A regularized Kellerer theorem in arbitrary dimension
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909630580391936 |
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| author | Pammer, Gudmund Robinson, Benjamin A. Schachermayer, Walter |
| author_facet | Pammer, Gudmund Robinson, Benjamin A. Schachermayer, Walter |
| contents | We present a multidimensional extension of Kellerer's theorem on the existence of mimicking Markov martingales for peacocks, a term derived from the French for stochastic processes increasing in convex order. For a continuous-time peacock in arbitrary dimension, after Gaussian regularization, we show that there exists a strongly Markovian mimicking martingale Itô diffusion. A novel compactness result for martingale diffusions is a key tool in our proof. Moreover, we provide counterexamples to show, in dimension $d \geq 2$, that uniqueness may not hold, and that some regularization is necessary to guarantee existence of a mimicking Markov martingale. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_13847 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A regularized Kellerer theorem in arbitrary dimension Pammer, Gudmund Robinson, Benjamin A. Schachermayer, Walter Probability 60G44, 60J60, 60H10, 60J25 (Primary) 60F99 (Secondary) We present a multidimensional extension of Kellerer's theorem on the existence of mimicking Markov martingales for peacocks, a term derived from the French for stochastic processes increasing in convex order. For a continuous-time peacock in arbitrary dimension, after Gaussian regularization, we show that there exists a strongly Markovian mimicking martingale Itô diffusion. A novel compactness result for martingale diffusions is a key tool in our proof. Moreover, we provide counterexamples to show, in dimension $d \geq 2$, that uniqueness may not hold, and that some regularization is necessary to guarantee existence of a mimicking Markov martingale. |
| title | A regularized Kellerer theorem in arbitrary dimension |
| topic | Probability 60G44, 60J60, 60H10, 60J25 (Primary) 60F99 (Secondary) |
| url | https://arxiv.org/abs/2210.13847 |