M. Mania^{∗} and R. Tevzadze^{∗∗}

∗A. Razmadze Mathematical Institute, Georgian Academy of Sciences

Tbilisi, Georgia ∗∗Institute of Cybernetics, Georgian Academy of Sciences Tbilisi, Georgia

AN EXPONENTIAL MARTINGALE EQUATION

Let (Ω, F, P) be a probability space with continuous filtration F = (F_{t}, t ∈ [0, T]). Denote by M a stable subspace of the space of square integrable martingales and let M^{⊥} be its strongly orthogonal complement.

We consider the following exponential equation

E_{T} (m)E_{T}^{α}(m^{⊥}) = c exp{η},  (1)

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.

Equations of such type arose in mathematical finance and they are used to characterize optimal martingale measures.

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:

A) η is an F_{T} -measurable random variable of the form η = ξ + γA_{T} , where ξ ∈ L^{∞}, γ is a constant and A = (A_{t}, t ∈ [0, T]) is a continuous F - adapted increasing process such that E(A_{T} − A_{τ} /F_{τ}) ≤ C for all stopping times τ for a constant C > 0.

Note that if α ≠ 0 (if α = 0, solution of (1) does not exist in general), equation (1) is equivalent to the following semimartingale backward equation with the square generator

Y_{t} = Y_{0} − \frac{γ}{2} A_{t} − ⟨L⟩_{t} − \frac{1}{α} ⟨L^{⊥}⟩_{t} + L_{t} + L_{t}^{⊥}, Y_{T} = \frac{1}{2} ξ.  (2)

Theorem 1. Let α ≠ 0 and condition A) 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 c ∈ R_{+}, m ∈ BMO ∩ M, m^{⊥} ∈ BMO ∩ M^{⊥}, that satisfies equation (1).

If the filtration generated by 2-dimensional Wiener processs (w, w^{⊥}) and

ξ = 2g(w_{T} , w_{T}^{⊥}), A_{t} = ∫_{0}^{t} f(s, w_{s}, w_{s}^{⊥})ds for some continuous, bounded func-

tions g(x, y), f(s, x, y), equation (2) is equivalent to the PDE

V_{t}+ \frac{1}{2} V_{xx}+ \frac{1}{2} V_{yy} +|V_{x}|^{2}+ \frac{1}{α} |V_{y}|^{2}+ \frac{γ}{2} f(t, x, y)=0, V (T, x, y)=g(x, y) (3)

and solutions of (2) and (3) are related as Y_{t} = V (t, w_{t}, w_{t}^{⊥}).