The main problem of the paper is to find a martingale on a manifold with a fixed random terminal value; this can be done by solving a backward SDE with quadratic growth (which means, that classical Lipschitz conditions etc.for existence and uniqueness of solutions fail). Let \(M\) be a manifold with connection \(\Gamma\) and a related exponential map \(\exp\). The author considers the following backward stochastic differential equation (in infinitesimal form):
\[ X_{t+dt} = \exp_{X_t}(Z_t dW_t + f(B_t^y, X_t, Z_t)dt), \quad X_T = U, \]
where \(W_t\) is \(d\)-dimensional Brownian motion. Assuming that \(M\) can be represented by a global chart, this becomes \[ dX_t = Z_t dW_t + (-\tfrac 12 \Gamma_{jk}(X_t)\langle [Z_t]^k , [Z_t]^j\rangle + f(B_t^y, X_t, Z_t)\,dt), \quad X_T =U, \] \([Z_t]^k\) denotes the \(k\)th row of the matrix \(Z_t\). The main result of the paper is the following: Let \(\omega\) be a relatively compact open subset of an open set \(O\subset M\), where \(O\) is such that there is a unique \(O\)-valued geodesic between any two points of \(O\). If \(f\) is outward pointing on the boundary of \(\bar\omega\), if \(U\in\bar\omega = \{\chi\leq c\}\) with a strictly convex \(\chi\) and if \(f\) satisfies some further technical conditions, then (i) the above backward SDE has a unique solution \((Z_t,X_t)_{0\leq t\leq T}\) such that \(X_t\) remains in \(\bar \omega\) if \(f\) does not depend on \(z\); (ii) if \(M\) is a Cartan-Hadamard manifold and if the Levi-Cività connection is being used, then the solution of the backward SDE has a unique solution \((Z_t,X_t)_{0\leq t\leq T}\) with \(X\) in \(\bar\omega\).