Let ${𝔽}_{}^{}$F be a filtration andτbe a random time. Let ${𝔾}_{}^{}$G be the progressive enlargement of ${𝔽}_{}^{}$F withτ. We study the following formula, called the optional splitting formula: For any ${𝔾}_{}^{}$G-optional processY, there exists an ${𝔽}_{}^{}$F-optional processY′ and a function Y′′ defined on [0,∞] × (ℝ+ × Ω) being $ℬ[0,\infty ]\otimes 𝒪\left({𝔽}_{}^{}\right)$ℬ[0,∞]⊗x1d4aa;(F) measurable, such that $Y=Y^{\text{'}}{1}_{[0,\tau )}^{}+Y^{\text{'}\text{'}}\left(\tau \right){\mathbb{1}}_{[\tau ,\infty )}^{}.$Y=Y′1[0,τ)+Y′′(τ)1[τ,∞). (This formula can also be formulated for multiple random timesτ1,...,τk). We are interested in this formula because of its fundamental role in many...