On one-dimensional stochastic differential equations involving the maximum process

Details On one-dimensional stochastic differential equations involving the maximum process On one-dimensional stochastic differential equations involving the maximum process

2010-03-30

arxiv.org/abs/1003.5844v1

Stochastics and Dynamics 2009, Vol. 9, No. 2, 277-292

Mathematics

Probability

16 pages, published in at this http://www.worldscinet.com/sd/09/0902/S0219493709002671.html Stochastics and Dynamics

Scientific paper

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}\sigma (s,X_{s})dW_{s}+\int_{0}^{t}b(s,X_{s})ds+\alpha \max_{0\leq s\leq 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_{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=\xi+\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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