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arXiv · 1003.5844

On one-dimensional stochastic differential equations involving the maximum process

Rachid Belfadli·Said Hamadéne·Youssef Ouknine
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Abstract

We prove existence and pathwise uniqueness results for four different types of stochastic differential equations (SDEs) perturbed by the past maximum process and/or the local time at zero. Along the first three studies, the coefficients are no longer Lipschitz. The first type is the equation \label{eq1} X_{t}=\int_{0}^{t}σ(s,X_{s})dW_{s}+\int_{0}^{t}b(s,X_{s})ds+α\max_{0\leq s\leq t}X_{s}. The second type is the equation \label{eq2} {l} X_{t} =\ig{0}{t}σ(s,X_{s})dW_{s}+\ig{0}{t}b(s,X_{s})ds+α\max_{0\leq s\leq t}X_{s}\,\,+L_{t}^{0}, X_{t} \geq 0, \forall t\geq 0. The third type is the equation \label{eq3} X_{t}=x+W_{t}+\int_{0}^{t}b(X_{s},\max_{0\leq u\leq s}X_{u})ds. We end the paper by establishing the existence of strong solution and pathwise uniqueness, under Lipschitz condition, for the SDE \label{e2} X_t=ξ+\int_0^t \si(s,X_s)dW_s +\int_0^t b(s,X_s)ds +\al\max_{0\leq s\leq t}X_s +\be \min_{0\leq s \leq t}X_s.

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Rachid Belfadli, Said Hamadéne, Youssef Ouknine. 2010-03-30. On one-dimensional stochastic differential equations involving the maximum process. https://arxiv.org/abs/1003.5844

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