Symmetrical Solutions of Backward Stochastic Volterra Integral Equations and Their Applications

Backward stochastic Volterra integral equations (BSVIEs in short) are studied. We introduce the notion of adapted symmetrical solutions (S-solutions in short), which are different from the M-solutions introduced by Yong [17]. We also give some new results for them. At last a class of dynamic coherent risk measures were derived via certain BSVIEs.


Introduction
Let (Ω, F, F, P) be a complete filtered probability space on which a d-dimensional Brownian motion W (·) is defined with F ≡ {F t } t≥0 being its natural filtration augmented by all the P-null sets. In this paper, we consider the following stochastic integral equations: where g : ∆ c × R m × R m×d × R m×d × Ω → R m and Ψ : [0, T ] × Ω → R m are some given measurable mappings with ∆ c = (t, s) ∈ [0, T ] 2 | t ≤ s . Such an equation is referred as a backward stochastic Volterra integral equation (BSVIE in short) (see [14] and [16]).
When Ψ(·), g, Z(t, s) are independent of t, g is also independent of Z(s, t), BSVIE (1) is reduced to a nonlinear backward stochastic differential equation (BSDE in short) Y (t) = ξ + T t g(s, Y (s), Z(s))ds − T t Z(s)dW (s), which was introduced by Pardoux and Peng in [8], where the existence and uniqueness of F-adapted solutions are established under uniform Lipschitz conditions on g. Due to their important significance in many fields such as financial mathematics, optimal control, stochastic games and partial differential equations and so on, the theory of BSDEs has been extensively developed in the past two decades. The reader is referred to [3], [4], [6], [9], [12] and [11].
On the other hand, stochastic Volterra integral equations were firstly studied by Berger and Mizel in [1] and [2], then they were investigated by Protter in [13] and Pardoux and Protter in [10]. Lin [5] firstly introduced a kind of nonlinear backward stochastic Volterra integral equations of the form But there is a gap in [5]. At the same time Yong [14] also investigated a more general version of BSVIEs, as the type of (1), and gave their applications to optimal control. In this paper, we give a further discussion to BSVIE (1).
Based on the martingale presentation theorem, especially for Z(t, s), t ≥ s, Yong [16] introduced the concept of M-solutions for BSVIE (1). He gave some conditions that suffice BSVIE (1) is uniquely solvable. We realize that there should be other kinds of solutions for BSVIEs. In this paper, we introduce the notion of symmetrical solutions (called S-solutions) in the way of Z(t, s) ≡ Z(s, t), t, s ∈ [0, T ]. It is worthy to point out that S-solutions should be solved in a more general Hilbert space, which is different from the one for M-solution, see more detailed accounts in Section 2. We prove the existence and uniqueness of S-solutions for BSVIEs. Some properties such as the continuity of Y (t) are obtained. We then study the relations between S-solutions and other solutions. We give the notion of adapted solutions of (1) (g is independent of Z(s, t)) and obtain the existence and uniqueness by virtue of the results of S-solution, which cover the ones in [5] and overcome its gap. Some relations between S-solutions and M-solutions are studied. By two examples we show that the two solutions usually are not equal, especially their values in ∆ = (t, s) ∈ [0, T ] 2 | t > s . We also give a criteria for S-solutions of BSVIEs. At last by giving a comparison theorem for Ssolutions of certain BSVIEs, we show a class of dynamic coherent risk measures by means of S-solutions for certain BSVIEs. This paper is organized as follows: in the next section, we give some preliminary results. In Section 3, we prove the existence and uniqueness theorem of S-solutions of (1) and show some corollaries and some other new results on S-solution. In Section 4 we give a class of dynamic coherent risk measures by means of the S-solutions of a kind of BSVIEs.

Notations and Definitions
In this subsection we give some notations and definitions that are needed in the following. For any R, S ∈ [0, T ], in the following we denote . which is a Banach space under the norm: is the set of all adapted processes X : .
. We also define the norm of the elements in H 2 [R, S]: As to the norm of the element in * H 2 [R, S], Here We also define the norm of the elements in H 2 1 [R, S]: We now give the definition of M-solutions, introduced by Yong [16]. (1) holds in the usual Itô's sense for almost all t ∈ [S, T ] and, in addition, the following holds: In this paper, we introduce the concept of S-solutions as follows.

Some lemmas for S-solutions
(2) as a family of BSDEs on [S, T ], parameterized by t ∈ [R, T ]. Next we introduce the following assumption of h in (2).
Moreover, the following holds: Proof. The proof of Proposition 1 can be found in [16]. 2 Now we look at one special case of (2). Let R = S and define Then the above (2) reads: Here we define Z(t, s) for (t, s) ∈ ∆[S, T ] by the following relation Z(t, s) = Z(s, t), which is different from the way of defining M-solution, and that's why we call it Ssolution of (6). So we have the following lemma.
admits a unique adapted S-solution (Y (·), Z(·, ·)) ∈ * H 2 [S, T ], and the following estimate holds: Hereafter C is a generic positive constant which may be different from line to line. If h also satisfies (H1), Proof. From Proposition 1 the existence and uniqueness of S-solution in [S, T ] is clear.
As to the other estimates, the proof is the same as the one in [16]. 2 Let's give another special case. Let r = S ∈ [R, T ] be fixed. Define Then (2) becomes: and we have the following result.
, and the following estimate holds: t ∈ [R, S], Proof. From Proposition 1 the result is obvious. 2 3 Well-posedness of S-solutions for BSVIEs

The existence and uniqueness of S-solutions
In this subsection we will give the existence and uniqueness of S-solutions. For it we need the following standing assumption.
where we denote g 0 (t, s) ≡ g(t, s, 0, 0, 0). Moreover, it holds where L : ∆ c → R is a deterministic function such that the following holds: for some So we have: Proof. We split the proof into two steps.
Step1 Here we consider the existence and uniqueness of the adapted S-solution of Clearly, S 2 [S, T ] is a nontrivial closed subspace of * H 2 [S, T ]. In fact, we assume that there is a series of elements (y n (·), z n (·, ·)) in S 2 [S, T ], and the limit is (y(·), z(·, ·)), which belongs to * H 2 [S, T ]. We easily know that the limit also belongs to S 2 [S, T ].

From
Step 1 we have known that {Y (s); s ∈ [S, T ]} is solved, then by Lemma 2.6, (18) admits a unique adapted solution (Ψ S (·), Z(·, ·)) ∈ L 2 Here C is a constant depending on sup Next we prove the estimate in the theorem. First we can choose T 1 ∈ [0, T ] so that it satisfies max 8θ thus by the way of choosing T 1 we know that furthermore we have, ∀u ∈ [T 1 , T ], Similarly we can choose where Ψ T 1 (t) = Ψ(t) + By the definition of S-solution we have Z(t, s) ≡ Z(s, t) in (23), then from the proof in Step 2 above we have, ∀u ∈ [T 2 , T 1 ], then from (22) and (24), ∀u ∈ [T 2 , T 1 ], Here C depends on sup T t L(t, s) 2+ǫ ds. Since we are considering the symmetrical form of Z(·, ·), by the stochastic Fubini Theorem we get: From (21), (24), (25) and (26), we can estimate that, ∀u ∈ [T 2 , T 1 ], Thus we can repeat the argument above to obtain the estimate. 2

Some corollaries for S-solutions
In this subsection we give some corollaries. Similar to [16], we easily claim the following results. We omit their proof.

The relations between S-solutions and other solutions
Let us consider the following BSVIE which is a generalization of BSVIE in [5].
First we give a definition of the adapted solutions of BSVIEs. There is a gap in [5]. Now we can easily prove the existence and uniqueness of adapted solution of (41) which is a generalization of the result in [5] and overcome the gap in [5]. We can claim: where L : ∆ c → R is a deterministic function so that for some ǫ > 0,  ∀y, y, z, z, ζ, ζ ∈ R, we have g(t, s, y, z, ζ) − g(t, s, y, z, ζ) = f (t, s, y, , But Z 1 (t, s) and Z 2 (t, s) may be different in ∆. Now we give two examples to illustrate it. Let's consider the following BSVIE Here and g satisfies the assumption (H2).
It is obvious that Y (t) = t 2 W (t), Z(t, s) = ts satisfies (43), thus it is the unique S-solution of (43). We also know that BSVIE (43) has a unique M-solution (see [16]). But the M-solution is not equal to the S-solution of (43). In fact, if the unique S-solution of (43) also is the M-solution of (43), we have which means that W (t) = W (T 1 ) for any t ∈ [T 1 , T ]. Obviously it is a contradiction. Now we give the explicit M-solution for (43). Let Z(t, s) = ts, T 1 ≤ t ≤ s ≤ T , and then by Ocone-Clark formula (see [7]) and the definition of M-solution, we have
Therefore we obtain the M-solution of (43) as follows: Here We can define the norm of H 2 [R, S] as the norm of H 2 [R, S]. We can define Ssolution for (45). Obviously (45) has a unique S-solution which is the same as the one of (1). By the same method as in [16], we can also prove (45) admits a unique M'-solution defined as follows.
So we have Z 1 (t, s) ≡ Z 2 (s, t), t ≤ s. Because of the assumption of Z 1 (t, s) ≡ Z 2 (t, s), t, s ∈ [0, T ], we have Z 2 (t, s) ≡ Z 2 (s, t), (t ≤ s), and the solution (Y 2 , Z 2 ) is the S-solution of (47). By a similar method we can show (Y 1 , Z 1 ) is the S-solution of (46). 2 Remark 3.4 From the proposition above, we know that when two kinds of equations such as (46) and (47) have the same terminal condition and the same generator, they have the same solution if and only if both of the solutions are S-solution. Next we give an example to show this. We consider the following two BSVIEs Obviously (48) and (49) have the same S-solution, for i = 1, 2 however, the M-solution of (48) is not equal to the M'-solution of (49). In fact, from We can also determine Z 1 (t, s) (t > s) by and Z 2 (s, t) (t > s) by So we have Z 1 (t, s) = ts = s 2 = Z 2 (t, s) for t < s and Z 1 (t, s) = t 2 = ts = Z 2 (t, s) for t > s.

Dynamic risk measures by special BSVIEs
In this section, we assume m = d = 1 and f is independent of ω. We know that the following BSVIE admits a unique adapted M-solution and a unique adapted S-solution when the generator and the terminal condition satisfy certain conditions: From the definition of the M-solution and S-solution, we know that both of them which solve (52) in the Itô sense have the same value in the following part (Y (t), Z(t, s)), 0 ≤ t ≤ s ≤ T, t ∈ [0, T ], and the only difference between the two kinds of solutions is the value of Now we will give a comparison theorem on S-solution for the following BSVIE:  And we can use Girsanov theorem to rewrite (54) where W (t) = W (t) + t 0 (r 1 (s) + r 2 (s))ds is a Brownian motion under new probability measure P defined by d P dP (ω) = exp Before proving the comparison theorem for S-solution, we need the following proposition in [15].
Proof. The result is obvious. 2 We then have Theorem 4.1 Suppose f (t, s, y) = η(s)y, with η(·) being a deterministic bounded function, then ρ(·) defined by (59) is a dynamic coherent risk measure.
Proof. It is not difficult to obtain the conclusion by the above lemmas. 2