Backward stochastic differential equation

Okay, so picture this: You're playing a game where you have to go backwards. You start at the end and you have to figure out what you did to get there. That's what a Backward Stochastic Differential Equation, or BSDE, is like!

But instead of a game, it's a mathematical equation. Essentially, it's a way to figure out what happened in the past, given what we know about the present. It's used a lot in finance, where people are trying to figure out what stock prices were in the past based on what they know now.

Think about it like this. Imagine you're trying to figure out what the weather was like yesterday, but all you have is today's weather report. You could use that report, along with some knowledge of how the weather works (like temperature patterns and such), to make a guess about what things were like yesterday. A BSDE is a way to do something similar with financial data.

So how does it work? Well, it uses a few things. First, it uses something called a "stochastic process". That's just a fancy way of saying a process that involves some randomness. Think of it like rolling dice - you never know exactly what's going to happen.

Then, it uses a "differential equation". This is a way of describing how things change over time. You might use a differential equation to describe how a ball falls from the sky, for example - it would tell you how fast the ball is going at any point in time.

Finally, it goes backwards. Instead of trying to solve the equation starting from the present and moving forward, it starts at the end and works its way backwards. That way, you can figure out what things must have been like in the past to get to where you are now.

So, a BSDE is really just a way of figuring out what happened in the past based on what we know now, using some fancy math. It's like a game where you work backwards to figure out what happened, but with numbers and equations instead of toys and games!