We prove L 2 -maximal regularity of linear non-autonomous evolutionary Cauchy probleṁwhere the operator A(t) arises from a time depending sesquilinear form a(t, ., .) on a Hilbert space H with constant domain V. We prove the maximal regularity in H when these forms are time Lipschitz continuous. We proceed by approximating the problem using the frozen coefficient method developed in [9], [10] and [13]. As a consequence, we obtain an invariance criterion for convex and closed sets of H.

The concept of asymmetric copulas is revisited and is made more precise. We give a rigorous topological argument for opportunity to define asymmetry measures defined recently by K.F Siburg [6] through exhibiting at least three ordered classes of copulas according to a suitable equivalence relation. We define a process of ordering subcopulas which makes clearer the degree of asymmetry. As illustration, we treat the asymmetric Cobb-Douglas utility model.

An alternative proof of invariance of convex sets by the solution of non autonomous Cauchy problem is given. The proof is based on the recent integral approximation of time dependent operators A(t) acting on Hilbert space when they are associated with smooth sesquilinear forms a(t,.,.) defined on common dense domain and the known Chernoff Product Formula. An application to positivity of Black-Scholes operator is given.

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A sufficient topological condition to ensure maximal regularity is given. The result permits to retrieve classical results established recently on non autonomous evolution equations mainly those concerning well-posedness of the non autonomous Cauchy problem. We illustrate the result in the hilbertian context by an application to Black-Scholes operator.