 Today, I'm going to talk about Baylor-Stochastic differential equations, which is abbreviated as ESDE. So my actual goal is not actually about stochastic calculus, but about the connections between stochastic calculus and PDE. So let me get started on the connections, which are quite well known. Well, if you are not in analysis, well, you can learn some right now. So Ian Hous was here, and he was teaching PDE. The first thing they would learn is the main value of this abbreviated as MVP, good name. So it says that a function yield, the main value property, can form a center at x with radius r, and you take the average of the function to get the value of the function at the center. So if you take some derivatives, this is equivalent instead of over 4, but over the sphere at the center as well. Still, there's no PDEs. But if you assume, and you can take second order derivative, this is equivalent to the Lafacile of the yield is equal to 0. I don't want to talk about boundaries. So this is the main value property between PDE, which corresponds to this Laplace equation, or Poisson equation. Something very well known into a function that satisfies the main value property is that they have a stochastic correspondence, namely the Brownian motion. W stands for been a defined Brownian motion, like rigorously. Anyway, it's a Brownian motion. This is a martingale. And if you satisfy the main value property, you actually have that, not only that W itself is a martingale, but adding on W, this process is also a martingale, at least when you look at this process on a really large disk. That's great for a martingale. And why is that? Because of two things. First, is that a martingale is actually an average equal to itself. So the definition of a martingale is saying that you have a process to take the conditional expectation at a previous time. It so happens that the conditional expectation is the process at the previous time. So expectation is an average, and it equals to the process, the value of the process at a previous time. But somehow, I've noticed this average is equal to the first thing itself. So a martingale is an average. That's one thing. And the other thing is that, here, let's suppose that U is smooth. So recall, recall, it's transformed. The change is given by Helmholtz term, which is given by radii of U, because the process is so irregular that, to reciprocate, its differential, dwt, is of order t1 over half. So because of the roughness of the path, particular expansion, first order term, by looking at the second order of dwt. So at the Brownian motion, which is the first order term of the time, because the differential of Brownian motion is of order of 1 half. So this guy is actually first order of time. So we pick up this extra term of first order of time. The mean value property, the PDE of the mean value property, tells you that, well, yes, in general, we will have this extra term, but because the Laplace will manage over the entire space. So this first order term of time is actually 0. So the change of the new process, dwt, it just looks like dwt itself. Which is a Brownian way. Because this one, on average, the exception, I would say the first basic connection between PDE and sarcastic capital is that now you have a connection between the mean value property or harmonic function and Brownian values. Transformation of Brownian motion, which happens to be Brownian. The intention of doing this is to introduce the e-transformer. Can we have a martingale, continuous martingale? Could you cook up something to make it like a Laplacian to be 0? Not Laplacian, but some other object. Like, yeah, some variants of Laplacian, some of second order. Or some matrix, maybe. So basically it depends on how your martingale looks like. Because right here's a fact. You can always write it as continuous martingale. You can write it as a PDE, as some sarcastic integration of a Brownian motion. So what you would do is that eventually you pick up something like this in your operator instead of a Laplacian. Because here, in the simplest case of Brownian motion, this integrates actually the identity matrix, which you would get Laplacian. Does that answer your question? So in general, for general martingales, you pick up some extra values, which puts your differential operators. Brownian or sarcastic calculus, they are PDEs. And now let me give you another connection, which is quite a deal. So it says that elliptic or parabolic PDEs, they are actually just nothing but a sarcastic process. So let me tell you what an elliptic equation is. So you have this operator u, it's not sorry l, acting on function u. It's given by a ij over this partial, second order partial resolution. So in matrix notation, this is just e times gradient u for a b in a quarter of matrices. Let's say you have this operator, differential operator. A is, for simplicity, let's say it's z-magic. What is that? This means that the operator L is the elliptist. You can think of it as generated by a matrix that's positive, positive that way. An elliptic operator. I was thinking that this can be think of as a sarcastic process. Because if you conjecture that u is given by an integration of a kernel, then this kernel g, sorry, let me test this, I look at the equation on a bounded elliptic domain. You can conjecture that this solution for PDEs is actually given by an integration of this form where this g is called a kernel. So this g, you can think of it as probability type of process. This is saying that x of y is the probability density at the point y, which is on the boundary. It's given by this probability. Let me tell you how to generate this process tax. I'm just telling you that there's a magic process tax. S is indeed generated by this operator L. That is that it's dotted, this little x. And then you flow this sarcastic process according to this differential equation, sarcastic differential equation, with a drift b, depending on the common position of it. And a diffusion matrix sigma. And the diffusion term is driven by a E Brown equation. And you choose this sigma such that sigma times its transpose is a matrix A. You can always do that because I assume that A is symmetric and positive definite. So it's like a matrix times its transpose, you can think of it as a matrix, and positive definite allows that you can do this thing, such that it is such a sigma. This is our conjecture, so let's verify that. Again, we will use L's lemma. An expectation of a function on the position that the process keeps the boundary, or x's the boundary. And the expectation is conditioned on the process, the process starts at the position x. Now let's look at this differential equation. The future dynamics of this process, how the process changes, only depends on the current time and the current position. No time depends. That's just a space of it. It depends on the current position. So the future does not depend on the history, but only on the current. So probably it has some t here. Probably a process here instead of a position. According to the current position, which is a sarcastic, which is a random variable, given by the sarcastic process. Apart from the x of t, what the process x is marked off, this is just this expectation, conditioned on the iteration and the time t. Futureation means the information, which is up to time t. The position that x, s is the domain, that's a fixed random variable. In a condition on the current information, because of the power property of conditional expectation, meaning that conditional expectation of conditional expectation is conditional expectation. So that's a marking there, because if you take a further conditional expectation, condition on a previous time, let's say f little s, that will pop out ux of x. So that's a marking there. Now it shouldn't have any drift. It only has this line. Diffusion time depends on the blinding motion. So when you use etan's formula again, you will have here other term times the summing over all the e-desks. And to do a little bit of algebra, this has a time component. First of all, the time component is even by b, b times gradient of u. And then there's just some diffusional term, which can be written by blinding motion. It's cross term. It does not have any diffusional and blinding motion term. It only has the first order term of the time, because dt times de does t square. We just throw that away. That's too small. And dt times dwt, that's the prehouse, order of t raised to the power of prehouse, which we will throw that away. So it only has this dwt times dwt. Which we will keep. It's the first order term of time. Here, we will have all the e-desks, and then the second order derivative of u times sigma i times sigma i k times sigma j k. And then we are summing over all the e-desks. That's all the k's. That's just the matrix capital A. The j term is actually the electric operator M, acting on function, which is the first order term of time. But we said that this process is modeling there. So this is for better variation. So lv is equal to zero. So things I have convinced you that the conjecture makes sense is that the solution to this PDE is given by integral with respect to a transition probability. You start at f's, and then you'll access the domain, which is a probability density. And the process is given by this stochastic differential equation with a drift b and a diffusion function. The reason for g is the PDE gaol called the influence function, the g set. But it's actually a probability density. This is called the Feynman-Katz formula where it sets down a solution to differential equations that we've verified. Solutions to differential equations is given by expectations of a functional of a stochastic process. The y's goes down. Let me give you one more example before I move on to something non-sophistic which is the method of characteristic. So I was putting this little piece here. I was writing my lifted equations. So it goes wrong because a lifted equation it only has a state variable that does not have a time variable. Now let's put the time variable back. Let's assume that we have a steady state. We have an extra term, time. So u changes the time frame. And this is called a parabolic equation because this is a second order of data and this is a first order of data. And a parabolic is given by this. So second order of data. That's what it's called a parabolic equation. We give the Feynman-Katz for parabolic equations and then we move on to methods of stochastic. So methods of cutting risks. Solutions of this equation is given by again anymore. So let's assume that this is defined for the space. It's not like that. It's the exit time, which is random. So I think that if I define a VGF like this then it has to solve this equation. So you're just cutting off the process at some given time? Yes. Let's say this way. You're given this fixed time first and then you produce a process. The process is like depending on PDE. So you're given the PDE first and then you cook up a process. Usually this is coming from social science. It's really unfeasible. So let's say this capital T is the time of your graduation. Let's say UTX is something like depending on your level of working. This is like the double paper or whatever. The salary that you get at the time of graduation. And you want to somehow optimize, let's say your salary. You have this PDE is the critical condition that you want to optimize your salary. So you have something, a goal that you aim for, which is your time is fixed. And then you generate your process. You generate your X, which is like, let's say the process of how are you doing. And the same idea. X is the space where it was fixed. But let me plot in the process here, in the random variable. This guy has a new process. It's going to be a margin gap because still it's marker. The future does not depend on the history of the current position. So it's a conditional expectation of the fixed random variable. And by the same principle of conditional expectation of conditional expectation is conditional expectation. This is a margin gap. And then you compute this line. How into the computation? And you will believe that the drift term, which will be set to zero, is given by the left-hand side of this PDE, a parabolic operator. So again, this V will verify the PDE. And this representation of a solution to PDE given by expectation of a function acting on a process that's called finding cap. This is the finding cap formula for equations and parabolic equations. The solution can be given by a stochastic process. And now I'm looking at something that's non-stochastic, non-stochastic version in baseball. And you see why I really introduced that. And then we'll move on to the bevel stochastic differential equation. So methods of characteristic is a way of writing PDEs of first order. By solving the systems of ODEs, you can reproduce the solutions to the first order PDE. Why is that? Because ODEs are easy to solve. And second, it's quite physical. There's a meaning attached to the ODEs, namely the flows. So ODEs is actually, you can think of it as a change of variable of space time. You'll see what I mean when I get to that point. I will draw a picture for you. PDEs are solved and then we solve ODEs first and then we cook up the solutions to PDE. And this PDE, why is it hard to solve? It's because this is really not the reason. So that's why we take a D tool to ODEs first and then solve the PDE. And by the way, the equations are just going down for the subcutive version. They're all linear because L operator is just a differential. And the truth is it's linear. Here, I'm looking at something that's more advanced. It's nonlinear. And that's why we are taking this D tool. If it's more linear, then you can just do the techniques of integrating factors. Now you guys have a city of yield first. And let's say for example, you're given the terminal condition. Again, let's say you try to optimize something at the time of the target time. You're given the system of PDE and you try to solve it, try to find a representation. So what you do is that you're looking for an ODE. What you want is that when you put x in the position of x, z in the position of u, and p in the position of gradient u, q in the position of the time that you want this to be true. You want the original PDE to be true. So what you want is that your... And heuristically, p will be gradient of u acting on. So the ODE you're looking for is that ODE satisfied by x, z, and also this. Where your product in would get this equality. Then the time derivative, see? What you would get is the time derivative of u. And now by chain rule, you get 3 of u, but this guy is your key. Because you want that to be true. This looks promising. It looks like a system of ODEs. Now, when you take the derivative of p and q, they're not so lucky. Because you'll get this cross term. It's high on the terms. If you take the derivative of p and q, you want to know the calculation because we don't need that. And it's bad. Because it's bad, so you don't need the calculation. Instead, we have another equation that we haven't used, which is saying that q plus this whole f acting on the jumps is 0. So we can take the derivative of that, which is also 0, time derivative of q, and then now by chain rule, we have partial derivative x times derivative x partial z partial p. So let me plug in z dot over there. Now I see an x dot here. I see an x dot here. What if I can create another x dot here? And I can cut them together. So let this part be x dot. Let x dot be partial f partial p. And now, in that terms, I will have z dot is just, if I let my ODEs to be this, to be x dot being partial p, this part, q dot is then z is to be 0. And also z is all h things to be 0. Then I get a closed system. OK. I'm giving the initial range. z is to be 0. So p dot is going to be negative partial f partial x partial z partial p. This looks horrible, but look at the system. You can uniquely determine it. On the left-hand side is the derivative of this vector. On the right-hand side is given by, you see a q here. It's a p here. Then you see p here. You see a q here. So that's actually a first-order OD, that's all the ODE inductance. So if I choose the right initial conditions, a solve is ODEs. And d of t, in particular, that will give me the equation. That's why I'm giving the value of t of x. So what I'm saying is that, if I want to know the value of u t x of this type with the appropriate initial condition, so let's call the solution initial value. The OD of this form, in particular, can plot me the value. This is going to be, this is going to be, I have to finish this part because of the boundary data. u capital t is greater than half. This by definition is going to be c. So now I solve the second equation with the terminal condition. So in particular, I plot in little t here. And the terminal condition I plot in this line. This expression is going to give me u little t hat with the right initial condition. You'll be confused. So let's draw a picture and then give an example. I'll explain what this OD is actually doing. It's a change of coordinate. This is what OD is doing. So as the space at... This is just local, right? It's just local, right? Yes, it's local. Or let's say if you want to explain the local, you'll have to find some specific place. That's why it's called Bolshevist. It's a guess. It's an answers of what the solution should look like. It never asserts that the solution exists, even exists. But the good thing is that if you know that the solution exists, then it's unique. Then it better be discovered. It cannot make anything else. So it's a guess of the form of the solution instead of trying to prove that the solution actually exists. The PDE could be so imposed that you don't have a solution. Or the solution is not even there. So this is a space at the time of time. This is a space at some previous time. So we want to know UT of X here. Instead of looking at the coordinates given by the space, instead we use an OD, use a flow, go back to eternal time, which is what the OD of X is doing. It's like you look at flow, make a px, and then you let it flow and goes back to, goes forward to time capital T, to eternal time. And now, because in eternal time, you look at the emotional value of this function, eternal condition of Z. So you solve the OD of Z, how you want the value of U, a little teacher. So that's what this OD is doing. Instead of using the space coordinates here, you use the space coordinates given by this flow, which is what the PDE does. This is called the O'Leven coordinates, and this is called the Lagrangian coordinates. So what the PDE is doing is that let's look at the coordinates in Lagrangian sense instead of the O'Leven sense. So that's what the characteristic is, and this system of OD is called characteristic. You touch these two, and maybe you touch in general, but in linear case, you don't need these two. C times Z. C and Z are both scalars. This is a simplified app that you can get as linear. So the first OD is X going on F with respect to P. Let's just leave. So my flow is actually a straight line, and it goes back to the terminal time. It's going to be Pt plus Pt, but you know that there's something impressive here which is Qt times P. So Qt plus P times Qt of C times Zt minus C times Z. This by 0. So in a linear case, you don't actually need... You don't need the second and the third. Sorry, you don't need the third and the fourth OD. So you only need the first one. You'll have a closed system that you can solve. Also, this is just an exponential, but your C is a constant. So in particular, now with the terminal value, you can just do something like this. See, there's something that is given by this one. Straight line. So the thing is that to get to write a real Tx should align back to T, capital T, and then find what Zt is, which is... and then multiply by exponential. Get what you want. So here's something interesting. Suppose that this line is not a straight line, but you add some points. When I say C is 0, we don't have this exponential T. We're not getting back the fixed position, but it's a random variable. So your best guess may be the expectation of all these variables. So realistically, what you should think is if this guy starts at... You always want to think about, well, how about the average? This is finite count. This is just a... instead of being an ODE, it's actually a sub-cultivation. It actually does the connections here. So the interpretation of finite count, I think, should really be this flow interpretation. So what is the sub-cultivation differential equation? What it does is actually floating the information, floating the coordinates with the terminal time, time space. And then... because there's noise. So the best you can do is take the interpretation, which, by the way, is not... but the methods of characteristic, yes, that's in advertising or any other PDEs. Finite count, yes, that's in almost every sub-cultivation assessment. But I didn't see why people connecting both, connecting the first one and the second one. So now I think I'm convinced that these two are actually the same. So finite count is actually the relative version of methods of characteristics. And this diffusion, the sub-cultivation differential equation, is actually the characteristic sub-cultivation differential of this PDE. Because here, this depends on PDE. If we add noise, that should be the second order. PDE should be second order. Yeah, there's a second order today. So there should be something related to this PDE here. So instead of putting a grad u here, possibly you should finish the talk by introducing what backwards sub-cultivation... So let's go back to the case of sub-cultivation. In finite count formula, we never express it while down with the PDE. So let me write down what the PDE is. Let's say it's given by this sub-cultivation differential equation. Instead of writing explicitly the differential of utxt, let's say we call this grad ye. We are thinking ye is u tx. The real commutation is e times gradient u times the differential of the drawing motion. Instead of calling this ye greater than u, let's call it t of t. Let's do it again. And all the rest, no matter how ugly it is, that only depends on x. I can write it as yt. Now suppose that we want to change this jump. We want this drift. Not only it depends on x, but it also depends on y side. It also depends on z. So let's say you have your boundary terminal condition. Well, I can't put t to remove it. So what the terminal condition is, but instead of learning to be a function of depending on x, we just learn to be able to grow. So you look at this equation. You have a terminal condition of a backward. And you have a stochastic condition of a stochastic differential function. So this is what a backward stochastic differential function is. Because now suppose that this, like, is formally a differential equation makes sense. Suppose that I can do countless of these. So while t with this process is going to solve called this real functionality, vt of x. This is found vt of x in this movie. vt of x is going to solve a nonlinear second-order differential equation. Instead of a linear second-order differential equation. So what the equation is, is there something as usual, a differential operator, a parallel differential operator. Now you have this equation. What should I put in this terminal condition? You have to t. Why do you solve this nonlinear equation? The nonlinear equation comes from this. You risk it because if you plug in v in the random variable, plug in the process of x of t. This is yt. Yt goes with x, the initial condition, xt being x. So look at this differential equation. We integrate it. We would get, of course, you don't start from zero. You start from the terminal condition of t. Because you know this line. The right-hand side is going to be, which is going to be martingale. And now if you take conditional expectations of t, this is a martingale that will vanish. The main move is negative g to the left because that's why it doesn't have a theorem. Because this martingale part vanish. It is not a fixed random variable. If you plug integration from zero to little t, this guy is going to be a fixed random variable. If you take conditional expectations, that's going to be martingale. So yt, which you think, is not martingale. If you plug this extra integration term, that's going to be martingale. So this g term is going to be martingale. And that's where this guy, this nonlinear term is coming from. It's coming from this integration. So if you do the same thing, we take the differential of this guy, we have a parabolic operator. And the differential of this integration, we have this extra g term here. And this entire thing is going to be a new g. It's set to zero because that's a martingale. So we get this nonlinear differential equation. The solution to the nonlinear differential equation is given by this convoluted construction. First, you flow x back to time capital T. We're trying to solve this backwards stochastic equation, the time letter t. That gives you the value of the p. The last thing, which is going to be rigorous, is how to make sense of this equation. Sorry, this particular equation. That's just something that's normal. So what's the solution? Let's say it's a sum of n. First thing is becoming tactical. So I want several conditions. First, the terminal condition, the terminal random variable to be squared. Why is that? Because I only want to study hyperspace for laziness. There's one thing. In order to be squared integral, all of my integration is better to be squared. I want this guy. What do we want? Let's do it with the root term. So the solution is a pair, y, z, such that these two are integrability inequality satisfies. And also, y and z, they are adapted. Y is given by differential equations, so y is continuous. And also, differential equation is actually an integral equation. Basically, I'm writing this integral equation down, such that this one holds, this equality holds almost surely. I'm just integrating that equation, and then rearranging something. What's the solution means? This is obvious. What's not obvious is that we really want this squared in order to have units. Think about it. Actually, parabolic equations, the solution to parabolic equations, is not unique unless you look at some nice functions. Maybe this is stupid, but where do you... So, given y and z as a solution, do you get x back? Or do you need to... You'll see that you know x. Here, you can just suppress x. Let's say, you can suppress x, and let's say g, instead of mapping from time to state. Think of it. So, the thing of g is random. You're suppressing it. And the thing is that this allows more general dynamics. The dynamics may not be generated by... The randomness may not come from x. It may not come from the diffusion, the stochastic flow. So, the inequality is not something observed if you think about parabolic equations. Actually, parabolic equations, the solution is not unique unless you just think that the parabolic equation is bounded by a solve by the end of the solution. Now, here's a theorem which will look familiar to PoE analysis because the first thing when you look at it, you will know what to do with PoE. Suppose that uniformly, if there's not a delicious constant, there's not an all. You assume that you are coming to your YV to make a solution which is great because people didn't study this object. But this is published in 1999. In French, I do this delicious condition Why do you want to do that? Let me sketch the proof. I won't do the calculations, but you know what to do. That one is just to define the map. Given U and E, so U is in the position of Y, E is in the position of Z. So, just formally connects. U in the position of Y, V in the position of Z on the left-hand side and the output on the right-hand side is this side. That's the output. So, given U and V, and what you produce on the right-hand side is Y. Y is easy to produce by just taking conditional expectations because this guy is a multi-viewer. So, if you take conditional expectation, you get Y. So, how do you cook up V? Well, that's because of the multi-viewer application. Also, because what is L2 is handy, is you have a Hilbert space, a sonochie, of the terminal value. So, what I'm talking about is that there's a Hilbert space that's isomagic, this Hilbert space. So, this is a Hilbert space of the integrand. This is the Hilbert space of the integrand. It says that this map is a Hilbert space on a sonochie. So, once you know U and V, you integrate it from 0 to capital T that gives you a random variable here. And because of the sonochie, which is absolute nonsense, you get an integrand automatically and you need an integrand. So, you have this map that's well defined. It's that one that becomes such a wrap. And it's F2 because G is uniform religious and plays a middle game. And your spinal conjunction is a spinal conjunction under some certain norms. But the fixed-form variable tells you that it's in the expansion. That's why we're imposing the expansion. Of course, if you're going to relax this condition, sometimes when you choose a node, it's not really a question. It's a sharp condition. It's not sharp. That's what I'm saying. So, you can just prove the goal. The first thing we show is that if G is uniform religious, if G is not religious, there's a case that we notice never going to have a solution. Which is that if G is super quadratic, we're saying that G is super quadratic, something like this. Yeah, we can show that in general, there's not going to be a solution. So, it's just like this. If G is quadratic, there's something not very good. But this is well known because it's using a classical tool. I think my intention of this tool is to suppress this type of detail. Can you do more... That's just in general. Can you do more... By doing it more difficult or using more processes at the same time, can you do something even more complicated than, say, a second-order G? G, where G is second-order... The relation to G of the BSD to the PDE is that G is second-order in this case. You mean the differential equation? Yeah, the differential equations. Second-order equations. But to be honest, I don't know. But... Are you emotionally correct for a second-order? Yeah, that's one thing. But I'm not sure if there's some other... So, I was thinking that you couldn't do more than second-order by wrongly motion. But then you're introducing more processes. Like, you're doing pairs of processes or if you take more processes... Okay, well, okay. I don't just say. So, the thing is that now I have X and Y. Let's say I add another process. Let's say... Write down another differential equation. It won't increase the order because it's still a wrongly motion case. So, it's still a second-order PDE. However, it will get you to a more... growth here. So, if you add something more, it's something like this. But you have to have two backwards forecasting, which they call two ESP. So, it's called second-order backwards forecasting. Because here, now the dynamics of graph depends on this command. So, you have to write down instead of write down a backwards... differential equation. You have to write down another backwards differential equation for graph. So, the first thing is not fully nominated. That's this. This one? Yeah. This one is a seven. Because the highest term, this is still it. But if I'm talking about this, I don't know. But the thing is that by throwing more processes, we increase the nonlinearity of the equation. It's the same thing as the first order methods of characteristics. For the linear case, we only need... F is not linear. Then you have to Japanese and this line. There's a fork. There's a representation of that. So, it's... Two things. It's how do you cook up Z of multiple applications. And the other thing is drawing.