 Hey guys, it's MJ the student-actree and we're going to be looking at the two-state Markov model this is chapter 4 for subject CT4 and Hopefully we have enough time to get through all six of these topics transitional diagrams probabilities assumptions the Kolmogorov forward differential equation the likelihood function for mu and the maximum likelihood Estimated for mu so without further ado, let's get into it. Okay transitional diagrams these things are visual representations of the Markov models and They're very simple You have your state that you write here You have the border can be a square or a circle and then you have an arrow that indicates the transition and The arrow is the weight of that arrow is the force so the greater That number is the more likely things are gonna go from this state into that state So what we have here is very simple. I can be alive Or I can be dead and there is a force that is pushing me from being alive to dead and that is known as the force of mortality and What we've got over here or what this formula, you know, I think we all are saying what the force of mortality is But what does yellow formula over here is showing? Let's get a chair it's saying that it is the the probability of Now we know that qx is the probability of someone dying So it's the probability of someone dying of a very very very very small Interval and we're gonna see later how this is the that whole derivatives and This is why mathematics is important for exterior science as you're gonna be needing it to derive these results But this will all become simpler Once we just you know recap on probabilities Essentially, there are two types of probabilities within Mark of models you have the transitional probability. This is the probability that I'm in one state and I move into another state. So in our example tqx This represents that if I'm alive at time x Also, so transitional probability says that this is the probability that I'm dead at times x plus t Given that I was alive at time x So transitional probability, I'm leaving that state to go to another state or There is the probability that I don't die that I stay alive and this is known as occupancy probability Tpx and this is the probability that I'm alive at x plus t given that I was alive at x Now before we can move any further We need to make some Assumptions, okay with regards to this whole mathematical theory, okay With mark of models we make the assumption that they have the mark of prop, you know proper Properties, sorry, and they have the mark of property and that is that the probabilities depend only on the current state So it doesn't matter what your history is everything All information is contained in the present state And so the past does not influence the present that pushes to the future Okay Then we have This assumption over here and what this says is that the probability that I die In states or that I'm dead by time x plus t plus h Given that I'm alive at x plus t is Equal to the force of mortality at h x plus t times by this h Okay, and now that's just that mathematical, you know, that's a function that goes straight to zero as h when you use h but the thing is is to remember here is that h is incredibly small and Again, this is where a background in mathematics does help as it becomes quite philosophical You know, what is a small beta of h anyway if we're not going to get into that too much The final assumption that we're going to make is that the force of mortality or these forces are constant Throughout the year. So you x plus t is a constant mu Okay, as long as t is between zero and one and that's just to make the mathematics simpler and because You know the force does change ever so slightly in a year We can get away with this assumption, but we are losing a little bit of accuracy Okay, now we get on to this thing known as the Kolmogorov forward differential equation and First time looking at this it is difficult. It is intimidating, but let's have a look at it Okay, we have that the probability that I'm alive at h t plus h plus x Given that I'm alive at probability x is equal to the probability that I'm alive at time x plus t given that I was at x Times the probability that I'm alive at time h plus x plus t given that I was at probability x plus t So what what we basically doing is we splitting so and I guess an easy way to say is this the probability that I'm alive in 10 years time is equal to the probability That I'm alive in five year time times the probability that five year older me is alive in a further five years time So you can split probabilities like that Then what we do is we use that assumption The mark of assumption we break it up into oh, sorry what we do is we just use the That relationship that you know one minus p is equal to q or one minus q is equal to p You know if I'm not leaving then I'm staying or if I'm not staying then I'm leaving so that's where and you know Probability is equal to one. So that's how we get that then we use assumption to to Transform the probability into the force like we have there We then rearrange this But yeah, and you'll see that this takes on This year from maths one is very similar to the derivative Sign so when we take the limit of h2o, we get that the derivative of the occupancy probability is Equal to the negative occupants probability times bar the force now This when you think about it it makes sense But you have to think about it this you're not gonna get this straight away by watching this video You know, you're not gonna have to be like oh, wow I understand it now this is something you have to think about and you just have to wrestle with by doing examples Mathematic numerical examples, it will become clearer It does get a bit confusing when you're just using t plus h plus x and all that Anyway, we continue with the Kolmogorov for differential equation We can solve the equation using just the separation method you do some mathematics and You get this very important piece of information, okay We use the assumption the third assumption just to simplify it But this should be looking familiar from subject ct1 when we did the force of Interest, you know, we add each negative Delta, you know Delta was the force of interest type of thing so it shouldn't be brand new stuff to you guys, but it is Difficult, but it is very important like the rest of the course builds upon this result and I wish I could give a better Explanation, but it is difficult and I did this last year. So I'm a little bit rusty, but anyway, that's very important Then I just did some Examples myself and some old study notes, then there's this whole likelihood function and If you've done subject ct3 Likelihood functions should come naturally to you If anyone is struggling with them, let me know and I'll maybe make a ct3 video on the likelihood functions and Then the maximum likelihood estimated, you know, you basically take the log Differentiate set it equation to zero and check the second derivative, but they are very Simple once you understand the concept, you know They're basically saying you count the amount of deaths divided by the amount of transitions to get the force of mortality type of thing So yeah, I know I'm jumping very quickly through this But I don't want this video to be too long and this is very much subject ct3 work What is cool? I'll leave it on a cool note is I mean you can see here the question paper in subject We are in 2012 for the subject had a cool question about aliens Sacrifice becoming dead and then zombies So the examiners do get quite creative with this work, but you can see how There is some fun. This is this is the fun part of actuarial science is these Markov models So, yeah, I know Maximum likelihood estimators. This is subject ct3. So you should be cool with it comogra for differential equations are something new but you should get this feeling from ct1 when we did the force of interest and Yeah learn your assumptions Remember the probabilities. I mean you're gonna use that also a lot in subject ct5 and Yeah, we're gonna be seeing a lot more transitional diagrams But y'all that is chapter for the two-state Markov model for subject ct4. I'm MJ the student actuary. Thank you for watching. Cheers