• R. White (2016). On the Exits of Multidimensional Renewal Processes from Hyperrectangles in n Dimensions (manuscript in preparation)

    • This paper establishes an n-dimensional result yielding a simple expression for a joint probability transform of the exit position and time of a renewal process from a hyperrectangle. The transform is under an arbitrary composition of n Laplace-Carson and/or modified z-transforms, and can yield explicit results for special cases with analytic or numerical inversion. I am currently working on one last piece of the result and formalising the completed results for submission to a journal. 

  • J. H. Dshalalow and R. White (2016). Random walks on random lattices with applications (submitted)

    • This paper extends the previous paper's results to arbitrarily many dimensions with arbitrarily many random times of interest for independent and stationary increment processes. These results are used to improve the results of the stochastic network problem and applied to additional topics.

  • J. H. Dshalalow and R. White (2016). Time sensitive analysis of independent and stationary increment processes. Journal of Mathematical Analysis and Applications.
    doi:10.1016/j.jmaa.2016.05.063

    • This paper first develops results depending on real time as well as upon an observation process as in previous papers, but for a much wider class of processes (independent and stationary increments) in 1 space dimension. These are applied to probabilistically "interpolate" observation-dependent results to predict the probabilistic status of the stochastic process at times between observations to get a better idea of what happens in random time vicinities of a threshold crossings.

Academic Papers

RESEARCH

Random Walks on Random Lattices and Their Applications (May 2015) [PDF] [Defense Slides]

 

Analysis of multidimensional stochastic processes with mutually dependent components and their fluctuations about thresholds. The process experiences multidimensional increments of random magnitude (for each component) upon random times, while the values of the components are viewed by an independent delayed random process rather than in real-time. I consider these types of processes particularly in the context of successive losses of weighted nodes and edges in random graphs as well as more general marked point processes.

Dissertation
  • J.H. Dshalalow and R. White (2013). On the reliability of stochastic networks (jointly with J. H. Dshalalow). Neural, Parallel, and Scientific Computations, 21: 141-160.

    • ​Detection and prediction of losses due to attacks randomly removing weighted network components under delayed observation. Probabilistic functionals of the predicted status of the network upon observed intermediate and critical threshold crossings (and just before such crossings), predicted time remaining before such crossings, and the like. We demonstrate the analytical tractability of more specific results for some versatile special cases.

Presentations
  • Random Walks on Random Lattices and Their Applications. FIT MTH/ORP Graduate Student Seminar. 21 Apr 2015

  • Time Sensitive Analysis of Multivariate Marked Random Walks. SIAM Conference on Computational Science and Engineering. Salt Lake City, UT. 14-18 Mar 2015 [Slides]

  • Time Sensitive Analysis of ISI Processes. FIT Mathematical Sciences Symposium. 8 Jan 2015 [Slides]

  • Random Walks on Random Lattices: An Operational Calculus Approach. FIT MTH/ORP Graduate Student Seminar. 30 Sept 2014 [Slides]

  • Stochatic Analysis of Strategic Networks. 38th Annual SIAM Southeastern Atlantic Section Conference. Melbourne, FL. 29-30 Mar, 2014 [Slides]

Non-technical description

Consider some counters that increase over time, each with a threshold. The counters increase at random times by random amounts such that each is related to one another.

 

We answer questions such as

  • Which threshold is likely to be crossed first?

  • How long will it likely take for this first threshold crossing to occur?

  • By how much will it likely surpass its threshold?

  • What can we adjust to most effectively increase (or decrease) the time until a threshold crossing?

  • J. H. Dshalalow, K. Iwezulu, and R. White (2016). Discrete Operational Calculus in Delayed Stochastic Games. Neural, Parallel, and Scientific Computations, 24: 55-64.

    • This article deals with classes of antagonistic games with two players. A game is specified in terms of two “hostile” stochastic processes representing mutual attacks upon random times that exert casualties of random magnitudes. The game ends when one of the players is defeated, that is, when the amounts of casualties to the players cross respective tolerance thresholds. We target the first passage time of the defeat and the amount of casualties to either player upon this time. We derive analytic tractability in formulas obtained under various transforms.

  • J.H. Dshalalow and R. White (2014). On strategic defense in stochastic networks. Stochastic Analysis and Applications, 32(3): 365-396. [Prepublication PDF] [Full text]

    • This paper lays out the basis of a two-component (mutually dependent) stochastic process associated with a losses of weighted graph components, prediction of time of entry to a critical state, the extent of losses upon observation of entry of the critical state, and other probabilistic information. System simulation demonstrating the accuracy of mathematically derived results for special cases.

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