Nội dung được dịch bởi AI, chỉ mang tính chất tham khảo
Các Thuật Toán Xấp Xỉ Học Máy cho Các Phương Trình Vi Phân Không Tuyến Tính Cao Chiều và Các Phương Trình Vi Stochastic Ngược Bậc Hai
Tóm tắt
Các phương trình vi phân riêng phần (PDE) cao chiều xuất hiện trong nhiều mô hình từ ngành tài chính, chẳng hạn như trong các mô hình định giá phái sinh, mô hình điều chỉnh định giá tín dụng, hoặc các mô hình tối ưu hóa danh mục đầu tư. Các PDE trong các ứng dụng này là cao chiều vì số chiều tương ứng với số tài sản tài chính trong một danh mục đầu tư. Hơn nữa, các PDE này thường là hoàn toàn không tuyến tính do cần phải đưa vào một số hiện tượng không tuyến tính trong mô hình như rủi ro vỡ nợ, chi phí giao dịch, sự không chắc chắn về độ biến động (sự không chắc chắn theo kiểu Knight), hoặc ràng buộc giao dịch trong mô hình. Các PDE hoàn toàn không tuyến tính cao chiều này rất khó để giải quyết vì công sức tính toán cho các phương pháp xấp xỉ tiêu chuẩn tăng theo cấp số nhân với số chiều. Trong công trình này, chúng tôi đề xuất một phương pháp mới để giải quyết các PDE không tuyến tính bậc hai hoàn toàn cao chiều. Phương pháp của chúng tôi có thể được sử dụng đặc biệt để lấy mẫu từ các kỳ vọng không tuyến tính cao chiều. Phương pháp này dựa trên (1) liên kết giữa các PDE không tuyến tính bậc hai hoàn toàn và các phương trình vi stochastic ngược bậc hai (2BSDEs), (2) một định dạng kết hợp của PDE và bài toán 2BSDE, (3) một phân discret hóa tạm thời theo tiến của 2BSDE và một xấp xỉ không gian thông qua các mạng nơ-ron sâu, và (4) một quy trình tối ưu hóa theo kiểu gradient ngẫu nhiên. Các kết quả số thu được bằng cách sử dụng TensorFlow trong Python minh họa hiệu quả và độ chính xác của phương pháp trong các trường hợp phương trình Black–Scholes–Barenblatt 100 chiều, phương trình Hamilton–Jacobi–Bellman 100 chiều, và một kỳ vọng không tuyến tính của chuyển động Brown G 100 chiều.
Từ khóa
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