A brief look at the circuit reveals that two analogue voltages are needed. They are fixed +12V & 0-10V. The shortest timing is only 70us which is very comfortable for a PIC to work on. You can't generate the voltages using PIC alone but you can generate any voltage between 0-5V using a PIC with ease.
Then all you'll need is a voltage amplifier of gain 2.4X to amplify it into 12V.
With the amplifier, 12V becomes +5V and the 10V control signal now ranges from 0-4.2V. If you use a R-2R DAC network connects to a 8-bit output port, then a value of 255 would give you very near +5V. For the 0-4.2V range, you have the binary value of 0-215 to play with. If you settle for 0-200, then that would translate into a resolution of 0.5% for each possible step in your final output brightness. If that doesn't do, you can always increase the resolution by using an extra bit in another output port.