Mean-field backward stochastic differential equations: A limit approach
Mean-field backward stochastic differential equations: A limit approach
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DOI：
10.1214/08-aop442
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发表时间：
2007-11
影响因子：
2.3
通讯作者：
R. Buckdahn;Boualem Djehiche;Juan Li;S. Peng
中科院分区：
文献类型：
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作者：
R. Buckdahn;Boualem Djehiche;Juan Li;S. Peng
数学平均场方法在物理和化学的不同领域发挥着重要作用，但在最近的著作中也发现它们在经济学、金融学和博弈论中的应用。我们论文的目的是研究纯随机方法中的特殊平均场问题：对于由具有解 X 的 McKean-Vlasov 型正向随机微分驱动的平均场后向随机微分方程的解 (Y, Z)，我们研究了一些解耦前向-后向方程的解 (X-N, Y-N, Z(N)) 的特殊近似，该方程的系数由 (X-N, Y-N, Z(N))。我们表明，该近似的收敛速度为 1/根 N。此外，我们对近似的特殊选择允许表征根 N(X-N - X, Y-N - Y, Z(N) - Z) 的极限行为。我们证明这个三元组在定律上收敛于某些前向-后向的解。平均场型随机微分方程，不仅受布朗运动控制，而且受独立高斯场控制。
Mathematical mean-field approaches play an important role in different fields of Physics and Chemistry, but have found in recent works also their application in Economics, Finance and Game Theory. The objective of our paper is to investigate a special mean-field problem in a purely stochastic approach: for the solution (Y, Z) of a mean-field backward stochastic differential equation driven by a forward stochastic differential of McKean-Vlasov type with solution X we study a special approximation by the solution (X-N, Y-N, Z(N)) of some decoupled forward-backward equation which coefficients are governed by N independent copies of (X-N, Y-N, Z(N)). We show that the convergence speed of this approximation is of order 1/root N. Moreover, our special choice of the approximation allows to characterize the limit behavior of root N(X-N - X, Y-N - Y, Z(N) - Z). We prove that this triplet converges in law to the solution of some forward-backward. stochastic differential equation of mean-field type, which is not only governed by a Brownian motion but also by an independent Gaussian field.