Mean-variance hedging based on an incomplete market with external risk factors of non-Gaussian OU processes

In this paper, we prove the global risk optimality of the hedging strategy of contingent claim, which is explicitly (or called semi-explicitly) constructed for an incomplete financial market with external risk factors of non-Gaussian Ornstein-Uhlenbeck (NGOU) processes. Analytical and numerical examples are both presented to illustrate the effectiveness of our optimal strategy. Our study establishes the connection between our financial system and existing general semimartingale based discussions by justifying required conditions. More precisely, there are three steps involved. First, we firmly prove the no-arbitrage condition to be true for our financial market, which is used as an assumption in existing discussions. In doing so, we explicitly construct the square-integrable density process of the variance-optimal martingale measure (VOMM). Second, we derive a backward stochastic differential equation (BSDE) with jumps for the mean-value process of a given contingent claim. The unique existence of adapted strong solution to the BSDE is proved under suitable terminal conditions including both European call and put options as special cases. Third, by combining the solution of the BSDE and the VOMM, we reach the justification of the global risk optimality for our hedging strategy.


Introduction
In this paper, we justify the global risk optimality of the hedging strategy of contingent claim, which is explicitly constructed for an incomplete market defined on some filtered probability space (Ω, F, {F t } t≥0 , P ). The financial market has d + 1 primitive assets: one bond with constant interest rate and d risky assets. The price processes of the assets are described by a generalized Black-Scholes model with coefficients driven by the market regime caused by 1 Partial results and graphs are briefly summarized and reported in 2012 Spring World Congress of Engineering and Technology. This enhanced version with extension and complete proofs of results is a journal version of the short conference report. leverage effect, etc. The financial market model includes the Barndorff-Nielsen & Shephard (BNS) volatility model proposed by Barndorff-Nielsen and Shephard [3] and further studied in Benth et al. [4], Benth and Meyer-Brandis [5], Lindberg [36], etc. as a particular case. Our model is closely related to the one considered in Delong and Klüppelberg [17]. As pointed out in Barndorff-Nielsen and Shephard [3], these models fit real market data quite well. Nevertheless, such models also induce incompleteness of the financial markets, which means that it is impossible to replicate perfectly contingent claims based on the bond and the d primitive risky assets. A rule for designing a good hedging strategy is to minimize the mean squared hedging error over the setΘ of all reasonable trading strategy processes, where H is a random variable representing the discounted payoff of the claim, D is the discounted price process of d risky assets, v is the initial endowment and T is the time horizon. Mathematically speaking, one seeks to compute the orthogonal projection of H − v on the spaceΘ of stochastic integrals.
To solve the mean-variance hedging problem (1.1), we explicitly construct a trading strategy for the financial market and justify it to be the global risk-minimizing hedging strategy by using the following procedure.
First, we explicitly construct the square-integrable density process of a variance-optimal martingale measure (VOMM) Q * . As a result, the set of equivalent (local) martingale measures with square-integrable densities, i.e., U e 2 (D) ≡ Q ∼ P : dQ dP ∈ L 2 (P ), D is a Q-local martingale (1.2) is nonempty. Hence, our market is arbitrage-free (e.g, Delbaen and Schachermayer [16]). Second, we derive an BSDE with jumps and external random factors of non-Gaussian Ornstein-Uhlenbeck (NGOU) type for the mean value process of the option H (i.e., E Q * [H|F t ]). The unique existence of adapted solution to the BSDE is proved under suitable terminal conditions including both European call and put options as special cases. Third, by combining the solution to the BSDE and the VOMM, we get the optimal hedging strategy for our market.
A closely related (local) risk minimizing problem was initially introduced by Föllmer and Sondermann [20] under complete information, who also suggested an approach for the computation of a minimizing strategy in an incomplete market by extending the martingale approach of Harrison and Kreps [24]. The basic idea of the approach was to introduce a measure of riskiness in terms of a conditional mean square error process where the discounted price process is a square-integrable martingale. Furthermore, the answer to the hedging problem is provided by the Galtchouk-Kunita-Watanabe decomposition of the claim. Then, this concept of local-risk minimization was further extended for the semimartingale case by Föllmer and Schweizer [21], and Schweizer [45,46], where the minimal martingale measure and Föllmer-Schweizer (F-S) decomposition play a central role. Interested readers are referred to Föllmer and Schweizer [22], Schweizer [48] for more recent surveys about (local) risk minimization and mean-variance hedging.
Owing to the fact that one cares about the total hedging error and not the daily profit-loss ratios, the solution with respect to global-risk minimization of the unconditional expected squared hedging error presented in (1.1) was considered (e.g., surveys in Pham [40] and Schweizer [48]). Then, the study on global-risk minimization was further developed by Cȇrný and Kallsen [7], who showed that the hedging model (1.1) admits a solution in a very general class of arbitrage-free semimartingale markets where local-risk minimization may fail to be well defined. The key point of their approach is the introduction of the opportunity-neutral measure P * that turns the dynamic asset allocation problem into a myopic one. Furthermore, the minimal martingale measure relative to P * coincides with the variance-optimal martingale measure relative to the original probability measure P . Recently, to overcome the difficulties appeared in Cȇrný and Kallsen [7] (i.e., a process N appeared in Definition 3.12 is very hard to find and the VOMM Q * in Proposition 3.13 is notoriously difficult to determine), the authors in Jeanblanc et al. [31] developed a method via stochastic control and backward stochastic differential equations (BSDEs) to handle the mean-variance hedging problem for general semimartingales. Furthermore, the authors in Kallsen and Vierthauer [33] derived semi-explicit formulas for the optimal hedging strategy and the minimal hedging error by applying general structural results and Laplace transform techniques. In addition to these works, some related studies in both general theory and concrete results in specific setups for the mean-variance hedging problem can be found in, such as, Arai [2], Chan et al. [9], Duffie and Richardson [18], Gourieroux et al. [23], Heath et al. [25], Laurent and Pham [37], and references therein.
Comparing with the above studies, our contribution of the current research is threefold. First, we firmly prove the no-arbitrage condition to be true for our financial market, i.e., the set defined in (1.2) is nonempty. This condition is used as an assumption for the existence of the VOMM in existing discussions (e.g., Arai [2], Cȇrný and Kallsen [7], Chan et al. [9], Jeanblanc et al. [31], Kallsen and Vierthauer [33]). In doing so, we explicitly (or called semiexplicitly) construct a measure through identifying its explicit density by the general structure presented in Cȇrný and Kallsen [7]. Then, we justify it to be the VOMM for our market model by proving the equivalent conditions given in Cȇrný and Kallsen [8]. Second, in applying our VOMM to obtain the optimal hedging strategy, we derive an BSDE with jumps for the mean value process of the option H. Here, we lift the requirements that the contingent claims are bounded (e.g., Heath and Schweizer [26], Cȇrný and Kallsen [8]) or satisfy Lipschitz condition (e.g., Roch [42], Chan et al. [9]) to guarantee the corresponding integral-partial differential equation (IPDE) to have a classic or viscosity solution. Furthermore, the unique existence of an adapted solution to our derived BSDE is firmly proved under certain conditions while in the recent study of Jeanblanc et al. [31], such existence of an adapted solution to their constructed BSDE is only showed as an equivalent condition to guarantee the existence of an optimal strategy. More importantly, our BSDE can be solved by developing related numerical algorithms through the given terminal option H (see, e.g., Dai [15]). Third, from the purpose of easy applications, our discussion is based on a multivariate financial market model, which is in contrast to existing studies (e.g., Cȇrný and Kallsen [7], Chan et al. [9], Jeanblanc et al. [31], Kallsen and Vierthauer [33]). Therefore, unlike the studies in Hubalek et al. [27] and Kallsen and Vierthauer [33], our option H is generally related to a multivariate terminal function and hence a BSDE involved approach is employed. Actually, whether one can extend the Laplace transform related method developed in Hubalek et al. [27] and Kallsen and Vierthauer [33] for single-variate terminal function to our general multivariate case is still an open problem.
Note that our study in this paper establishes the connection between our financial system and existing general semimartingale based study in Cȇrný and Kallsen [7] since we can overcome the difficulties in Cȇrný and Kallsen [7] by explicitly constructing the process N and the VOMM Q * as mentioned earlier. Furthermore, our objective and discussion in this paper are different from the recent study of Jeanblanc et al. [31] since the authors in Jeanblanc et al. [31] did not aim to derive any concrete expression. Nevertheless, interested readers may make an attempt to extend the study in Jeanblanc et al. [31] and apply it to our financial market model to construct the corresponding explicit results.
Finally, when the random variable H in (1.1) is taken to be a constant (e.g., a prescribed daily expected return), the associated hedging problem reduces to a mean-variance portfolio selection problem as studied in Dai [10] by an alternative feedback control method. In this case, the optimal policies can be explicitly obtained by both the feedback control method in Dai [10] and the martingale method presented in the current paper. In the late method, the related BSDE is a degenerate one. From this constant option case, we can construct two insightful examples to provide the effective comparisons between the two methods. More precisely, our newly constructed hedging strategy can slightly outperform the feedback control based policy. However, the performance between the two methods is consistent in certain sense.
The remainder of the paper is organized as follows. We formulate our financial market model in Section 2 and present our main theorem Section 3. Analytical and numerical examples are given in Section 4. Our main theorem is proven in Section 5. Finally, in Section 6, we conclude this paper with remarks.
The financial market under consideration is a multivariate Lévy-driven OU type stochastic volatility model, which consists of d + 1 assets. One of the d + 1 assets is risk-free, whose price S 0 (t) is subject to the ordinary differential equation (ODE) with constant interest rate r ≥ 0, The other d assets are stocks whose vector price process S(t) = (S 1 (t), ..., S d (t)) ′ satisfies the following stochastic differential equation (SDE) for each t ∈ [0, T ], Here and in the sequel, the diag(v) denotes the d × d diagonal matrix whose entries in the main diagonal are v i with i ∈ {1, ..., d} for a d-dimensional vector v = (v 1 , ..., v d ) ′ and all the other entries are zero. Y (t) is a Lévy-driven OU type process described by the following SDE, where Λ = diag(λ) and y 0 = (y 10 , ..., y h0 ) ′ . Now, define Thus, we can impose the following conditions related to the coefficients in (2.2)-(2.3): C1. The functions b(y) and σ(y) are continuous in y and satisfy that, for each y ∈ R h c , where the norm A takes the largest absolute value of all components of a vector A or all entries of a matrix A, and C2. The derivatives ∂b(y) ∂y i and ∂(σ(y)σ(y) ′ ) −1 ∂y i for all i ∈ {1, ..., h} are continuous in y and satisfy that, for each y ∈ R h c , ∂b(y) We now introduce the conditions for each subordinator L i with i ∈ {1, ..., h}, which can be represented by (e.g., Theorem 13.4 and Corollary 13.7 in Kallenberg [32]) Here and in the sequel, ) denotes a Poisson random measure with deterministic, time-homogeneous intensity measure ν i (dz i )ds. I A (·) is the index function over the set A. ν i is the Lévy measure satisfying with C taken to be a sufficiently large positive constant to guarantee all of the related integrals in this paper meaningful. Note that the condition in (2.10) is on the integrability of the tails of the Lévy measures (readers are referred to Dai ([10,11,12,13,14]) for the justification of its reasonability).
Definition 2.1 An R d -valued trading strategy u is called simple if it is a linear combination of strategies ZI (τ 1 ,τ 2 ] where τ 1 ≤ τ 2 are stopping times dominated by σ n for some n ∈ N and Z is a bounded F τ 1 -measurable random variable. Furthermore, the set of all such simple trading strategies is denoted by Θ(D). Definition 2.2 A trading strategy u ∈ L(D) is called admissible if there is a sequence {u n , n ∈ N } of simple strategies such that: (u n · D) (t) → (u · D)(t) in probability as n → ∞ for any t ∈ [0, T ] and (u n · D) (T ) → (u · D)(T ) in L 2 (P ) as n → ∞. Furthermore, the set of all such admissible strategies is denoted byΘ(D).

Main Theorem
First, for each y ∈ R h c , define Note that the process a(·) presented in (3.5) is corresponding to the adjustment process defined in Lemma 3.7 of Cerny and Kallsen [7]. Furthermore, the processẐ(·) presented in (3.6) is associated with the density process defined in Proposition 3.13 of Cerny and Kallsen [7]. In addition, here and in the sequel, E(N ) = {E(N )(t), t ∈ [0, T ]} denotes the stochastic exponential for a univariant continuous semimartingale N = {N (t), t ∈ [0, T ]} (e.g., pages 84-85 of Protter [41]) with where [·, ·] denotes the quadratic variation process of N .
Second, let L 2 F ,p ([0, T ], R d , P ) denote the set of all R d -valued predictable processes (see, e.g., Definition 5.2 in page 21 of Ikeda and Watanabe [28]) and let L 2 where, e i is the h-dimensional unit vector with the ith component one. Then, we define To impose suitable condition on the option H, we use L γ F T (Ω, R d , P ) for a positive integer γ to denote the set of all F T (Ω, R, P ) and there exists a sequence of random variables H τn ∈ L 2 F T ∧τn (Ω, R, P ) satisfying H τn → H in L 2 as n → ∞ and H τn (ω) = H(ω) for all ω ∈ {ω, τ n (ω) ≥ T }, where {τ n } is a sequence of nondecreasing {F t }-stopping times satisfying τ n → ∞ a.s. as n → ∞.
As pointed out in Dai [12], under conditions C1, C2, and (2.10), the discounted European call and put options satisfy Assumption 3.1. Now, we can state our main theorem of the paper as follows.  (3.12). Then, the optimal hedging strategy φ ∈Θ(D) for (1.1) is given by where, the pure hedge coefficient ξ is given by In addition, Ψ is the unique solution of the SDE The process V (·) appeared in Theorem 3.1 is actually the conditional mean value process, Since it is not easy to be computed directly as the Markovian based conditional process O(t, Y (t)), we turn to use the BSDE in (3.12) to evaluate it, which is convenient for us to design the optimal hedging policy as explained in Introduction of the paper.
The proof of Theorem 3.1 will be provided in Section 5.

Performance Comparisons
The material in this section is partially reported in the short conference version of the current paper (see, Dai [12]). To be convenient and clear for readers, we refine it here. Note that the interest rate r in (2.1) here is taken to be zero. Furthermore, the financial market is assumed to be self-financing, which implies that X(t) = v + (u · D)(t). In addition, the terminal option H is taken to be a constant p, i.e., H = p. In this case, the optimal policies can be explicitly obtained by the feedback control method studied in Dai [10] and the martingale method presented in the current paper. In the late method, the related BSDE is a degenerate one, which can be easily observed from (3.19) in Remark 3.1. However, from this constant option H = p, we can construct two insightful examples to provide the effective comparisons between the two methods.
More precisely, by (18) in Theorem 3.1 of Dai [10], we know that the terminal variance under the optimal policy stated in (15) of Theorem 3.1 of Dai [10] is given by In addition, by using Theorem 3.1 in the current paper and Theorem 4.12 in Cȇrný and Kallsen [7], we know that the hedging error under the optimal policy in (3.14) is given by For the purpose of performance comparisons, we calculate the differences between the optimal terminal variances in (4.1) and the optimal hedging errors in (4.2), i.e., The result shown in the last inequality of (4.3) is intuitively right since the optimal strategy in (3.14) is taken over a general decision set given in Definition 2.2 and the one in (15) of Theorem 3.1 of Dai [10] is taken in an ad-hoc approach. Nevertheless, the errors are very small as displayed in the following numerical examples.
Example 4.1 Here, we suppose that the financial market is given by the Black-Scholes model where α and β are given constants. Owing to Definition 2.1.4(b) in pages 273-274 of ∅ksendal [39], the option H = p (a positive constant) is not attainable and hence the associated hedging error can not be zero if the initial endowment v = p. However, by the simulated results displayed in Figures 1 and 2, we see that the absolute error between the optimal variance based on the policy in (15) of Theorem 3.1 of Dai [10] and the optimal hedging error based on the strategy in (3.14) approaches zero as the terminal time increases. The rate of convergence is heavily dependent on the volatility β. If β is relatively large, the difference requires more time to reach zero. Nevertheless, if the millisecond is employed to represent the time unit in a supercomputer based trading system, the required time for the convergence makes sense in practice. Example 4.2 Here, we assume that the financial market is presented by the BNS model where α and β are given constants. Furthermore, owing to the remarks to the condition in (2.10) and owing to the discussions in Dai [11], we suppose that the driving subordinator L(λ·) with λ = 1 to the SDE in (2.3) is a compound Poisson process. The interarrival times of the process are exponentially distributed with mean 1/µ and the jump sizes of the process are also exponentially distributed with mean 1/µ 1 . By the simulated results displayed in Figure 3, we see that the similar illustration displayed in Example 4.1 also makes sense for the current example, where δ appeared in Figure 3 is the length of equally divided subintervals of [0, T ]. In addition, by the simulated results, we also see that, by perfect hedging is impossible in an incomplete market, the mean-variance hedging errors can be very small in many cases when terminal time increases.

Proof of Theorem 3.1
The proof consists of four parts presented in the subsequent four subsections: the justification of a proposition related to the discounted price process, the demonstration of a proposition related to the VOMM, the illustration of unique existence of solution to a type of BSDEs with jumps, and the remaining proof of Theorem 3.1.  where M D (·) and B D (·) are an {F t }-martingale and a predictable process of finite variation respectively. Furthermore, D(·) is locally in L 2 F ([0, T ], R d , P ) in the sense as stated in (2.12).

The Proposition
We divide the proof of the proposition into two parts. First, we have the following lemma.
Lemma 5.1 Under (2.10), the unique adapted solution to the SDE in (2.3) for eacht > t, i ∈ {1, ..., h}, and y ∈ (0, ∞) h is given by Furthermore, under conditions C1, C2, and (2.10), there is a unique solution (S 0 (t), S(t) ′ ) for (2.2)-(2.3), which is an {F t }-adapted and continuous semimartingale with In addition, for each m ∈ {1, ..., d}, Proof. The claim concerning (5.2) directly follows from pages 316-317 in Applebaum [1]. Furthermore, owing to conditions C1 and C2, we know that our market given by (2.2)-(2.3) satisfies the conditions as required by Lemma 4.1 in Dai [10]. Thus, our market has a unique solution, which is {F t }-adapted, continuous, and mean-square integrable as stated in Lemma 5.1. In order to prove (5.4), let Then, by condition C1, there exists some nonnegative constant D 1 such that where we have used the facts that L(λt) is nonnegative and nondecreasing in t, the independence assumption among L i (λ i ·) for i ∈ {1, ..., h}, and Similarly, we can show that Then, it follows from Definition 4.1.1 in ∅ksendal [39] and the associated Itô's formula (e.g., Theorem 4.1.2 in ∅ksendal [39]) that S m (t) given in (5.4) for each m is the unique solution of (2.2). Now, we show that S m (·) for each m ∈ {1, ..., d} is a square-integrable {F t }-semimartingale. To do so, we rewrite (2.2) in its integral form σ mn (Y (s − ))dW n (s). (5.11) Then, the third term on the right-hand side of (5.11) is a square-integrable {F t }-martingale. In fact, it follows from (5.2) that, for each i ∈ {1, ..., h} andt > t, where the last equality in (5.12) holds in distribution. Thus, it follows from Condition C1 and (5.4) in Lemma 5.1 that where C is some positive constant and we have used Theorem 39 in page 138 of Protter [41] and the condition (2.10). Therefore, by Theorem 4.40(b) in page 48 of Jacod and Shiryaev [30], we know that the third term in (5.11) is a square-integrable {F t }-martingale. Furthermore, by the same method, we can show that the second term on the righthand side of (5.11) is of finite variation a.s. and is square-integrable over [0, T ]. Therefore, we conclude that S m (·) for each m ∈ {1, ..., d} is a square-integrable {F t }-semimartingale. Hence, we complete the proof of Lemma 5.1.  [30] that M D is an {F t }-martingale. Furthermore, it follows from a similar explanation with the end of the proof for Lemma 5.1 that B D is a predictable process of finite variation and square-integrable. Thus, we know that D is a continuous {F t }-semimartingale. Moreover, it is locally in L 2 (P ) since we may take σ n ≡ inf{τ : D 2 (τ ) ≥ n} as the sequence of localizing times. Hence, we complete the proof of Proposition 5.1.

A Proposition Related to the VOMM
First of all, we use P D (Θ)(D) to denote the set of all signedΘ-martingale measures in the sense that Q(Ω) = 1 and Q ≪ P with dQ dP ∈ L 2 (P ) and E dQ dP (u · D)(T ) = 0 for a signed measure Q on (Ω, F) and all u ∈Θ(D). Then, we have the following proposition. 3. The measure Q * is the VOMM in the sense that We divide the proof of the proposition into demonstrating six lemmas as follows.
Proof. It follows from conditions C1, C2, and (5.2) that, for each i ∈ {1, ..., h}, whereĀ i for i ∈ {1, 2, 3, 4} are some nonnegative constants, A ρ and B ρ are given by Then, based on an idea as used in Benth at al. [4], we can prove Lemma 5.2 by the following four steps. First, by direct calculation, we know that P (t, y) is finite for any (t, y) ∈ [0, T ] × R h c , i.e., where the nonnegative constant K 1 is given by Second, we prove that P ∈ C 0,1 [0, T ] × R h c , R 1 and the mapping (t, y) → ∂P ∂y i (t, y) for each i ∈ {1, ..., h} is continuous.. The continuity of P (·, y) for each y ∈ R h c can be shown as follows. Owing to the condition (2.4) and the fact (5.12), we know that By (2.10) and (5.9), we know that the function on the right-hand side of (5.24) is integrable for each fixed y ∈ R h c . Then, it follows from the Lebesgue's dominated convergence theorem that P (t, y) for each y is continuous in terms of t ∈ [0, T ].
Next, we show that ∂P ∂y i (t, ·) with i ∈ {1, ..., h} for all t ∈ [0, T ] exist and are continuous. In fact, consider an arbitrary but fixed point y and take a compact set U ⊂ R h c such that y is in the interior of U . Note that all points in U can be assumed to be bounded by some positive constant M . Thus, by (5.22), (5.2), (5.8) and (5.7), we have, for all s ≥ t, where Y t,y (s) denotes the process with the initial value y at time t. Owing to (2.10) and (5.9), the function on the right-hand side of (5.25) is integrable. Thus, it follows from Theorem 2.27(b) in Folland [19] that the partial derivative of T t ρ(Y t,y (s))ds in terms of y i for each i ∈ {1, ..., h} exists. Hence, we have Again, by (2.10) and (5.9), we know that the function on the right-hand side of (5.26) is integrable. Therefore, by Theorem 2.27(b) in Folland [19], we can conclude that P (t, y) is differentiable with respect to y ∈ R h c . Furthermore, by (5.2), (5.26) and the Lebesgue's dominated convergence theorem, we obtain that the mapping (t, y) → ∂P ∂y i (t, y) for each i ∈ {1, ..., h} is continuous. Hence, P (t, y) ∈ C 0,1 [0, T ] × R h c , R 1 . Third, we prove the square-integrable property (5.20) to be true. In fact, it follows from condition (2.10) that ν i (·) (i ∈ {1, ..., h}) is a σ-finite measure since ν i ([ǫ, ∞)) < ∞ for any ǫ > 0. In addition, it is easy to see that the nonnegative function |P (t, Y (t − )+z i e i )−P (t, Y (t − ))| 2 is a measurable one on the product space [0, T ] × R h c × Ω. Hence, by the mean value theorem, (5.25), (5.26), the Jensen's inequality, and the differentiability of P (t, y) in y, we have where K 3 and K 4 are some positive constants. Furthermore, it follows from (5.23), (5.8), and (2.10) that (5.19) is true. Fourth, we prove that P (t, y) satisfies the IPDE (5.17). In fact, for each t ∈ [0, T ), it follows from the time-homogeneity of Y that g(T − t, y) ≡ E 0,y e − T −t 0 ρ(Y (s))ds = E t,y e − T t ρ(Y (s))ds = P (t, y). (5.28) Since P (t, y) ∈ C 0,1 [0, T ] × R h c , it follows from the Itô's formula (see, e.g., Theorem 1.14 and Theorem 1.16 in pages 6-9 of ∅ksendal and Sulem [38]) that, for each fixed t, g(T − t, Y 0,y (l)) (5.29) .., h} and ω ∈ Ω. Then,ĝ is {F t }-predictable. Thus, owing to (5.20) (here we need to use an arbitrary but fixed y to replace y 0 ), it follows from Theorem 4.2.3 in Applebaum [1] (or the explanation in page 61-62 of Ikeda and Watanabe [28]) that the last term in (5.29) is a semimartingale. Thus, taking expectations on both sides of (5.29), we get Then, by letting l ↓ 0, we know that P (t, ·) is in the domain of the infinitesimal generator of Y , which is denoted by A, that is, Now, by (5.23), we see that g(T − t, y) = P (t, Y 0,y (l)) ∈ L 2 (Ω, P ) for each t ∈ [0, T ) and all l in a neighborhood of zero such that t − l ≤ T . Thus, we have where the second equality in (5.31) follows from the Markov property of Y (e.g., Proposition 7.9 in Kallenberg [32]). Then, we have Since the function in the left-hand side of (5.33) is uniformly bounded by an integrable function, it follows from the dominated convergence theorem that the right-derivative of g(T − ·, y) at t exists and satisfies Ag(T − t, y) = ρ(y)g(T − t) + ∂g ∂t (T − t, y). Hence, by (5.28) and (5.35), we know that P (t, y) satisfies (5.17). In addition, we have where K 5 is some positive constant. Thus, by the Lebesgue's dominated convergence theorem, we can conclude that is continuous in t. Therefore, it follows from (5.17) that ∂P ∂t (t, y) is continuous in t ∈ [0, T ), which implies that P ∈ C 1,1 [0, T ) × R h c , R 1 . Hence, we complete the proof of Lemma 5.2.
Then, K is an {F t }-semimartingale and has the following canonical decomposition where, F (t, z i , ω)) is defined in (3.10).
Proof. First, we show that O is an {F t }-semimartingale. In fact, it follows from the Ito's formula (see, e.g., Theorem 1.14 and Theorem 1.16 in pages 6-9 of ∅ksendal and Sulem [38]) and Lemma 5.2 that Then, by Lemma 5.2 and the claim in pages 61-62 of Ikeda and Watanable [28], we know that the third term in the right-hand side of (5.38) is an {F t }-martingale. Furthermore, by (5.21) and the similar proof as used for Lemma 5.1, we know that the second term on the right-hand side of (5.38) is of finite variation a.s. Hence, we get that O is an {F t }-semimartingale. Thus, it follows from (5.38) and the definition of K(t) that (5.37) is true. Second, M K defined as follows is an {F t }-martingale, In fact, by the mean-value theorem, (5.21), (5.2) (2.10), and the fact that ν i (·) (i ∈ {1, ..., h}) is a σ-finite measure since ν i ([ǫ, ∞)) < ∞ for any ǫ > 0, we have Thus, it follows from (5.40) and the claims in pages 61-62 of Ikeda and Watanable [28] that M K is an {F t }-martingale. Therefore, we can conclude that K is an {F t }-semimartingale. Hence, Lemma 5.3 is true.
Lemma 5.4 Let b D and c D be the drift and the covariance matrix processes associated with D, b K is the drift process associated with K. Then, under conditions C1, C2, and (2.10), we have Furthermore, the process a defined in (3.5) satisfies the following relationship, Proof. First of all, it follows from Lemma 5.1 and Lemma 5.
Then, by simple calculations, we know that (5.41) and (5.42) are true. Hence, we complete the proof of Lemma 5.4.
For convenience, we will use C D ij ≡ [D i , D j ] to denote the co-quadratic variation processes with i, j ∈ {1, ..., d} for the process D and write interchangeably c D i D j ≡ c D ij and c D i = c D ii . Furthermore, similar notations are also used for other processes related in the following discussions. Proof. First, we show that a ∈ L(D). In fact, it follows from the condition C1, (5.2), and (5.1) that Y (t − )) ≥ min{y i0 e −λ i T , i = 1, ..., h} > 0 for any t ∈ [0, T ]. Then, for m, n ∈ {1, ..., d}, we havē where Cρ is some positive constant. Thus, it follows from the Kunita-Watanable inequality (e.g., Theorem 25 in page 69 of Protter [41]) that where a m and M D m with m ∈ {1, ..., d} are the mth components of a and M D respectively. Furthermore, it follows from (5.1) that  [30], we know that a ∈ L(D). Thus, (a · D)(T ) is well defined.
In addition, it follows from Theorem 4.5(a) in page 180 of Jacod and Shiryaev [30] that, for each u ∈ L(D), we have, (5.49) where the limit in the first term on the right-hand side of (5.49) corresponds to the convergence in probability uniformly on every compact set of [0, T ]. Therefore, by (5.1), (2.10), (5.14)-(5.15), (5.49), and the Lebesgue dominated convergence theorem, we know that where the second equality follows from the facts that A(t) = t, K(0) = 0 and the independence among driving Brownian motions and Lévy processes. The third equality follows from Lemma 5.4. Furthermore, M K and M D are given by (5.39) and (5.15), which are {F t }-martingales. Hence, is also an {F t }-martingale. Thus, it follows from Theorem 4.61 in page 59 of Jacod and Shiryaev [30] thatẐ is an {F t }-local martingale.
Second, we prove thatẐ is of class (D), i.e., the set of random variables is uniformly integrable (e.g., Definition 1.46 in page 11 of Jacod and Shiryaev [30]).
In fact, consider an arbitrary finite-valued {F t }-stopping time τ ≤ T and an arbitrary constant γ > 0. Then, we have where we have used the facts that 0 < O(·) ≤ 1 and D is continuous. Furthermore, let where the third inequality follows from the optional sampling theorem, the fact that e −2U 2 (t) is a submartingale by the Jensen's inequality, and Theorem 39 in page 138 of Protter [41]. The last inequality follows from conditions C1-C2. Therefore, it follows from (5.56) that sup τ E Ẑ (τ ) ≤ K 1 , where K 1 is some positive constant. Thus, by the Markov's inequality, we have that for all stopping time τ ≤ T . Therefore, it follows from (5.53)-(5.57) thatẐ is of class (D). Hence, it follows from (5.51) and Proposition 1.47(c) in page 12 of Jacod and Shiryaev [30] thatẐ is a uniformly integrable {F t }and P -martingale. Proof. First, we use P t to denote the restriction of P to F t for each t ∈ [0, T ]. Then, we define dQ * t ≡Ẑ(t)dP t and dQ * ≡Ẑ(T )dP . Owing to (3.3)-(3.6), we know thatẐ(t) > 0 for each t ∈ [0, T ]. Furthermore, note thatẐ is a {F t }and P -martingale. Hence, it follows from the discussion in page 166 of Jacod and Shiryaev [30] that Q * is equivalent to P with the density processẐ.
Next, we show that D is an Q * -martingale. In fact, since D is an P -semimartingale with the decomposition given in (5.1), it follows from Girsanov-Meyer Theorem (e.g., Theorem 35 in page 132 of Protter [41]) that D is also an Q * -semimartingale with the decomposition D =D +D. The processD is an Q * -finite variation process. For each m ∈ {1, ..., d}, whereB n (Y (s − )) is defined in (3.9). The second equality in (5.58) follows from Theorem 29 in page 75 of Protter [41], the proof of Corollary in page 83 of Protter [41], the fact that W is continuous, Theorem 4.52 in page 55 of Jacod and Shiryaev [30], and the explanation in page 70 of Protter [41]. The third equality in (5.58) follows from the Ito's formula for multi-dimensional semimartingales (e.g., Theorem 33 in pages 81-82 of Protter [41]), and the associated function f is taken to be f (O, U ) = Oe U . Furthermore, a r is the rth component of a, and U is defined by U (t) ≡ −a · D(t) − 1 2 [a · D, a · D](t). Thus, we havē where s i for each i ∈ {1, ..., d} is the initial price as given in (2.2).
Therefore, to show that D is an Q * -martingale, it suffices to show thatD is an Q *martingale. More precisely, by the last equation in the proof of Theorem 35 in pages 132-133 of Protter [41], we have that Then, we can show that the both terms on the right-hand side of (5.60) are Q * -martingales.
Then, we can show that each of the three terms on the right-hand side of (5.61) is an Q * -martingale.
The claim that the first term on the right-hand side of (5.61) is a Q * -martingale can be proved as follows. First, it follows from the similar argument as used in (5.64) that M D is a square integrable P -martingale. Second, by the Tonelli's Theorem (e.g., Theorem 20 in page 309 of Royden [43]) and the Hölder's inequality, we have whereK is some positive constant. The last inequality in (5.63) follows from the similar arguments as in (5.56) and (5.13). Thus, it follows from Theorem 4.40(b) in page 48 of Jacod and Shiryaev [30] that the first term on the right-hand side of (5.61) is an {F t }and P -martingale. The claim that the second term on the right-hand side of (5.61) is an Q * -martingale can be proved as follows. It follows from (5.13) and Exercise 3.25 in page 163 of Karatzas and Shreve [34] that Then, by (5.64), the Hölder's inequality and the similar method as used in (5.63), we know that the second term on the right-hand side of (5.61) is an {F t }and P -martingale.
The claim that the third term on the right-hand side of (5.61) is an Q * -martingale can be proved as follows. It follows from the Tonelli's Theorem (e.g., Theorem 20 in page 309 of Royden [43]) that where K 1 is some positive constant. The inequalities in (5.65) follow from the similar proofs as used in (5.56), (5.64), the Hölder's inequality, the proof of (5.27), and the fact that Then, it follows from (5.65) and the argument in pages 61-62 in Ikeda and Watanable [28] that the third term on the right-hand side of (5.61) is also an {F t }and P -martingale. Therefore, by summarizing the discussions for the three terms on the right-hand side of (5.61), we know that the process given by (5.61), is an {F t }and P -martingale. Moreover, by applying Proposition 3.8(a) in page 168 of Jacod and Shiryaev [30], we can conclude that the first term on the right-hand side of (5.60) is an Q * -martingale.
For the second term on the right-hand side of (5.60), we can show that it is also an {F t }and Q * -martingale. In fact, sinceẐ is a density process of Q * in terms of P and 1 Z Ẑ = 1 (that is an P -martingale), it follows from Proposition 3.8(a) in page 168 of Jacod and Shiryaev [30] that 1 Z is an Q * -martingale. Furthermore, it follows from the Ito's formula (e.g., Theorem 32 in page 78 of Protter [41]), (5.62) and the calculation of dẐ(t) in the last equality in (5.61) that d 1 whereB(Y (t − )) is defined in (3.9). Thus, it follows from (5.66) that 1 Z is squarely integrable under Q * , i.e., where the second inequality in (5.67) follows from the Doob's martingale inequality (e.g., Theorem 2.1.5 in page 74 of Applebaum [1]) since 1 Z is an Q * -martingale. The last inequality of (5.67) follows from the similar argument as in (5.56).
Therefore, to show that the second term on the right-hand side of (5.60) is an Q *martingale, it suffices to show that the following expectation under Q * is finite owing to (5.67) and Theorem 4.40(b) in page 48 of Jacod and Shiryaev [30], The first term on the right-hand side of (5.68) is finite since (Y (s − ))ds (5.69) where K 1 is some positive constant. The first inequality in (5.69) follows from the Doob's martingale inequality (e.g., page 74 of Applebaum [1]). The second inequality in (5.69) follows from the similar arguments as in (5.53) and (5.13). Similarly, the second term on the right-hand side of (5.68) is also finite, which can be proved along the line of the discussion as in (5.69). Thus, it follows from the finiteness of (5.68) that the second term on the right-hand side of (5.60) is an Q * -martingale. Therefore, by combining this fact with (5.60) and (5.61), we know that D =D +s displayed in (5.59) is an Q * -martingale (i.e. Q * is an equivalent martingale measure). Finally, by applying the similar discussion as used in (5.63), we conclude that dQ * dP ∈ L 2 (P ), which implies that Q * ∈ U e 2 (D).

Proof.
It suffices to justify that all conditions stated in Theorem 3.25 of Cȇrný and Kallsen [7] are satisfied. First of all, for any stopping time τ , we can show that In fact, it follows from the proof of Lemma 5.6 that U e 2 (D) is nonempty. Furthermore, since D is a continuous P -semimartingale, it is sufficient to prove that the three equivalent conditions stated in Theorem 2.1 of Cȇrný and Kallsen [8] are satisfied for (5.70), which can be done by tedious computations similarly as before. In addition, we can show that OE((−aI (τ,T ] ) · D) is of class (D). Therefore, by combining this claim with Lemma 5.4, (5.70), and Theorem 3.25 in Cȇrný and Kallsen [7], we know that O and a are the opportunity and adjustment processes in the sense defined Section 3 of [7]. Thus, it follows from Proposition 3.13 in Cȇrný and Kallsen [7] that Q * is the VOMM. Hence, we complete the proof of Proposition 5.2.

The Unique Existence of Solution to A Type of BSDEs
Consider the following q-dimensional BSDE with jumps and a terminal condition H Furthermore, for anyṽ ∈ L 2 ν (R h + , R q×h ), the associated norm is defined by F T (Ω, R, P ) by H ∈ L 2 F T (Ω, R, P ) in Assumption 3.1.
where K n depending on n are positive constants. Then, the BSDE in (5.71) has a unique solution where V is a càdlàg process. The uniqueness is in the sense: if there exists another solution (U,Ū ,Ũ ) as required, then, Proof. First, for each n ∈ {1, 2, ...}, we define τ n ≡ inf{t > 0, L(λt) > n}. (5.78) Then, it follows from Theorem 3 in page 4 of Protter [41] and condition (2.10) that {τ n } is a sequence of nondecreasing {F t }-stopping times and satisfies τ n → ∞ a.s. as n → ∞ since P {τ n ≤ t} = P { L(λt) > n} ≤ E L(λt) 2 n 2 → 0 as n → ∞ for any given t ∈ [0, ∞), where we have used (2.10), (5.9), (5.7), and the fact that L(λt) is a h-dimensional nonnegative and nondecreasing càdlàg process. Second, for each n, consider the following BSDE with a random terminal time σ n ≡ T ∧τ n and a terminal condition H τn , Then, by slightly generalizing the discussion as in Yong and Zhou [53] and Tang and Li [51] (see also El Karoui et al. [35], Situ [49], Yin and Mao [52] for related discussions), we know that (5.79) has a unique adapted solution as required over [0, σ n ].
Now, for each t ∈ [0, T ] and B K (t) = t 0 ρ(Y (s − ))ds, we define the density process . (5.85) Then, the corresponding probability P * ∼ P . Thus, it is the opportunity-neutral probability measure in the sense of Definition 3.16 in Cȇrný and Kallsen [7]. Furthermore, by Corollary 8.7(b) and equation (8.19) in pages 135-138 of Jacod and Shiryaev [30], we can rewrite Z P * in (5.85) as for each t ∈ [0, T ], where K is defined in (5.36) and M K is defined in (5.39). Then, by a similar method as used in the proof of Proposition 5.2(2), we know that Z P * is a bounded positive martingale. Thus, for each pair of i, j ∈ {1, ..., d} and t ∈ [0, T ], we have where the first equality in (5.87) is owing to the continuity of D, Theorem 5.52 in page 55 of Jacod and Shiryaev [30], Theorem 4.47(c) in page 52 of Jacod and Shiryaev [30], the equivalence between P * and P , and Girsanov-Meyer Theorem in page 132 of Protter [41]. The second equality follows from Theorem 4.47(a) in page 52 of Jacod and Shiryaev [30] since Z P * is bounded and Girsanov-Meyer Theorem in page 132 of Protter [41]. Furthermore,c D * ij in the last equality is defined in (3.16). Now, note that D is continuous. Then, by Theorem 4.52 in page 55 of Jacod and Shiryaev [30] (or the proof of Corollary in page 83 of Protter [41]), we know that [D i , V ](t) and [D i , V ] c (t) for each i ∈ {1, ..., d} under P or P * have the same compensator. Hence, we have Then, it follows from (5.87)-(5.88), Definition 4.6, and equation (4.8) in Cȇrný and Kallsen [7] that (3.15) is true.
Finally, the unique existence of solution to (3.18) is owing to Theorem 6.8 in Jacod [29] and the proofs of Lemma 4.9 and Theorem 4.10 in Cȇrný and Kallsen [7]. Thus, by Theorem 4.10 in Cȇrný and Kallsen [7], we know that the mean-variance hedge strategy is given by (3.14). Hence, we complete the proof of Theorem 3.1.
In this paper, we prove the global risk optimality of the hedging strategy explicitly constructed for an incomplete financial market. Owing to the discussions in Pigorsch and Stelzer [50] and references therein, our discussion in this paper can be extended to the cases that the external risk factors in (2.3) are correlated in certain manners. For the simplicity of notation, we keep the presentation of the paper in the current way. Furthermore, our study in this paper establishes the connection between our financial system and existing general semimartingale based study in Cȇrný and Kallsen [7] since we can overcome the difficulties in Cȇrný and Kallsen [7] by explicitly constructing the process N and the VOMM Q * . In addition, our objective and discussion in this paper are different from the recent study of Jeanblanc et al. [31] since the authors in Jeanblanc et al. [31] did not aim to derive any concrete expression. Nevertheless, interested readers may make an attempt to extend the study in Jeanblanc et al. [31] and apply it to our financial market model to construct the corresponding explicit results. Finally, unlike the studies in Hubalek et al. [27] and Kallsen and Vierthauer [33], our option H is generally related to a multivariate terminal function and hence a BSDE involved approach is employed. Interested readers may take an attempt to study whether the Laplace transform related method developed in Hubalek et al. [27] and Kallsen and Vierthauer [33] for single-variate terminal function can be extended to our general multivariate case.