, \quad 
\xi_k = \frac{B_{t_{k}} - B_{t_{k-1}}}{t_k - t_{k-1}} , 
\quad k \in -\N$$ satisfy the equation . Conversely, we can reconstruct the process $ (X_t) $ from the processes $ (\eta_k) $ and $ (B_t) $. Hence we may say that all properties of the SDE can be deduced from those of the equation .

To generalize the stochastic equation , we introduce a class of stochastic equations in discrete negative time, which we call [*Tsirelson–Yor equations*]{}. Let $ S $ be a Polish space and $ G $ a compact group. We introduce an operation of $ G $ on the state space $ S $ through a measurable map $ \psi:G \times S \to S $ and consider a measurable map $ \theta:S \to G $. Then our Tsirelson–Yor equation is of the form $$\eta_k = \psi {\left(  \theta(\eta_{k-1}),\xi_k  \right)} , 
\quad k \in -\N. 
\label{TYE}$$ The key to the generalization is the following [*commutation*]{} condition[^1]: $$\theta {\left(  \psi( g,s )  \right)} = g \cdot \theta(s), \quad g \in G, \ s \in S. 
\label{UeniC}$$ This condition allows us to reduce the equation to the equation $$\theta(\eta_k) = \theta (\eta_{k-1}) \cdot \theta(\xi_k) , 
\quad k \in -\N. 
