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Financial institutions and insurance companies that analyze the evolution andsources of profits and losses often look at risk factors only at discretereporting dates, ignoring the detailed paths. Continuous-time decompositionsavoid this weakness and also make decompositions consistent across differentreporting grids. We construct a large class of continuous-time decompositionsfrom a new extended version of It\^o's formula and uniquely identify apreferred decomposition from the axioms of exactness, symmetry andnormalization. This unique decomposition turns out to be a stochastic limit ofrecursive Shapley values, but it suffers from a curse of dimensionality as thenumber of risk factors increases. We develop an approximation that breaks thiscurse when the risk factors almost surely have no simultaneous jumps.

Profit and loss decomposition in continuous time and approximations