A closed-form transition density expansion for elliptic and hypo-elliptic SDEs

We introduce a closed-form expansion for the transition density of elliptic and hypo-elliptic multivariate Stochastic Differential Equations (SDEs), over a period Δ∈(0,1), in terms of powers of Δj∕2, j≥0. Our methodology provides approximations of the transition density, easily evaluated via any software that performs symbolic calculations. A major part of the paper is devoted to an analytical control of the remainder in our expansion for fixed Δ∈(0,1). The obtained error bounds validate theoretically the methodology, by characterising the size of the distance from the true value. For elliptic SDEs, closed-form expansions are available, with some works identifying the size of the error for fixed Δ, as per our contribution. Our methodology allows for a uniform treatment of elliptic and hypo-elliptic SDEs, when earlier works are intrinsically restricted to an elliptic setting. We show numerical applications highlighting the effectiveness of our method, by carrying out parameter inference for hypo-elliptic SDEs that do not satisfy stated conditions. The latter are sufficient for controlling the remainder terms, but the closed-form expansion itself is applicable in general settings.