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Linear inverse problems for diffusions

Banff International Research Station for Mathematical Innovation and Discovery

Let \( (P_t)_{t \geq 0} \) be the semigroup corresponding to a one-dimensional diffusion, \( X \), with speed measure \( m \). After giving a brief motivation from market microstructure theory I will discuss \( L^2 \) solutions of the integral equation \( g=P_T f \) for a given deterministic \( T>0 \) and \( g \in L^2(m) \). If \( g \) is a probability density, the solution of this problem amounts to finding an initial distribution for \( X \) so that its law at time-\( T \) is defined by \( g dm \). In such a setting this inverse problem can also be viewed as an alternative to finding a martingale that vanishes at time-0 and has the required distribution at time-\( T \). The crucial difference between the two is that the dynamics of the martingale is already given in the formulation of the inverse problem and one looks for an appropriate initial distribution. Although fixing the martingale might be desirable for the modeller for numerical or empirical reasons, the catch is that the inverse problem does not always have a solution since the inverse of \( P_T \) is an unbounded operator. In this talk I will give a necessary and sufficient condition for the existence of an \( L^2 \) solution to the inverse problem described above and present a formula for the inversion.

Mathematics; Game theory, economics, social and behavioral sciences; Probability theory and stochastic processes

video/mp4

Author affiliation: London School of Economics

2017-01-26

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/59812

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/