On one-dimensional stochastic differential equations involving the maximum process

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 longerLipschitz. The first type is the equation label{eq1} X_{t}=int_{0}^{t}sigma (s,X_{s})dW_{s}+int_{0}^{t}b(s,X_{s})ds+alpha max_{0leq sleq t}X_{s}. The second type is the equation label{eq2} {l} X_{t} =ig{0}{t}sigma (s,X_{s})dW_{s}+ig{0}{t}b(s,X_{s})ds+alpha max_{0leq sleq t}X_{s},,+L_{t}^{0}, X_{t} geq 0, forall tgeq 0. The third type is the equation label{eq3} X_{t}=x+W_{t}+int_{0}^{t}b(X_{s},max_{0leq uleq 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=xi+int_0^t si(s,X_s)dW_s +int_0^t b(s,X_s)ds +almax_{0leq sleq t}X_s +be min_{0leq s leq t}X_s.