Project Details
Abstract
Wider research context
Going back to the seminal works of Strassen (1965) and Kellerer (1972) the following ``mimicking problem'' has continued to attract attention: Given the marginal distributions of a (sub-) martingale, construct a simpler Markovian (sub-) martingale that mimics (or emulates) the original process in the sense that it has the same one-dimensional marginal distributions.
Besides its theoretical appeal, the problem represents the quintessential version of the ubiquitous calibration problem in mathematical finance where the task is to construct martingales that satisfy marginal constraints imposed by market data.
Research questions
Despite the long history of the problem, seemingly simple questions have remained open.
While several groups have revisited Kellerer's theorem on the existence of Markov martingales for given marginals on the real line, the challenge to prove a multi-dimensional counterpart is still standing wide open after almost 50 years.
Another example of an important problem in the field of Mathematical Finance consists in establishing the existence of solutions to the so called stochastic local volatility model where the goal is to mimic marginal distributions as well as dynamics. Despite substantial interest by financial practitioners, rigorous mathematical foundations are lacking almost entirely.
Approach
In the proposed project we approach these problems based on the emerging theory of probabilistic transport optimal transport. Specifically, we will combine results on adapted Wasserstein distances and their connection with the mimicking challenge established by the authors and their collaborators with insights on the (multi-dimensional) martingale transport problem achieved by Ghoussoub-Kim-Lim, de March-Touzi, and Obloj-Siorpaes.
Innovation
Some of the proposed goals of the project represent natural problems in stochastic analysis that have been open since almost 50 years. Others do not have a comparable history but are of specific importance due to their relevance in applications. The project is innovative in that it applies methods that have been emerging in the last ten years to challenges that have been difficult to approach using approaches from a classical stochastic analysis viewpoint.
Primary researchers involved
This proposal was drafted by Mathias Beiglböck and Walter Schachermayer in close collaboration. Both authors have actively contributed to research areas relevant for the proposed project (stochastic analysis, optimal transport, mathematical finance) and have also collaborated on questions that are closely related to the proposed project. We also have accumulated experience in guiding PhD-students and Postdocresearchers as well as in leading third party funded projects.
Going back to the seminal works of Strassen (1965) and Kellerer (1972) the following ``mimicking problem'' has continued to attract attention: Given the marginal distributions of a (sub-) martingale, construct a simpler Markovian (sub-) martingale that mimics (or emulates) the original process in the sense that it has the same one-dimensional marginal distributions.
Besides its theoretical appeal, the problem represents the quintessential version of the ubiquitous calibration problem in mathematical finance where the task is to construct martingales that satisfy marginal constraints imposed by market data.
Research questions
Despite the long history of the problem, seemingly simple questions have remained open.
While several groups have revisited Kellerer's theorem on the existence of Markov martingales for given marginals on the real line, the challenge to prove a multi-dimensional counterpart is still standing wide open after almost 50 years.
Another example of an important problem in the field of Mathematical Finance consists in establishing the existence of solutions to the so called stochastic local volatility model where the goal is to mimic marginal distributions as well as dynamics. Despite substantial interest by financial practitioners, rigorous mathematical foundations are lacking almost entirely.
Approach
In the proposed project we approach these problems based on the emerging theory of probabilistic transport optimal transport. Specifically, we will combine results on adapted Wasserstein distances and their connection with the mimicking challenge established by the authors and their collaborators with insights on the (multi-dimensional) martingale transport problem achieved by Ghoussoub-Kim-Lim, de March-Touzi, and Obloj-Siorpaes.
Innovation
Some of the proposed goals of the project represent natural problems in stochastic analysis that have been open since almost 50 years. Others do not have a comparable history but are of specific importance due to their relevance in applications. The project is innovative in that it applies methods that have been emerging in the last ten years to challenges that have been difficult to approach using approaches from a classical stochastic analysis viewpoint.
Primary researchers involved
This proposal was drafted by Mathias Beiglböck and Walter Schachermayer in close collaboration. Both authors have actively contributed to research areas relevant for the proposed project (stochastic analysis, optimal transport, mathematical finance) and have also collaborated on questions that are closely related to the proposed project. We also have accumulated experience in guiding PhD-students and Postdocresearchers as well as in leading third party funded projects.
| Status | Active |
|---|---|
| Effective start/end date | 1/12/21 → 30/11/25 |
Keywords
- stochastic analysis
- optimal transport
- Mathematical Finance
- martingales