| In this thesis,deep learning is applied to solve several different issues,including numerical methods of backward stochastic differential equations and financial asset pricing problems.Since deep learning can effectively overcome the "curse of dimensionality" in numerical methods for nonlinear partial differential equations and backward stochastic differential equations,it has become one of the most important research directions in numerical computations in recent years.The numerical algorithms for solving high-dimensional forward-backward doubly stochastic differential equations and high-dimensional mean-field backward doubly stochastic differential equations are proposed in this thesis,in which deep neural networks are introduced as a critical step to obtain the approximate solution.Based on the numerical methods for backward stochastic differential equations,the g-pricing modeling method driven by the option data is investigated and the model performance is validated with SPX options.Finally,a model for mining high-frequency order book price movements is developed with the data mining capability of deep learning,and empirical analysis with high-frequency order book data is conducted.First,the introduction chapter briefly introduces the research background and outlines the main contents of this thesis.In Chapter 2,the numerical schemes of solving high-dimensional(decoupled)forward-backward doubly stochastic differential equations(FBDSDEs)is studied.FBDSDEs are converted into the equivalent stochastic optimal control problems with u and Z as controls,and a deep learning-based numerical algorithm is proposed.The neural networks are used to approximate the controls u and Z and a recursive formulation of Y can be constructed.Meanwhile,the constraint penalties of u and Z ensure the Ft-measurability of Y in the recursive process.For the high-dimensional case,this method avoids curse of dimensionality with the use of Monte Carlo simulation.According to the stochastic nonlinear Feynman-Kac theorem,a class of highdimensional stochastic partial differential equations(SPDEs)can thus be converted into corresponding FBDSDEs and calculated by applying the proposed deep learning algorithm.Two 100-dimensional numerical experiments are given and the convergence rates of the algorithm are also estimated.In Chapter 3,numerical solutions of a class of mean-field backward doubly stochastic differential equations(MF-BDSDEs)are studied.Two numerical methods are presented.The first method,backward explicit scheme dependent on conditional expectations is proposed,for which we prove the convergence rate and give 1-dimensional numerical experiments;For the other method,based on the deep learning method for FBDSDEs in Chapter 2,a deep learning numerical method for MF-BDSDEs applicable to the high-dimensional case is proposed.A group of neural networks is introduced to approximate the generator function represented by the mathematical expectation,and Monte Carlo estimation under different random path sampling is used to constrain the neural network.Numerical experiments are presented to illustrate the convergence of the algorithm by giving error analysis for different dimensions of Brownian motion for d=1,10,50,100.In Chapter 4,deep learning methods are incorporated with option pricing methods based on BSDEs,which is called“deep learning g-pricing methods." According to the theory of g-pricing mechanism proposed by Peng[1],the option market price corresponds to some generator function g.Therefore,the idea of deep learning BSDEs numerical algorithm proposed in Chapter 2 and Chapter 3 is introduced to train the solutions of the generator function g and its corresponding BSDE simultaneously with the real data of the options market.In this chapter,neural networks are introduced to approximate the generator functions in BSDEs and construct the backward recursive formulations of BSDEs.Experiments conducted on SPX European options show that it is feasible to train a pricing model driven by the BSDE directly through options market data.Under the given experimental conditions,the trained g-pricing model can give a price that is closer to the option market than the Black-Scholes formula.In Chapter 5,the problem of searching asset price impact mechanisms from limit order book data based on deep neural networks is considered.To mine the early or delayed response mechanisms that driving price changes in high-frequency data,a Cross-Interval Price Impact Model(CIPIM)is proposed,including a feature extractor for order book events in the form of a long short-term memory layer(LSTM),and a fully connected layer(MLP)to model the price change response mechanism.By introducing this two-level structure,the endpoints of the time interval can be randomly selected to achieve cross-interval effects.The performance of CIPIM is demonstrated empirically.On the given high-frequency dataset,the proposed size-price combined CIPIM model obtains a higher contemporaneous Avg-R2.The CIPIM model has significantly higher prediction performance than OFI for the event factors defined only by the order sizes.In addition,the classification version of CIPIM can effectively predict the direction of price changes. |
