by taking conditional expectations in 3.22 under Q. = E D 2 [ 1 G t ξ 1 ], t T 5.19 We are now in a position to identify the process ϑ H appearing in 5.16, 5.18 and state the following result, which simplifies and generalizes Theorems 5, 6 of Rheinländer & Schweizer Theorem 5.1. Suppose that the semimartingale X S 2 P has continuous paths, and that 5.7, 5.11 hold. Then the optimal process ϑ c Θ for Problem 2.2 takes the form [ ] ϑ c = ζ H + E DH c + E D 2 d L H s ξ Ds in the notation of 5.14, Sketch of Proof : The Q-martingales Ñt = t 1/ Ds d L H s, t T and D of 5.19 are orthogonal, since Ñ, D = d i=1 ξ i s Ds d L H, X i s from 5.15 and Thus, integration by parts gives L H T = T Ds dñs = T DT ÑT Ñs ξ s dxs = DÑT G T Ñ ξ,

20 and transforms 5.17 into [ E G T ϑ H + Ñ ξ ] G T ϑ ] = [ÑT Ẽ GT ϑ, ϑ Θ But the right-hand-side of 5.22 vanishes, since ] Ẽ [ÑT GT ϑ = Ẽ = Ẽ [ T T d i=1 d L H s Ds d i=1 T ϑ i s dx i s ϑ i s Ds d L H, X i s = thanks to Thus the left-hand-side of 5.22 also vanishes for every ϑ Θ, which suggests ϑ H = Ñ ξ = E D 2 ξ 1 d L H s Ds and leads to 5.2 after substitution into In order to justify the legitimacy of the above argument, particularly the steps leading t to 5.22, one needs to show that sup t T Dt 1/ Ds d L H s belongs to L 2 P ; this is carried out on pp of Rheinländer & Schweizer ] References [1] Ansel, J.-P. & Stricker, C Lois de martingale, densités et décomposition de Föllmer-Schweizer. Ann. Inst. Henri Poincaré 28, [2] Arai, T. 22 Mean-Variance Hedging for Discontinuous Semimartingales. Tokyo J. Mathematics, to appear. [3] Arai, T. 22 Mean-Variance Hedging for General Semimartingales. Preprint, Tokyo University of Science. [4] Clark, J.M.C. 197 The representation of functionals of Brownian motion as stochastic integrals. Annals of Mathematical Statistics 41, Correction, [5] Duffie, D. & Richardson, H.R Mean-variance hedging in continuous time. Annals of Applied Probability 1, [6] Delbaen, F., Monat P., Schachermayer, W., Schweizer, M. & Stricker, C. [DMSSS] 1997 Weighted norm inequalities and hedging in incomplete markets. Finance & Stochastics 1, [7] Delbaen, F. & Schachermayer, W The variance-optimal martingale measure for the continuous processes. Bernoulli 2,

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22 [26] Schweizer, M. 1995b Variance-optimal hedging in discrete time. Mathematics of Operations Research 2, [27] Schweizer, M Approximation pricing and the variance-optimal martingale measure. Annals of Probability 24, [28] Schweizer, M. 21 A guided tour through quadratic hedging approaches. In Handbook of Mathematical Finance: Option Pricing, Interest Rates and Risk Management E.Jouini, J.Cvitanic and M.Musiela, eds., Cambridge University Press. 22