We consider a prototype class of Lévy-driven stochastic differential equations (SDEs) with McKean–Vlasov (MK–V) interaction in the drift coefficient. It is assumed that the drift coefficient is affine in the state variable, and only measurable in the law of the solution. We study the equivalent functional fixed-point equation for the unknown time-dependent coefficients of the associated linear Markovian SDE. By proving a contraction property for the functional map in a suitable normed space, we infer existence and uniqueness results for the MK–V SDE, and derive a discretized Picard iteration scheme that approximates the law of the solution through its characteristic function. Numerical illustrations show the effectiveness of our method, which appears to be appropriate to handle the multi-dimensional setting.

我们考虑在漂移系数中具有McKean–Vlasov（MK–V）相互作用的Lévy驱动的随机微分方程（SDE）的原型类别。假设在状态变量中漂移系数是仿射的，并且只能在解法则中进行测量。我们研究了相关线性马尔可夫SDE的未知时间相关系数的等价函数不动点方程。通过证明功能映射在合适的范数空间中的收缩性质，我们推断MK–V SDE的存在性和唯一性结果，并推导离散化的Picard迭代方案，该方案通过其特征函数近似解的定律。数值插图显示了我们方法的有效性，这似乎适合处理多维设置。