How to approach this problem!
Numerically. Trying to find some closed form equation to spit things out is extremely difficult, especially in the face of how well this lends itself to numeric methods. Solving this numerically is better in every way.
I was able to get a proof of concept out at 1 am in a few minutes, and causative was able to make a program that iterated numerically with full features in not much time at all. Coming up with how big things are, how fast, how accurate, and how good your sensors are will take a lot of research if you want to postulate the future, and with that basis you can 'turn knobs' for a look and feel you'd like if you're writing a story.
Basic C# code numerically solving this
https://pastebin.com/tq3d2TEr Seems that, yes, you can rather easily, if you don't have a factor to account for missiles being hard to spot. Interesting.
I'm trying to make a formula for rough estimates to find the "time to kill" of an inbound missile with a starting range, terminal range, the missile's initial closing speed, its dodge dV, its cross section, how many hits to kill (could be 1), and the jitter of your weapons that are trying to fire at it.
For people who write Sci-Fi and like to include numbers, or just people interested by the problem space, having a way to 'turn the knobs' of "what if this is better, what if that is worse" to see what works and what doesn't would be helpful. The problem is, such an undertaking is beyond my present abilities.
The first thing that came to mind is "oh, integrate probability over time!" but I have no clue how to do this, being decades out of school. While I have formulas for likeliness of hitting a target based on jitter, range, and so on, at a fixed distance, all the dependent variables of moving through space, and integrating probability, is something I have zero experience with.
Another thing I have to clear about is that the point of this is to have multiple terms to plug in, given that these are all assumptions. Given that this is to help fiction writers, not the space force, there's no right or wrong answers, as much as "let's assume this is better than that, what shakes out?"
Finally, given that this is inherently general, if I plugged in a very low velocity but a large cross section, that is, a ship, and not a missile, this formula or something very similar could determine time-to-hit on the ship firing the missiles, and thus determine how far away they need to be to not get chewed up by anti-missile weapons themselves.
Missile: a maneuvering, disposable vehicle that either crashes into something or carries a ranged warhead.
Laser: A weaponized pulse laser, likely with a very large aperture (100+ meters!) and UV wavelengths. These damage by ablating what they shoot. At very long ranges, due to diffraction, it doesn't drill as much as flash away a wide, flat region of what it does shoot.
Particle beam: I'm assuming the use of either neutralized or relativistic particle beams so you don't have to worry about bloom. You still have to worry about hitting anything. These cause damage by irradiating the insides of whatever they hit, and depending on the specifics of the particle beam, cause damage not unlike making a needle to pencil sized cross section of whatever they intersect with behave not unlike it was turned into det-cord. Some particle beams may well deposit their energy in a shallow cross section, too, but that isn't so important for this exercise.
Jitter: The ultimate limit of precision that your weapons and sensors have to deal with. Measured in micro to nano-radians.
Particle beams can be approximated as points. Lasers, however, not so much, since even with UV lasers and 200meter apertures, spot sizes can be in centimeters or meters over the distances I'm thinking of, which goes from .1 to 3 light seconds. So, you can have a cross section intersecting another cross section.
Formulas I've found through research
If you know your jitter J, and your range R, the diameter of the region your beam wanders over is, roughly: $D_r=2\cdot R \cdot tan(J/2)$.
For very small ranges, you can approximate with $RJ$.
If your target has a profile area A, you can approximate the chance of hitting with: $A / ( (\pi/4) * 2\cdot R \cdot tan(J/2) )$
I do not know how to account for a profile of your spot size, or for that matter, LaTeX.
Now it gets hard
How do you go about integrating this?
I'd need to include a firing rate for pulsed weapons, as much as how many of them I have, so one would have some finite number of shots over time. You'd start from the outside of the envelope (let's say 3 light seconds for now) and it would end with the missile hitting the ship trying to shoot it down, or reach the minimum range for a warhead like a reactor-pumped laser or a bomb-pumped particle beam, casaba howitzer, or whatnot.
That's where I just stare blankly at my screen and wish I did more stats back in school.
Oh, let's make it harder
Uncertainty radiuses due to jamming or other EW methods. I wouldn't ask about specific modeling, I'd want some simple polynomial term that can be used, and a cutoff for "burn through", that being your own active sensors 'burn through' the jamming. Jamming in space could be many things, such as the ship that fired the missile having a spot size big enough on their lasers they can't miss and dazzling your sensors, or the missiles being multi stage, and the n-1 stage backlighting them with radar, lidar, lasers, or whatnot.
Put another way, if a missile's cross section is, say, a meter, your sensors won't necessarily be able to precisely know where it is exactly with the enemy spamming your sensors with nonsense to dazzle them, so there would be a multi-meter blob that you'd think it's in, at least until it gets close enough your own active sensors can get returns.
The laser spot size getting smaller as the missile approaches. This is roughly 1.22(Distance)*(Wavelength/Aperture). Instead of flashing the surface, you can start drilling higher and higher aspect ratio holes into the target. On the other hand, if you have enough time, why not just flash the entire thing and not miss?
The Space Force is now interested
Active maneuvering? An earlier stage acting as a missile bus coming in obliquely and taking pot shots as it arcs away? I know 'path integrals' are a thing but I remember little about that part of Calc 3
And now something completely different
Is this even viable? Could making a program to actually simulate be easier?
"I don't think lasers work"
Based on what? While not relevant to the question, it's a matter of aperture size (mirror or lens), laser wavelength, and power. In this setting, ships are going to be big, and powerful - terawatt or hundred-gigawatt effective power will be what is trying to shoot at missiles over multiple light seconds. Many calculators are online to compute drilling speed over time, or per-pulse-train. If you want something to start with, go with an excimer laser deep in the UV (157 nm), an aperture size of 200 meters, and a power of 1TW. Yes, that's big. Yes, that's powerful! The setting I have in mind has goes big.
In the first place, once you reach a certain intensity, it does not matter what it's made out of. The electric field from that spot will be so intense that nothing chemical will withstand it. Electrons will be smacked off of nuclei, and then things go crazy. I believe it is a coulomb explosion. Once you get into extremely high intensities, you actually make a reflective plasma that means you don't put all of the energy in the target, but I'm not qualified to speak to it. Yes, you can make a "plasma mirror" or "plasma lens" with this effect, but you run into efficiency issues and other complexities outside of "it can work." No, this isn't really useful as an armor.
https://en.wikipedia.org//wiki/Laser_damage_threshold If you want to go into it, this briefly discusses how lasers do damage, such as dielectric and avalanche breakdown.
Prior art calculators to play with
http://panoptesv.com/SciFi/LaserDeathRay/DamageFromLaser.php This also comes with a lot of information if you want to scroll down and read it all.
Space is big
If a missile is traveling multiple light seconds while you shoot at it, unless it's juking and dodging like a space squirrel on space crack, you're going to smack it quite a few times. Not only that, but the spot size of a laser could easily be as big as the entire nose of the missile, perhaps by design.
Valid, but as some have already said, drones scattered around your side of space (and maybe a few shot at the enemy) with line of sight laser communications that are cryogenically cold are useful. While they can't be too big lest they make odd holes in the sky, or you handwave meta materials and hope they don't occult a star, this will help. Also, messing up datalink would basically require glassing the outside of the ship by getting a nuke close enough. If they've got almost an hour and a half to do so, that's not so guaranteed, now is it?
Oh, indeed. You want to have screens of some sort. Drones (glorified missile buses, or, a big stage that has multiple terminal stages) out ahead of you, if you know the enemy is coming, are a great idea. The same formula I want to make would be useful to see how the enemy could shoot them down.