FBSDE general case

Stochastic simulations
1

Hello,

Let’s consider the Forward-Backward SDE composed of one Forward SDE:

dx = a(x, y) * dt + b(x, y) * dW

and one Backward SDE:

dy = c(x, y) * dt + z * dW

where a, b and c are some functionals. Thanks to Dynare, I can solve this FBSDE using a second order approximation near its steady state using

dt = 1/12;

model;
x - x(-1) = a(x(-1), y) * dt + b(x(-1), y) * dWt;
y(+1) - y = c(x(-1), y) * dt;
end;

shocks;
var dWt = dt;
end;

Let’s assume now the more general case where a, b and c depend on z. I would need to define z as an endogenous variable, but what equation should be associated with it?

In other words, how can I access the volatility of an endogenous forward-looking variable?

Thanks a lot for your help!

Best regards,
Thibaut

2

What is z here and what is its relation to the

?
But at higher order, it is possible to define conditional variances. See e.g. DSGE_mod/Basu_Bundick_2017/Basu_Bundick_2017.mod at master · JohannesPfeifer/DSGE_mod · GitHub

4

Hello Johannes,

Thank you for your reply.

z is precisely the volatility of the endogenous forward-looking variable y, as is made clear by the equation for the dynamic of y:

dy = c(x, y) * dt + z * dW

Ultimately, a FBSDE consists of two equations

dx = a(x, y, z) * dt + b(x, y, z) * dW
dy = c(x, y, z) * dt + z * dW

for three unknown variables x, y and z.

As is well explained for example by Carmona & Delarue in their textbook on Mean Field Games:

“For the reader who is not familiar with the theory of forward-backward equations, it may sound rather strange to ask for the well posedness of a system with three unknowns but two equations only. Actually, the reader must remember the fact that the triple (X; Y; Z) is required to be progressively measurable with respect to F. In particular, it should not anticipate the future of W. The role of the process Z is precisely to guarantee the adaptedness of the solution with respect to the filtration F. In the end, the forward-backward system actually consists of two equations and a progressive measurability constraint.”

Dynare, for its part, requires the same number of equations and variables.

So how can this be sorted out?

Thanks a lot in advance!

Best regards,
Thibaut

5

Sorry, I am not an expert in this field. But as far as I understand, your are discretizing a differential equation to work with difference equations. But in that case, something needs to take the place of the

6

Yes indeed that is exactly what I am doing.

Let me ask you my question another way then:

Still considering my model with a predetermined variable x and a forward looking variable y, I am looking to access the volatility of y (named z) in the model block.

The missing equation would be something like

z = volatilityOf(y);

if only such a function existed.

But maybe we can make use of the policy function F (also known as decision rule) according to which the forward looking variable y can be express as a function of the predetermined one x:

y = F(x)

(see for example this paper from Dynare at the beginning of page 5: https://www.dynare.org/wp-repo/dynarewp002.pdf)

Applying Ito’s formula, we find

z = F’(x) * b(x,y,z);

where b is defined as in my previous messages.

That could be my missing equation if only I could access the derivative of the policy function F’(x) in the model block.

Do you know if that’s possible ?

Thanks a lot for your help !

Thibaut

7

When you say

what exactly do you mean? It’s not the unconditional variance of y as far as I can see.

Regarding your question, there may be no closed form way of doing this. But you may be able to solve this iteratively. Start with a guess for the policy function and its derivative. Compute the actual policy function based on that solution. Update the guess and keep iterating until you reach the fixed point.

8

Hello Johannes,

It is not the unconditional variance of y indeed, but the variance of y(+1) conditional on y. The missing equation I am looking for is actually:

z^2 = Var( y(+1) | y );

That I can compute following the example of the github link you sent me earlier.

However, that would give me z^2 only, and not the sign of z, which does matter in what I am doing.

Alternatively, I can write (b being defined as in my previous messages):

z * b(x,y) = Cov( y(+1), x | y, x(-1) );

How could I compute such a conditional covariance in the model block ?

Thank you!

Thibaut

9

I haven’t worked it out, but a covariance is simply a production of expectations. You should be able to define it using auxiliary variables.