Stochastic Processes and Applications

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The proposal involves: 1) replicating processes onto compact metric spaces for systematic study; 2) studying particle systems and measure-valued processes with new features; and 3) engineering of better particle algorithms. Synergies and interest will be gained through vertical integration. Motivation derives from tracking, prediction, signal processing, disease evolution, fraud detection, spot advertising, information spread, weather prediction, finance, insurance, stochastic equation simulation as well as the mathematics of long memory, random environments and bad spaces. ******General results for the: 1) existence and uniqueness of nonlinear filters, martingale problems, stochastic partial differential equations or Dirichlet forms; 2) existence and uniqueness of stationary processes described these ways; 3) tightness and convergence of finite-dimensional distributions, progressive processes, cadlag processes or approximate filters; or 4) general theory of Markov processes, Dirichlet forms, martingale problems or large deviations; are mostly only for Polish or compact spaces. One treats problems on exotic spaces on a case-by-case basis. We will extend general existence, uniqueness, tightness, convergence, stationary distribution, Markov process, Dirichlet form, filtering and large deviation results from compact Polish spaces to quite general topological spaces. ******Superprocesses, Fleming-Viot processes, interacting particle systems and branching processes are motivated biologically by interacting, branching and competing individuals of possibly multiple types but mostly lack real-world features. For example, the observation individuals basically disappear and then re-appearing elsewhere has been modeled by (random direction and) Levy flight times but a rough environment with valleys at preferred locations may explain the observation better than just heavy tails. We will study well-posedness, support properties, long-time behavior and simulation-facilitating representations for new and existing particle and measure-valued processes.******Sequential Monte Carlo (SMC) methods process big data in real time; are used for classifying, locating, tracking, predicting, model selection and parameter estimation; and have potential new application in medical imaging, fraud detection, hidden liquidity, network security, option pricing, insurance analysis and stochastic convergence. For example, option pricing often uses Monte Carlo simulation with slow and inaccurate Euler or Milstein methods. Alternatively, one can simulate incorrectly but quickly with importance sampling weights to correct but weight discrepancy and effective degeneration to few particles occurs. SMC is a solution: We interact, branch or resample to even weights, keeping the ancestral paths for pathspace pricing. We will produce new SMC algorithms, applications and analysis.