Abstract
Suppose \(Z=(Z_t)_{t\ge0}\) is a normal martingale which satisfies the structure equation
. By adapting and extending techniques due to Parthasarathy and to Kurtz, it is shown that, if α is locally bounded and β has values in the interval [-2,0], the process Z is unique in law, possesses the chaotic-representation property and is strongly Markovian (in an appropriate sense). If also β is bounded away from the endpoints 0 and 2 on every compact subinterval of [0,∞] then Z is shown to have locally bounded trajectories, a variation on a result of Russo and Vallois.
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A. C. R. Belton acknowledges support from the European Community’s Human Potential Programme under contract HPRN-CT-2002-00279, QP-Applications.
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Attal, S., Belton, A.C.R. The chaotic-representation property for a class of normal martingales. Probab. Theory Relat. Fields 139, 543–562 (2007). https://doi.org/10.1007/s00440-006-0052-z
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DOI: https://doi.org/10.1007/s00440-006-0052-z
Keywords
- Azéma martingale
- Chaotic-representation property
- Normal martingale
- Predictable-representation property
- Structure equation