The multiple Exp-function method is employed for searching the multiple soliton solutions for the new extended (3+1)-dimensional Jimbo-Miwa-like (JM) equation, the extended (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (eCBS) equation, the generalization of the (2+1)-dimensional Bogoyavlensky-Konopelchenko (BK) equation, and a variable-coefficient extension of the DJKM (vDJKM) equation, which contain one-soliton-, two-soliton-, and triple-soliton-kind solutions. The physical phenomena of these gained multiple soliton solutions are analyzed and indicated in figures by selecting suitable values.

The purposes of this paper are to improve the scientific processing level of risk management in the financial field, enrich the application range of mathematical models in financial calculations, and comprehensively discuss the theories and concepts of mathematical finance and stochastic differential equations. More importantly, the common option pricing issues in financial risk management have been researched using the forward–backward stochastic differential equation. The fully discrete and uncoupled forward–backward stochastic differential equation is employed to analyze the spread option and the better-of option, the complicated multi-asset options. Results demonstrate that the fully discrete and uncoupled forward–backward stochastic differential equations can effectively price the spread option and the better-of option. Simulation by the MATLAB software suggests that the value of spread option pricing is 0.0264, and the value of the better-of option pricing is 0.0251. The above results can provide scientific and useful references for the subsequent application research on forward–backward stochastic differential equations in the financial field; simultaneously, they also have important practical significance for researching on and developing the financial risk management.

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To further explore the research of financial time series prediction and broaden the application scope of Gaussian distribution probability density equation, based on the nonlinear differential equation of Gaussian distribution probability density, the semi-supervised Gaussian process model is taken as the research object to discuss the application of semi-supervised Gaussian process model in the stock market. Shanghai 180 Index, Shanghai (Securities) Composite Index and the yield of three stocks are studied in detail. The specific results are as follows. The prediction accuracy of Shanghai 180 Index is 83%, and the prediction accuracy of Shanghai (Securities) Composite Index is 78%. The stock yield series curves of the three stocks have the characteristics of peak and thick tail, which do not obey the normal distribution. Among the three models of semi-supervised Gaussian process model, initial Gaussian process model and SVM algorithm model, the prediction accuracy of semi-supervised Gaussian process model is the best, and the prediction accuracy of the yield of three stocks is 82.45%, 85.03% and 84.53%, respectively. The research results can fully prove that the semi-supervised Gaussian process model has good application effect in stock time series prediction. The research content can provide scientific and sufficient reference for the follow-up research of financial time series, and also has important significance for the research of Gaussian process model.