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Expected signature of Brownian motion up to the first exit time from a bounded domain. (English) Zbl 1350.60086

The authors study the expected signature of a Brownian motion up to the first exit time from a bounded domain in a Euclidean space. The main contribution of the study is to prove that the tensor series valued function satisfies an elliptic PDE system under a boundary condition. Based on an iterative technique, the authors justify the geometric bounds for the terms in the value function. This research extends existing results to a stopping time situation of a random terminal time and presents some open problems for future study, in terms of the law of the signature.

MSC:

60J65 Brownian motion
60J60 Diffusion processes
60G40 Stopping times; optimal stopping problems; gambling theory
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J35 Transition functions, generators and resolvents
47D07 Markov semigroups and applications to diffusion processes
47D03 Groups and semigroups of linear operators
35K05 Heat equation
35K08 Heat kernel
35K10 Second-order parabolic equations
35K51 Initial-boundary value problems for second-order parabolic systems
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References:

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