# Interpolation of $\mu(t,X(t))dt+\sigma(t,X(t))dW(t)$

Let's assume that we have SDE $$dX(t)=\mu(t,X(t))dt+\sigma(t,X(t))dW(t)$$ and we simulate it on a time grid which contains points $$t_k$$ and $$t_{k+1}$$. How can we then calculate value of $$X$$ at time $$t_k < t ?

• You could adjust your time grid, but presumably you would like to avoid this in order to save computation time. Interpolation could be conducted in many ways, the easiest would be just a linear interpolation. Nov 22, 2021 at 16:24
• The log of this process is a Brownian Motion with drift. So perhaps you could use a Brownian Bridge on the log. Nov 22, 2021 at 19:08
• mathworks.com/help/finance/sde.interpolate.html Nov 23, 2021 at 10:46

That is a tricky question because interpolation seems to be ok if you need one point $$\tau$$ between $$t_k$$ and $$t_{k+1}$$ but it is not.

The difficulty arise a direct way if you want two points inside $$[t_k,t_{k+1}]$$: you immediately see that there is no randomness between your two new points and the previous ones because they all belong to the same line: where is the $$dW$$ component of your SDE?

In fact even with one point it is not that simple, because you reduced the randomness in the $$t_k,\tau,t_{k+1}$$ sequence...

What you need is a Brownian bridge that will preserve the randomness.

But indeed what is the best is to simulate your SDE with more points from the start, because I guess that if you need one point soon you will need more. May be you are talking about sampling one new point between $$t_k$$ and $$t_{k+1}$$ for all your $$k$$s... that's too much because you will decrease the randomness of any statistic you will compute on this new "grid".

I do not know what is the usage of these new points,

• if you simply want to understand what is happening between two nodes of your grid and you do not want to throw away what you have (without paying the "complexity" of coding a Brownian bridge), may be running another simulation with a grid that is translated by $$(t_{k+1}-t_k)/2$$ (that I suspect is constant) could be a solution. Now you will have two independent simulations and may be you can learn more with 2 than with one that has twice the number of points.
• if you want to understand what happens between any two $$t_k$$ and $$t_{k+1}$$ independently of $$k$$, or for a specific value of $$(X_k=x,\mu_k=m,\sigma_k=s)$$ you can run a lot of simulations starting from this specific $$(x,m,s)$$ state with a finer step.

(It is alway good to start by the usage of a simulation to understand how to extend it optimally).