Suppose $X$ is a Markov diffusion process on $R^p$, or more generally on a manifold $N$. The diffusion variance of $X$ induces a semi-definite metric $\langle\cdot\mid\cdot\rangle$ on the cotangent bundle, a version of the Levi-Civita connection $\Gamma$ and a Laplace-Beltrami operator $\Delta$. We may treat $X$ as a diffusion on $N$ with generator $\xi + (1/2)\Delta$, where $\xi$ is a vector field. For sufficiently small $\delta > 0$, $X_\delta$ has an ``intrinsic location parameter'', defined to be the non-random initial value $V_0$ of a $\Gamma$-martingale $V$ terminating at $X_\delta$. It is obtained by solving a system of forward-backwards stochastic differential equations (FBSDE): a forward equation for $X$, and a backwards equation for $V$. This FBSDE is the stochastic equivalent of the heat equation (with drift $\xi$) for harmonic mappings, a well-known system of quasilinear PDE. Let $\{\phi_t: N \rightarrow N, t \geq 0\}$ be the flow of the vector field $\xi$, and let $x_t \equiv \phi_t(x_0) \in N$. Our main result is that $\exp^{-1}_{x_\delta}V_0$ can be intrinsically approximated to first order in $T_{x_\delta}N$ by $$ \nabla d\phi_\delta(x_0)(\Pi_\delta) - \int_{0}^{\delta}(\phi_{\delta - t})_{*}(\nabla d\phi_t(x_0))d\Pi_t $$ where $\Pi_t = \int_{0}^{t}(\phi_{-s})_{*}\langle\cdot\mid\cdot\rangle_{x_s}ds \in T_{x_0}N \otimes T_{x_0}N$. This is computed in local coordinates. More generally, we find an intrinsic location parameter for $\Psi(X_\delta)$, if $\Psi:N \rightarrow M$ is a $C^2$ map into a Riemannian manifold $M$.

These formulas have immediate application, discussed in a separate article, to the construction of an intrinsic nonlinear analog to the Kalman Filter.