The other thing I want to say is that this is work that I have developed with many people

and among those people, among those collaborators, I would like to mention my student Vinicius

Albany but I'll give credit at the end to these collaborators. Okay, so my talk,

Local volatility estimation, the presence of jumps. Since I realize not everybody here comes

from UBS problems, not everybody here comes from PDEs, we have a mixed audience, I prefer to start

with some motivations and I hope I'll be convincing on these motivations. It's a bit on the business

side and I hope you forgive me for talking about this dirty thing called money but it helps some

people, not everybody, and allows us to travel for example. So what I show here is a very interesting

graph of something called derivative markets, it's not DDX please, it's derivatives. How many

people have heard about derivatives here by the way? Okay, okay, good, good. So I'm not...

No, not Caputo, not fractional, although it can be risky, it can be more risky than Caputo. Even

Caputo that is minister somewhere in... Why minister? I don't know. Anyway, okay, you never know,

might be minister again. So no, derivatives are really part of the world finance and the graph

that I show here is with time, in dollars, in billions of dollars, the amount of traded derivatives

worldwide from 2004 and 2022 and you see that this is a very, very big growth and we're talking

about a lot of money and these are amazingly strong numbers. The top banks in the US, in Europe,

have a tremendous exposure to derivatives, notional value of order of 632 trillion at the end of June

22. This is when I took these numbers, I gave up renewing these numbers every time I talk about it.

The gross market of outstanding over-the-counter derivatives, some in positive and negative values,

rose in the first half of 2022 to 18.3 trillion. Deutsche Bank, I guess you guys know Deutsche Bank,

right? Yeah, has more than 90 to the 9 euros potential exposures in 2020. So we're talking

about serious money here. What I'm saying is basically is that options and derivatives are

fundamental parts of the world economics. They really, really play an important role. So let me

give you, and I hope you don't... Well, you can sleep a little bit, that's why I wanted to close

the windows a little bit. But this future, I'll give you a little bit of an explanation of these

things so that you get a feeling. Futures are a type of derivative that you basically have the

right to buy or sell a specific commodity. What's a commodity? Things like cotton, oil, sugar, beer.

Beer is a commodity. At a set future date for a set price. So for example, Eric and I decide that

in one year and a half, I'm going to deliver 1,000 gallons of beer in San Carlos, right? And then we

set a price and there is a way of trading so that... Well, we have to decide the price and there is a

way of computing this price provided you take also in account the transportation. Futures are

crucial in commodities and energy trading. These are immense markets. They are interconnected with

fixed income markets like interest rates. So even if you hate those things and you have to buy a

house, you have to worry about these things because fixed income is basically related to how much

interest you pay for, for example, when you buy a house or an apartment or some land. And this is

interconnected with currency markets. So when you come and trade and exchange money here in,

let's say, Euros for the eyes, you in a certain sense, you are related to these things. Okay,

but what's the math behind it and why am I, as a mathematician, interested in this? And you have

to trust me, but I'll try to give you an idea of that. What is behind these things? Diffusive

processes, expected values of diffusive processes and control. Now, since I gave this talk also to

audiences in, let's say, machine learning community, I have to also please my other community. So it's

related to model estimation selection. In financial language and economics, sometimes I talk to

economics people, actually I try to talk as much as possible. You have to find the risk premium

associated to a contract. So for example, if I go into a contract with Enrique, like an option or

a future, there's a risk associated to that and somehow there is going to be a risk premium and

that's what's going to make a difference when, you know, I sell it in the market. This talk is

related also to Professor Kasten's talk, very nice talk by the way, and although my point of view is

of inverse problems that are ill-posed and we should discuss the possibility of model reduction

here, but I don't want to go into that now, at least, because I have a talk all prepared on

ill-posed problems. Anyway, financial data is a highly complex phenomenon. We are talking about