An additive to a fuel oil for a Diesel engine having a Diesel particulate filter (DPF), which comprises a molybdenum compound and having the function of improving the combustion property of a particulate matter (PM) trapped with DPF; a fuel oil comprising the additive; a lubricating oil composition for a Diesel engine having DPF, which has a sulfated ash content of 1.0% by weight or smaller, a sulfur content of 0.3% by weight or smaller and a molybdenum content of 100 ppm or greater; and DPF for removing PM in combustion gas discharged from a Diesel engine vehicle by trapping and burning PM, DPF comprising a filter supporting a molybdenum compound. The combustion property of PM trapped with the filter of DPF is improved, PM is burned at a low temperature with stability, the efficiency of removal of PM is improved and the life of DPF is increased.