Abstract
Strong solutions of p -dimensional stochastic differential equations d X t = b ( X t , t ) d t + σ ( X t , t ) d W t , X s = x that can be represented locally in explicit simulation form X t = ϕ x , s ( ∫ s t V s , u d W u , t ) are considered. Here; W is a multidimensional Brownian motion; u → V s , u , ϕ x , s are continuous functions; and b , σ , ϕ x , s are locally continuously differentiable. The following three-way equivalence is established: 1) There exists such a representation from all starting points ( x , s ) , 2) V s , u , ϕ x , s satisfies a set differential equations, and 3) b , σ satisfy commutation relations. (For generality, the function V s , t is allowed to depend upon ϕ x , s via V s , t = U s , t ϕ x , s for some operators U s , t .) Moreover, construction theorems, based on a diffeomorphism between the solutions X and the strong solutions to a simpler Itô integral equation, with a possible deterministic component, are given. Finally, motivating examples are provided and its importance in simulation methods, including sequential Monte Carlo, financial risk assessment and path-dependent option pricing, is explained.
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