Brownian Bridges for Sampling Between Simulated SDE Time Steps
Summary
The document considers how to estimate a stochastic process value between two simulated time-grid points for an SDE with drift and diffusion. Straight-line interpolation may give an intermediate value, but it does not preserve the process’s random evolution. Adding multiple interpolated points can make them share the same path segment and distort statistics by removing the appropriate Brownian variation.
A Brownian bridge is proposed to retain randomness when inserting points between existing simulated values. The response also suggests choosing the simulation design according to the goal: simulate on a finer grid from the outset when intermediate observations are expected, or run a separate simulation with a shifted grid for exploratory insight. For a particular state and local drift and volatility, repeated simulations with a finer step can examine possible outcomes. The document offers conceptual guidance rather than a derivation or comparison of numerical methods, and the best choice depends on why additional points are needed.
Key ideas
- Linear interpolation between SDE grid points does not reproduce the process’s intervening Brownian randomness.
- Adding several interpolated points can distort statistics by imposing a shared straight path segment.
- A Brownian bridge can add intermediate values while preserving randomness conditional on the simulated endpoints.
- When intermediate observations are expected, simulating on a finer grid from the start may be more appropriate.
- The sampling method should be chosen based on how the added points will be used.
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# Interpolation of $\mu(t,X(t))dt+\sigma(t,X(t))dW(t)$
# 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 $t_k < t <t_{k+1}$?
## Answer by lehalle (score 6)
https://quant.stackexchange.com/a/68879
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).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.