Research

• Choi, B. S. and T. M. Cover (1984) An information theoretic proof of Burg’s maximum entropy spectrum, Proceedings of the IEEE, vol. 72, pp. 1094-1095.

Tom Cover was my mentor, teacher, Ph.D. advisor, and friend. He previously served as President of the IEEE Information Theory Society. In 1990 he received the Claude E. Shannon Award, regarded as the highest honor in information theory; in 1997 he received the IEEE Richard W. Hamming Medal; and in 2003 he was elected to the American Academy of Arts and Sciences. 

Professor Cover himself was very proud of this paper. He had shared with me some laudatory letters regarding this paper from his admirers. In the letters, Professor Cover and I were described as geniuses. At the time of its publication, the journal’s circulation had already surpassed 200,000 (nowadays, over 425,000). It was also one of the finalists in the journal’s Best Paper for the year 1984 and is the main body of Chapter 12 of the book Elements of Information Theory by Cover and Thomas.

• Csiszár, I., T. M. Cover and B. S. Choi (1987) Conditional limit theorems under Markov conditioning, IEEE Trans. Information Theory, vol. IT-33, pp. 788-801.

I had the privilege of learning information theory from Professor Thomas M. Cover and Professor Imre Csiszár, a Hungarian mathematician and recipient of the 1996 Claude E. Shannon Award and 2015 IEEE Richard Hamming Medal.

The results of this paper supports the “maximum entropy” or “minimum discrimination information” principle: if new information requires “updating” of an original probability assignment, the new probability assignment should be the closest possible to the original in the sense of Kullback-Leibler information divergence.

As co-authors with Professor Csiszár, Professor Cover and I have Erdös Number of two. This paper became a part of Chapter 11 in the book Elements of Information Theory by Cover and Thomas, and presents theoretical motivation to cross entropy loss frequently used in Deep Learning today. 

• Choi, B. S. (2003) A fundamental theorem of algebra, spectral factorization, and stability of 2-D systems, IEEE Trans. Signal Processing, vol. 51, pp. 853-863.

In his 1797 doctoral dissertation, my great-grand academic ancestor Carl Friedrich Gauss proved the fundamental theorem of algebra, which states that any 1-D polynomial of degree n with complex coefficients can be factored into a product of n polynomials of degree 1. Since then, the question of factorizing a 2-D polynomial into a product of basic polynomials has remained an open problem.

In this paper I finally solved it. When I sent the manuscript to the journal, I got a very detailed and useful reviewer’s report about this paper. The reviewer commented, “ ... It presents a nice unified point of view concerning the problem of factorization of 2-D power spectra. This problem is known to be important and hard. It has plagued MDSP folks for more than thirty years. ...”

• Choi, B. S. (2003) Model specification of a non-causal 3-D AR process using a causal 3-D AR model on the nonsymmetric half-space, Multidimensional Systems and Signal Processing, vol. 14, pp. 319-341.

I mention this paper in this list because I hope it will be popularly read and used in 3-D image analysis, particularly in Machine Learning. 

In this paper a formula is presented to relate the AR coefficients of the non-causal 3-D AR process with those of the causal 3-D AR process on the non-symmetric half-space, which I call the autocorrelation equivalence relation. Also, the 3-D Yule-Walker equations for the causal 3-D AR model and a computationally efficient Levinson-type order-recursive algorithm are proposed. Using the relation, the equations, and the algorithm, one can specify a good model of a 3-D image through a non-causal 3-D AR process.

• Choi, B.S, and M.Y. Choi (2018) General solution of the Black-Scholes boundary-value problem, Physica A, vol. 509, pp. 546-550. https://doi.org/10.1016/j.physa.2018.06.095.

The Black-Scholes formula for a European option price, which resulted in the 1997 Nobel Prize in Economic Sciences, is known to be the unique solution of the Black-Scholes partial differential equation with the terminal condition corresponding to the European option. In this paper, we show that there exist infinitely many solutions to the Black- Scholes boundary-value problem. This conclusion conflicts with the law of one price, which is one of the most important axioms in Economics. This means the results of the famous Black and Scholes’ 1973 paper and Merton’s 1973 paper are wrong. Moreover, our result implies that any known price formula of financial derivatives such as options, swaps, and interest rate derivatives is not correct, as far as it is derived using either a partial differential equation or the Feynman-Kac theorem. What happens to a Nobel prize in such a case?

In 2015, we had originally directly derived the infinitely many solutions and sent the manuscript to more than ten top Finance and Economics journals. All of them rejected our paper. We were frustrated. Our guess is that the manuscript was rejected because the reviewers do not have the mathematical background to comprehend the derivation. For this reason, we changed our strategy. To persuade future reviewers, we showed that our solutions satisfy not only the Black-Scholes partial differential equations but also the boundary condition. We sent the revised manuscript to this physics journal, where the reviewers understand calculus and analysis. However, we wish to publish our original paper, where we derive the infinitely many solutions directly, in a good Finance or Economic journal. Until that time, we will write and publish some more papers as stepping stones to guide Economics and Finance scholars through our mathematical ideas. I think we need two more papers before we can publish the original one in either Finance or Economics journal.

• Choi, B.S., C. Kim, H. Kang, and M.Y. Choi (2020) General solutions of the heat equation, Physica A, vol. 539, 122914, https://doi.org/10.1016/j.physa.2019.122914.

In this paper, we show that one should not rely on a partial differential equation to represent a phenomenon. In his book The World as I See It, Albert Einstein said, “The partial differential equation entered theoretical physics as a handmaid, but has gradually become a mistress.” Einstein is emphasizing the importance of the role that the partial differential equation plays in Physics. Indeed, partial differential equation plays an important role in several fields, including natural sciences, engineering, social sciences, and management sciences. This follows from Philosophiæ Naturalis Principia Mathematica, in which Issac Newton laid down his framework for the theory of motion. Following Principia, people have believed that if the partial differential equation describing the motion of an object is known, one need only the initial condition in order to obtain the object’s trajectory at all times. The presupposition behind this is that the trajectory is unique. 

Our paper, in contrast, discloses that there can exist infinitely many trajectories satisfying the partial differential equation subject to the initial condition. This raises the need for a paradigm shift in the sciences. We hope this paper will initiate the process and play a seminal role in this needed paradigm shift.

• Jeong, D. and B.S. Choi (2020) Deriving general solutions of the heat-transfer and the Fokker-Planck boundary-value problems through modified separation methods, Physica A, vol. 556, 124831, https://doi.org/10.1016/j.physa.2020.124831.

We present modified separation methods to solve the heat equation, which provide the general solutions in terms of a non-separable basis. This basis not only covers the set of the Hermite polynomials but also provides a more general set of confluent hypergeometric functions. This series solution was introduced in the work of Bluman and Cole (1969), where only the Gaussian fundamental solution was chosen when the conservation of the heat was considered. However, in Choi, Kim, Kang, and Choi (2020), it is applied to various boundary-value problem of the heat equation, and the existence of infinitely many solutions is revealed. Modifying the closed form solutions of the heat equation with an exponential weight, we derive corresponding partial differential equations, which include simple cases of the Fokker-Planck equation. In conclusion, we provide a variety of diffusion equations, whose closed-form solutions are already known.

• Choi, B.S, and M.Y. Choi (2024) On Some Results of the Nonuniqueness of Solutions Obtained by the Feynman–Kac Formula, Mathematics, vol. 12, no. 1, 129; https://doi.org/10.3390/math12010129

The Feynman–Kac formula establishes a link between parabolic partial differential equations and stochastic processes in the context of the Schrödinger equation in quantum mechanics. Specifically, the formula provides a solution to the partial differential equation, expressed as an expectation value for Brownian motion. This paper demonstrates that the Feynman–Kac formula does not produce a unique solution but instead carries infinitely many solutions to the corresponding partial differential equation.