An Exponential Martingale Equation

We prove an existence of a unique solution of an exponential martingale equation in the class of BM O martingales. The solution is used to characterize optimal martingale measures.


Introduction
Let (Ω, F, P ) be a probability space with filtration F = (F t , t ∈ [0, T ]).We assume that all local martingales with respect to F are continuous.Here T is a fixed time horizon and F = F T .Let M be a stable subspace of the space of square integrable martingales H 2 .Then its ordinary orthogonal M ⊥ is a stable subspace and any element of M is strongly orthogonal to any element of M ⊥ (see, e.g.[3], [8]).We consider the following exponential equation where η is a given F T -measurable random variable and α is a given real number.A solution of equation ( 1) is a triple (c, m, m ⊥ ), where c is strictly positive constant, m ∈ M and m ⊥ ∈ M ⊥ .
Here E(X) is the Doleans-Dade exponential of X.
It is evident that if α = 1 then equation (1) admits an "explicit" solution.E.g., if α = 1 and η is bounded, then using the unique decomposition of the martingale E(exp{η}/F t ) it is easy to verify that the triple c = 1 E exp{η} , satisfies equation (1).Note also that if α = 0 then a solution of (1) does not exist in general.
In particular, in this case equation ( 1) admits a solution only if η satisfies (2) with m ⊥ (η) = 0. Other cases are much more involved and ( 1) is equivalent to solve a certain martingale backward equation with square generator.Equations of such type are arising in mathematical finance and they are used to characterize optimal martingale measures (see, Biagini, Guasoni and Pratelli (2000), Mania and Tevzadze (2000), ( 2003),( 2005)).Note that equation ( 1) can be applied also to the financial market models with infinitely many assets (see M. De Donno, P. Guasoni, M. Pratelli ( 2003)).In Biagini at al (2000) an exponential equation of the form was considered (which corresponds to the case α = −1).Assuming that any element of H 2 is representable as a sum of stochastic integrals H •W +K •M , where W is a Brownian motion and M is a martingale (not necessarily continuous) orthogonal to W , they identified the varianceoptimal martingale measure as the solution of equation ( 3).In so-called extreme cases (already studied in Pham et al. (1998), Laurent and Pham (1999) using different methods), when the market price of risk λ is measurable with respect to the σ-algebras generated by W and M respectively, they gave explicit solutions of (3)) providing an explicit form for the density of the variance-optimal martingale measure.These extreme cases correspond in our setting to the following conditions on the random variable η: respectively.It is easy to see that if (4) is satisfied then the triple c = 1 c(η) , m = (c(η) + m(η)) −1 • m(η), m ⊥ = 0 solves equation (1) and under condition (5) equation (1) Our aim is to prove the existence of a unique solution of equation (1) for arbitrary α = 0 and η of a general structure, assuming that it satisfies the following boundedness condition: B) η is an F T -measurable random variable of the form where η ∈ L ∞ , γ is a constant and for all stopping times τ for a constant C > 0.
The main statement of the paper is the following Theorem 1.Let condition B) be satisfied.Then there is a constant γ 0 > 0 such that for any |γ| ≤ γ 0 there exists a unique triple (c, m, m ⊥ ), where , that satisfies equation (1).
In case α = −1 and γ = 0 this theorem was proved in [14].In P. Grandits and T. Rheinländer (2002), T. Rheinländer (2005) and D. Hobson (2004) minimal entropy and q-optimal martingale measures are studied using a fundamental representation equation of type where and Z are independent Brownian motions and λ is fixed process.This equation is equivalent to (1) for suitable α, η and for the probability measure d P = E T (−q • 0 λ s dM s )dP , since Bt = B t + q t 0 λ s ds and Z t are independent Brownian motions w.r.t.P .Assumption B) guarantees the solvability of the equation either if ess sup τ Ẽ T τ q 2 λ 2 u du|F τ is small enough or if K T is bounded.One can show that equation ( 1) is equivalent to the following semimartingale backward equation with the square generator We show that there exists a unique triple (Y, L, L ⊥ ), where , that satisfies equation (7).If the filtration F is generated by a multidimensional Brownian motion and if A T is bounded, the existence of a solution of equation ( 7) follows from the results of M. Kobylanski (2000) and J.P. Lepeltier and J. San Martin (1998), where the BSDEs (Backward Stochasic Differential Equations) with generators satisfying the square growth conditions were considered.We prove existence and uniqueness of (7) (or (1)) by different methods.
In section 2, using the BM O norm for martingales (L and L ⊥ ) and the L ∞ ([0, T ] × Ω) norm for semimartingales Y , we apply the fixed-point theorem to show an existence of a solution first in case when L ∞ norm of η and constant γ are sufficiently small.Then we construct the solution for an arbitrary bounded η.
In section 3 we construct a solution of equation ( 1) using the value process of a certain optimization problem, for some values of the parameter α.We give also a necessary condition for equation (1) to admit a solution in the class BM O (or a bounded solution Y for equation (7)).This condition shows that we can't expect an existence of a bounded solution of (7) for arbitrary γ.

Proof of the main Theorem
We recall the definition of BMO-martingales and of a similar notion for the processes of finite variation.
The square integrable continuous martingale M belongs to the class BM O if there is a constant We say that the process B strongly dominates the process A and write A ≺ B, if the difference B − A ∈ A + loc , i.e., is a locally integrable increasing process.We shall use also the notation ϕ • X for the stochastic integral with respect to the semimartingale X.Let N ∈ BM O(P ) and dQ = E T (N )dP .Then Q is a probability measure equivalent to P by Theorem 2. 3 Kazamaki 1994).Denote by ψ = ψ(X) = X, N − X the Girsanov's transformation.It is well known that (see Kazamaki 1994 ) both H 2 and BM O are invariant under transformation ψ.Let M(Q) and M ⊥ (Q) be images of the mapping ψ for M and M ⊥ respectively, .Note that M(Q) and M ⊥ (Q) are stable orthogonal subspaces of the space H 2 (Q) of square integrable martingales with respect to Q.We shall need the following lemma to switch solutions of equation ( 1) for different final random variables.
Let Q be a probability measure defined by and assume that there exists Then there exists a solution of equation Proof.Note that , where m1 = m 1 − m 1 and m⊥ 1 = m ⊥ 1 − m ⊥ 1 are BM O martingales under Q.By Girsanov's theorem m 2 and m ⊥ 2 are special semimartingales under P with the decomposition , m⊥ 1 are BM O(P )-martingales according to Theorem 3.6 of Kazamaki (1994).Girsanov's theorem and the uniqueness of the canonical decomposition of special semimartingales imply that m2 , m Multiplying now equations (2.1) and (2.2) and using Yor's formula we obtain By equality (11)  Proof.Let (c, m, m ⊥ ) and (c , l, l ⊥ ) be two solutions of (1) from the class BMO.Then (1) implies that Recall that if M and N are continuous local martingales then (see, e.g., Jacod 1979) Therefore, from (13) we have that Let Q be a measure defined by 2 m ⊥ − α−1 2 l ⊥ are BM O-martingales with respect to the measure Q according to Theorem 3.6 of Kazamaki (1994) and corresponding Doleans-Dade exponentials are uniformly integrable Q-martingales.Therefore equality (14) holds for all t ∈ [0, T ], which implies that c = c and Hence m − l = α(l ⊥ − m ⊥ ) and since m − l is orthogonal to l ⊥ − m ⊥ , we obtain that m = l and m ⊥ = l ⊥ .Proposition 2a.Let η be a bounded F T -measurable random variable.Then there exists a triple (c, m, m ⊥ ), where Proof.It is evident that equation ( 1) is equivalent to the following martingale equation where α = 0.The latter equation can be also written in the following equivalent semimartingale form as a BSDE Let first show that there exists a solution (c, m, m ⊥ ) of equation ( 16) if |ξ| ∞ is small enough.For brevity we shall use the notation m tT = m T − m t for the square characteristic of a martingale m.
Let consider the mapping which transforms BMO-martingales l and l ⊥ into a triple (Y, L, L ⊥ ), where L and L ⊥ are BMO-martingales and Y is a semimartingale.Using L ∞ ([0, T ] × Ω) norm for semimartingales and BMO norms for martingales, we shall show that if the norm |ξ| ∞ is sufficiently small, then there exists r > 0 such that mapping ( 18) is a contraction in the ball Using the Ito formula for Y 2 T − Y 2 t and ( 18),( 19) we have For any l, l ⊥ ∈ BMO by condition B) and (19) Y is bounded and L and L ⊥ are square integrable martingales.Therefore, the stochastic integral Y • (L + L ⊥ ) is a martingale and taking conditional expectations in (20) we have , then there exists r satisfying the inequality Therefore Now we shall show that the mapping ( 18) is a contraction on the ball B r from the space BM O. Let Y i , L i , L ⊥ i , i = 1, 2 correspond to l i , l ⊥ i , i = 1, 2 by transformation ( 18),(19).Since Y 1 (T ) − Y 2 (T ) = 0 applying the Ito formula for (Y 1 − Y 2 ) 2 similarly to (21) we obtain On the other hand using the Kunita-Watanabe inequality, elementary inequalities (a + b) 2 ≤ 2(a 2 + b 2 ) and l + l ⊥ , l , the relations (23) and (24) imply the inequality Finally we remark that if |ξ| ∞ ≤ 1 4β and 1 8β 2 ≤ r 2 < 1 4β 2 then the inequalities (22) and rβ < 1 are satisfied simultaneously.Thus we obtain that if |ξ| ∞ is small enough, then the mapping ( 18) is a contraction and by fixed point theorem equation (17) (and hence equation ( 1)) admits a unique solution.In particular if |ξ| ∞ ≤ 1 4β then the BMO-norm of the solution is less than To get rid of the assumption that |ξ| ∞ should be small enough, let us use the Lemma 1.Let us take an integer n ≥ 1 so that equation admits a solution.Let dQ = E T (m 1 + m ⊥ 1 )dP , where (m 1 , m ⊥ 1 ) ∈ BM O(P ) is a solution of (25).Since the norm |ξ| ∞ is invariant with respect to an equivalent change of measure and since the Girsanov transformation is an isomorphism of BM O(P ) onto BM O(Q), similarly as above one can show that there exists a pair m 2 , m ⊥ 2 ∈ BM O(Q) that satisfies equation (25).Therefore by Lemma 1, there exists a solution of equation Using now Lemma 1 to equation ( 26) by induction we obtain that there exists a solution of equation (1).
Remark.For the solution (Y, m, m ⊥ ) of ( 17) the following estimate is true Indeed, since m, m ⊥ ∈ BM O, the process Y is a uniformly integrable martingale under Q, where dQ = E T (L + αL ⊥ )dP then (17).Therefore Let us consider now the case, where the final random variable is of the form η = γA T for a constant γ and an increasing process (A t , t ∈ [0, T ]).Proof.The proof is similar to the proof of Proposition 2a.The only difference is that in equation ( 19) ξ is replaced by γ(A T − A t ), where γ = 1 2 γ and equation ( 17) is replaced by the BSDE Therefore, applying the Itô formula for Y 2 t and after using the same arguments we obtain

Solution as a value process of an optimization problem
In this section we shall construct the solution of equation (1) using the value process of a certain optimization problem for some values of parameter α.Let A = (A t , t ∈ [0, T ]) be a continuous F -adapted increasing process.Assume that
s. for every stopping time τ .The smallest constant with this property (or +∞ if it does not exist) is called the BM O norm of M and is denoted by ||M || BM O .For the process of finite variation A we denote by var t s (A) the variation on the segment [s, t].
τ (A)|F τ ) ≤ C, P − a.s.for every stopping time τ , let us denote by |A| ω the smallest constant with this property.