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SUMMARY:A Closed-Form Transition Density Expansion for Elliptic and Hypo-E
 lliptic SDEs - Alexandros Beskos (University College London)
DTSTART:20241111T150000Z
DTEND:20241111T160000Z
UID:TALK223834@talks.cam.ac.uk
DESCRIPTION:Co-author: Mr Yuga Iguchi\n&nbsp\;\nWe introduce a closed-form
  expansion for the transition density of elliptic and hypo-elliptic multiv
 ariate Stochastic Differential Equations (SDEs)\, over a period $\\Delta\\
 in (0\,1)$\, in terms of powers of $\\Delta^{k/2}$\, $k\\ge 0$. Our method
 ology provides a tractable approximation of the true transition density\, 
 which can be easily evaluated via any software that carries out symbolic c
 alculations. A major part of the paper is committed to obtaining analytica
 l results about the size of the residual in our closed-form expansion for 
 fixed $\\Delta\\in(0\,1)$. The produced error bound validates the methodol
 ogy\, by providing a guarantee of increased precision when including extra
  terms in our proxy and by characterising the size of the distance from th
 e true value.&nbsp\;\nIt is the first time that such a closed-form expansi
 on becomes available for the important class of hypo-elliptic SDEs\, to th
 e best of our knowledge. For elliptic SDEs\, closed-form expansions are av
 ailable\, with previous works identifying the size of the error for fixed 
 $\\Delta$\, as per our own contribution. Our methodology follows an approa
 ch allowing for a uniform treatment of elliptic and hypo-elliptic SDEs\, w
 hen earlier works are intrinsically restricted to an elliptic setting.&nbs
 p\;\nWe show numerical applications that highlight the effectiveness of ou
 r method\, by carrying out parameter inference for SDE models that do not 
 necessarily satisfy stated conditions. The latter are sufficient for an an
 alytical control of the errors\, but the closed-form expansion itself is a
 pplicable in general settings.
LOCATION:Seminar Room 2\, Newton Institute
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