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- في الرياضيات، تعتبر معادلة تاناكا هي مثال على المعادلات التفاضلية العشوائية التي لا يوجد لديها حل مؤكد وإنما حل ضعيف. وسميت على اسم عالم الرياضيات الياباني هيروشي تاناكا. ومعادلة تاناكا هي معادلة تفاضلية عشوائية ذات بعد واحد : ومن تعريف المتعارف عليه للحركة البراونية B , بشرط X0 = 0, , ويدل sgn على دالة الإشارة وللتبسيط: وبالتالي: (ar)
- In mathematics, Tanaka's equation is an example of a stochastic differential equation which admits a weak solution but has no strong solution. It is named after the Japanese mathematician Hiroshi Tanaka (Tanaka Hiroshi). Tanaka's equation is the one-dimensional stochastic differential equation driven by canonical Brownian motion B, with initial condition X0 = 0, where sgn denotes the sign function (Note the unconventional value for sgn(0).) The signum function does not satisfy the Lipschitz continuity condition required for the usual theorems guaranteeing existence and uniqueness of strong solutions. The Tanaka equation has no strong solution, i.e. one for which the version B of Brownian motion is given in advance and the solution X is adapted to the filtration generated by B and the initial conditions. However, the Tanaka equation does have a weak solution, one for which the process X and version of Brownian motion are both specified as part of the solution, rather than the Brownian motion being given a priori. In this case, simply choose X to be any Brownian motion and define by i.e. Hence, and so X is a weak solution of the Tanaka equation. Furthermore, this solution is weakly unique, i.e. any other weak solution must have the same law. (en)
- Em matemática, a equação de Tanaka é um exemplo de equação diferencial estocástica que admite uma solução fraca, mas que não tem nenhuma solução forte. Recebe este nome em homenagem ao matemático japonês Hiroshi Tanaka. (pt)
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- في الرياضيات، تعتبر معادلة تاناكا هي مثال على المعادلات التفاضلية العشوائية التي لا يوجد لديها حل مؤكد وإنما حل ضعيف. وسميت على اسم عالم الرياضيات الياباني هيروشي تاناكا. ومعادلة تاناكا هي معادلة تفاضلية عشوائية ذات بعد واحد : ومن تعريف المتعارف عليه للحركة البراونية B , بشرط X0 = 0, , ويدل sgn على دالة الإشارة وللتبسيط: وبالتالي: (ar)
- Em matemática, a equação de Tanaka é um exemplo de equação diferencial estocástica que admite uma solução fraca, mas que não tem nenhuma solução forte. Recebe este nome em homenagem ao matemático japonês Hiroshi Tanaka. (pt)
- In mathematics, Tanaka's equation is an example of a stochastic differential equation which admits a weak solution but has no strong solution. It is named after the Japanese mathematician Hiroshi Tanaka (Tanaka Hiroshi). Tanaka's equation is the one-dimensional stochastic differential equation driven by canonical Brownian motion B, with initial condition X0 = 0, where sgn denotes the sign function i.e. Hence, and so X is a weak solution of the Tanaka equation. Furthermore, this solution is weakly unique, i.e. any other weak solution must have the same law. (en)
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- معادلة تاناكا (ar)
- Equação de Tanaka (pt)
- Tanaka equation (en)
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