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Exact and high-order discretization schemes for Wishart processes and their affine extensions. (English) Zbl 1269.65003

The paper is devoted to the simulation of the Wishart processes defined by the following stochastic differential equation (SDE): \[ X_{t}^{x}=x+\int_{0}^{t}(\alpha a^{\intercal}a+bX_{s}^{x}+X_{s}^{x} b^{\intercal})ds+\int_{0}^{t}(\sqrt{X_{s}^{x}}dW_{s}a+a^{\intercal} dW_{s}^{\intercal}\sqrt{X_{s}^{x}}),\tag{1} \] where \(\alpha\geq0\), \(a\), \(b\) are \(d\) square matrices, \(W_{s}\) is a \(d\) square matrix made of independent Brownian motions, \(x\), \(X\) are \(d\) square positive semidefinite matrices. If \(d=1\), (1) is the SDE of the Cox-Ingersoll-Ross process. The authors find a splitting for Wishart processes that makes it possible to sample exactly Wishart distributions. Moreover, they construct high-order discretization schemes for Wishart processes. The authors consider affine extensions of the Wishart processes as well.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
91B70 Stochastic models in economics
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J65 Brownian motion
35R60 PDEs with randomness, stochastic partial differential equations

Software:

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Full Text: DOI arXiv Euclid

References:

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