Whilst a solution image has been posted already, a solution path has not. Since this is a pretty tricky puzzle with some interesting deductions, I figure it is worth explaining to other potential solvers how exactly to see this puzzle through from start to finish, ultimately ending up with this solved state:
Step 1:
First, some obvious deductions by the basic rules of masyu:
Now consider the cells known to be in the grey loop.
If these connect to each other somehow by passing in between the pairs of red and blue loop cells, this leaves only three possible path sections around this loop for the red and blue loops - but we need four for those loops to both enter and exit this top-left region.
Thus the grey loop must pass outside the red and blue cells and occupy spaces around the edge of the board, forcing some of the red and blue loops...
Step 2:
For red to escape from the top-left corner without cutting across the grey loop, it must pass through R5C3-4. This requires the white circle in R4C2 to be part of the red loop.
It is also clear at this stage that the two green squares cannot be part of a loop that goes outside any of the others.
In fact, inspection reveals that it is required for each loop to circle the green loop fully (one part of the loop going round the leftmost green square, the other part going around the rightmost green square - no coloured loop can exit the top left corner of the grid with both parts passing by the same green 'end'), which in turn means that we are dealing with four 'concentric' loops where the green must be entirely contained inside the blue loop, which is inside the red loop, which is inside the grey. We can fill in more loop sections for each colour...
...and in fact, we can make several further logical deductions on the left-hand side concerning where the grey and red loops must pass, forcing more of the loops. Plus in that top-right corner we need to extend the green loop 2 spaces from the black circle in each direction, which has several other knock-on deductions.
Step 3:
We can keep extending these loop segments in the top-left and top-right, ultimately joining the two grey loop segments.
And we can now conclude that the green loop's black circle in R9C6 must have a segment leading downwards rather than up, as that route is now blocked. This enables us to resolve the top-left quadrant entirely...
...and we can also keep extending those segments in the top-right quadrant downwards, leading us to realise that the black circle in R10C15 must be part of the blue loop.
Step 4:
Focus now on the black circle in R14C4.
This must be part of the blue loop. If it were grey, then attempting to resolve the white circle above it with red would result in the red loop making a turn afterwards that blocks off blue.
Now there is only one way for the blue loop to squeeze its way out of the bottom-left corner whilst satisfying the white circle currently sandwiched between it and the green loop.
There now remains only one way to resolve the green loop fully:
Plus the white circle in R11C12 must be passed through vertically (to avoid closing off the blue loop too early).
Step 5:
Almost there now.
The white circle in R15C15 must be passed through horizontally, which forces the red and grey loops to completeness:
And now all that remains is to complete the blue loop, and we are done!
