Hydrostatic transmission for a work vehicle, comprising a hydraulic pump and a hydraulic motor hydraulically connected to each other, wherein the hydraulic pump is arranged to be mechanically coupled to an internal combustion engine of the vehicle, and the hydraulic motor is arranged to be mechanically connected to driving wheels of the vehicle, wherein the hydraulic pump is an electronically-controlled variable displacement hydraulic pump, and the hydraulic motor is a hydraulically-controlled variable displacement hydraulic motor, wherein the hydrostatic transmission further comprises an electronic control unit for controlling the operation of the hydrostatic transmission, characterized in that the electronic control unit is configured to generate drive currents (i d1, i d2) for adjusting the displacement of the hydraulic pump based on information, provided by suitable sensors of the vehicle, relating to the position of an accelerator pedal of the vehicle, the position of a brake pedal of the vehicle, the position of a FNR (Forward-Neutral-Reverse) switch of the vehicle, the speed of the vehicle and the speed of the engine, and wherein the electronic control unit is programmed in such a manner that when it establishes, based on the information received from said sensors of the vehicle, that the vehicle is braking or reversing downhill, it starts to generate suitable drive currents (i d1, i d2) for reducing the displacement of the hydraulic pump to zero, in case of braking, or for bringing the displacement of the hydraulic pump to a negative value passing through zero, in case of reversing downhill, and suitably modulate said drive currents (i d1, i d2) based on the speed of the vehicle so as to prevent the displacement of the hydraulic pump from going to zero as long as the speed of the vehicle is higher than a given threshold value (v 1). wherein the hydraulic pump is arranged to be mechanically coupled to an internal combustion engine of the vehicle, and the hydraulic motor is arranged to be mechanically connected to driving wheels of the vehicle, wherein the hydraulic pump is an electronically-controlled variable displacement hydraulic pump, and the hydraulic motor is a hydraulically-controlled variable displacement hydraulic motor, wherein the hydrostatic transmission further comprises an electronic control unit for controlling the operation of the hydrostatic transmission, characterized in that the electronic control unit is configured to generate drive currents (i d1, i d2) for adjusting the displacement of the hydraulic pump based on information, provided by suitable sensors of the vehicle, relating to the position of an accelerator pedal of the vehicle, the position of a brake pedal of the vehicle, the position of a FNR (Forward-Neutral-Reverse) switch of the vehicle, the speed of the vehicle and the speed of the engine, and wherein the electronic control unit is programmed in such a manner that when it establishes, based on the information received from said sensors of the vehicle, that the vehicle is braking or reversing downhill, it starts to generate suitable drive currents (i d1, i d2) for reducing the displacement of the hydraulic pump to zero, in case of braking, or for bringing the displacement of the hydraulic pump to a negative value passing through zero, in case of reversing downhill, and suitably modulate said drive currents (i d1, i d2) based on the speed of the vehicle so as to prevent the displacement of the hydraulic pump from going to zero as long as the speed of the vehicle is higher than a given threshold value (v 1).