# Why is Backward SDE more difficult than forward SDE?

I need to explain Backward Stochastic Differential Equation (BSDE) for some non-mathematicians. The audiences are most likely familiar with ODE/PDE as physicists.

One concern is probably that why BSDE is essentially different to FSDE (Forward SDE), while one can easily find change of variable in time to make equivalent between forward ODE and backward ODE in time.

To be more specific, let's consider following deterministic Backward ODE: $$d Y_t = Y_t dX_t, \ Y_T = \xi$$ where $X_t$ is a given smooth function in $t$. In this case, one can use $Z_t = Y_{T-t}$ to convert the equation into forward ODE: $$d Z_t = - Z_t d X_{T-t}, \ Z_0 = \xi.$$

However, similar approach does not work if $X_t$ is a Brownian motion. I want to explain this point in plain word without measure theory or filtration involved. In particular, I want to convince them $dX_t$ makes sense by Ito's definition, while $dX_{T-t}$ does not. Any good idea?

• I meant that a (stochastic) $Z_t$ as in your example is not adapted, so you can't establish existence using the usual framework as in FSDEs. – parsiad Mar 24 '16 at 17:18