Convex ordering of solutions to one-dimensional SDEs
arxiv(2023)
摘要
In this paper, we are interested in the propagation of convexity by the
strong solution to a one-dimensional Brownian stochastic differential equation
with coefficients Lipschitz in the spatial variable uniformly in the time
variable and in the convex ordering between the solutions of two such
equations. We prove that while these properties hold without further
assumptions for convex functions of the processes at one instant only, an
assumption almost amounting to spatial convexity of the diffusion coefficient
is needed for the extension to convex functions at two instants. Under this
spatial convexity of the diffusion coefficients, the two properties even hold
for convex functionals of the whole path. For directionally convex functionals,
the spatial convexity of the diffusion coefficient is no longer needed. Our
method of proof consists in first establishing the results for time
discretization schemes of Euler type and then transferring them to their
limiting Brownian diffusions. We thus exhibit approximations which avoid {\em
convexity arbitrages} by preserving convexity propagation and comparison and
can be computed by Monte Carlo simulation.
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