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Time homogeneous diffusions with a given marginal at a deterministic time
- Speaker(s)
- John M. Noble
- Affiliation
- Linköping University, Sweden
- Date
- May 12, 2011, 12:15 p.m.
- Room
- room 3130
- Seminar
- Seminar of Probability Group
This talk outlines a method for constructing a generalised martingale diffusion such that the process, at a given deterministic time t > 0, has a specified probability measure. The construction gives existence of a process for any measure with bounded support in R, but does not give uniqueness.
The outline of the proof is the following: firstly, it is straightforward to establish an explicit formula for finding an appropriate diffusion with a given marginal at an exponential time, for a finite state space, using the resolvent. This argument was presented by Jan Obłój in his seminar. It is reasonably straightforward to extend this, using a fixed point argument, to show existence of a process for a random time with Gamma distribution, using the fact that a Gamma distribution is the sum of independent exponential distributions. Then, keeping the expectation of the random time fixed and altering the parameters so that the coefficient of variation vanishes, the limiting process satisfies the required criteria. Finally, the result can be extended to arbitrary probability measures with bounded support.
The outline of the proof is the following: firstly, it is straightforward to establish an explicit formula for finding an appropriate diffusion with a given marginal at an exponential time, for a finite state space, using the resolvent. This argument was presented by Jan Obłój in his seminar. It is reasonably straightforward to extend this, using a fixed point argument, to show existence of a process for a random time with Gamma distribution, using the fact that a Gamma distribution is the sum of independent exponential distributions. Then, keeping the expectation of the random time fixed and altering the parameters so that the coefficient of variation vanishes, the limiting process satisfies the required criteria. Finally, the result can be extended to arbitrary probability measures with bounded support.
