Journal
ANNALS OF PROBABILITY
Volume 40, Issue 1, Pages 74-102Publisher
INST MATHEMATICAL STATISTICS
DOI: 10.1214/10-AOP616
Keywords
Large deviations; interacting random processes; McKean-Vlasov equation; stochastic differential equation; delay; weak convergence; martingale problem; optimal stochastic control
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Funding
- Army Research Office [W911NF-0-1-0080, W911NF-10-1-0158, W911NF-09-1-0155]
- NSF [DMS-10-04418, DMS-07-06003]
- US-Israel Binational Science Foundation [2008466]
- Air Force Office of Scientific Research [FA9550-09-1-0378, FA9550-07-1-0544]
- German Research Foundation (DFG)
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1008331] Funding Source: National Science Foundation
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1004418] Funding Source: National Science Foundation
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We study large deviation properties of systems of weakly interacting particles modeled by Ito stochastic differential equations (SDEs). It is known under certain conditions that the corresponding sequence of empirical measures converges, as the number of particles tends to infinity, to the weak solution of an associated McKean-Vlasov equation. We derive a large deviation principle via the weak convergence approach. The proof, which avoids discretization arguments, is based on a representation theorem, weak convergence and ideas from stochastic optimal control. The method works under rather mild assumptions and also for models described by SDEs not of diffusion type. To illustrate this, we treat the case of SDEs with delay.
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