First Advisor

Jong Sung Kim

Term of Graduation

Spring 2023

Date of Publication


Document Type


Degree Name

Doctor of Philosophy (Ph.D.) in Mathematical Sciences


Mathematics and Statistics




Investment analysis, Ito processes, Random times, Stochastic calculus, Stochastic filtering, Survival times



Physical Description

1 online resource (x, 213 pages)


The central statistical problem of survival analysis is to determine and characterize the conditional distribution of a survival time given a history of some observed health markers.

This dissertation contributes to the modeling of such conditional distributions in a setup where the health markers evolve randomly over time in a manner that can be represented by an Ito stochastic process, that is, a stochastic process that can be written as a sum of a time integral of some stochastic process and an Ito integral of some stochastic process, with both integrands subject to certain restrictions.

The random survival time is modeled as a deterministic function of a generating random variable that is related to the random evolution of the health markers, where the deterministic function is chosen so that the survival time has the desired distribution function.

The dissertation presents two families of such models. In the first family of models, the generating variable of the survival time is an Ito integral over the positive half-line, with the observable health marker at any given time represented by the same integral up to that time.

The second family of models involves a linear filtering framework, in which the generating variable affects linearly a number of observable health markers that evolve as Ito processes.

The dissertation offers formulas for conditional distribution, survival, and hazard functions of the survival time, and the relevance of each model is demonstrated with a simulation.

This application of a filtering model is not limited to the analysis of survival times. The dissertation shows that instead of a positive survival time we can use a return to a financial asset which can be positive or negative.

To apply the model in that context, we need to determine the distribution of log returns. The dissertation includes a goodness-of-fit investigation of some possible statistical distributions of a long history of log returns to the S & P 500 stock market index, concluding that we can use the generalized hyperbolic distribution to describe such returns.

With a number of investors, who think in terms of a normal distribution of log returns, providing the observable forecast markers, there is the problem of forcing the convergence of forecast markers to actual log returns at the end of the forecasting period. That problem is solved by using a multi-dimensional Brownian bridge process to model forecasting error.


©2023 Zhenzhen Li

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