Stochastic Way
The works below could be cross-listed elsewhere on this site, and have been regrouped here.
Most scholars of my generation have been instilled with the Newtonian view of an orderly, causal, and trajectorial world, and further framed by all the deterministic toolkits of Calculus, Linear or Commutative Algebra, Differential Equations, Differential Geometry, etc. It has become quite a challenge to engender a stochastic view of the complex world surrounding us, and to unleash the power of all the classical tools to the wild arena of stochasticity. The stochastic worldview was even hard for Einstein to reconcile to, some say.
From Imaging/Vision to Quant Finance, the stochastic approach offers a language, a model, and a solution.
[Disclaimer: The header background image is copied from NASA's James Webb space telescope]
(2021) J. Shen, "Bucketed PCA neural networks with neurons mirroring signals (arXiv)" (or via SSRN.3897477)
[Keywords] Interpretable AI, supervised learning, (artificial) neurons, DNN, mirroring, PCA, transforms, bucketing, error correction.
(2020) J. Shen, A stochastic LQR model for Child Order Placement (COP) in algorithmic trading. SSRN #3574365.
[Keywords] Child order placement, dynamic programming, LQR, delay cost, spread, impact cost, information leakage, Poisson hits, passive, aggressive, Bellman equation, optimal policy.
(2017) J. Shen, Hybrid IS-VWAP dynamic algorithmic trading via LQR. SSRN #2984297.
[Keywords] Dynamic programming, LQR, implementation shortfall (IS), VWAP, slippage, spread, delay cost, impact cost, stochastic price dynamics, Bellman equation, optimal policy.
(2009) J. Shen, Least-square halftoning via human vision system and Markov gradient descent (LS-MGD): Algorithm and analysis, SIAM Review, 51(3):567-589, 2009. The PDF file. A sample figure.
[Keywords] Halftoning, Human Vision System (HVS), mixing, entropy, least square, stochastic gradient descent, Markov random walk, random fields, convergence analysis, blue noise.
(2006) J. Shen, A stochastic-variational model for *soft* Mumford-Shah segmentation, Int'l J. Biomedical Imaging, vol. 2006, Article ID 92329, 2006 (Open Access). The PDF file (at nih.gov). Sample figure A. Sample figure B.
[Keywords] soft vs. hard, Mumford-Shah, pattern, fuzzy ownership, probability simplex, Modica-Mortola, phase-field, Egorov's theorem, existence theorems, AM algorithm.
(2005) J. Shen and Y.-M. Jung, Geometric and stochastic analysis of reaction-diffusion patterns , Int'l J. Pure Applied Math., 19(2):195-248, 2005. The PDF file.
[Keywords] data mining, pattern, Turing instability, reaction, diffusion, entropy, skewness, kurtosis, isoperimetric ratio, curvature measure.
(2001) J. Shen, On the singular values of Gaussian random matrices, Linear Alg. Appl., 326(1-3), 1-14, 2001.
[Keywords] Random matrices, singular values, Gaussian ensemble, Wishart ensemble, thermodynamic limit, pseudo-Coulomb gas, circle law, and quadrant law.
(2000) Gian-Carlo Rota and Jianhong Shen, On the combinatorics of cumulants, J. Comb. Theory (A) , 91(1), 283-304, 2000.
[Keywords] Cumulants, umbrae, exponential (moment) generating function, Schur symmetric functions, orthogonal polynomials, binomial sequence, Moebius inversion.
In memory of my beloved mentor and friend - Gian-Carlo Rota (1932-1999).
(2000) J. Shen, A geometric approach to ergodic non-homogeneous Markov chains, in "Wavelet Analysis and Multiresolution Methods", Lecture Notes in Pure and Applied Mathematics, 212, pp. 341-366, 2000.
[Keywords] Weak Ergodicity, Markovian, Scrambling Matrix, Hajnal, Simplex Transform, Contraction, Rota-Strang Joint Spectral Radius.
(1998) G.-C. Rota, J. Shen, and B. D. Taylor, All polynomials of binomial type are represented by Abel polynomials, Ann. Scuola Norm. Sup. Pisa. Cl. Sci. (IV), 25(3-4), pp. 731-738, 1998.
[Keywords] Moment generating functions, symbolic random variables (i.e. umbrae), binomial polynomials, Lagrange inversion. In memory of De Giorgi.