# Unexpected simulation behaviour of Geometric Brownian Motion with SciML

**URL:** <https://discourse.julialang.org/t/unexpected-simulation-behaviour-of-geometric-brownian-motion-with-sciml/108870>\
**Category:** Modelling & Simulations\
**Tags:** question, diffeq, sciml\
**Created:** [January 16, 2024, 3:03pm UTC](https://discourse.julialang.org/t/unexpected-simulation-behaviour-of-geometric-brownian-motion-with-sciml/108870 "2024-01-16T15:03:11Z")\
**Posts on this page:** 2\
**Page:** 1

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**Author:** ![VincentW](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/vincentw/32/202719_2.png) [@VincentW](https://discourse.julialang.org/u/VincentW)\
**Post date:** [January 16, 2024, 3:03pm UTC](https://discourse.julialang.org/t/unexpected-simulation-behaviour-of-geometric-brownian-motion-with-sciml/108870/1 "2024-01-16T15:03:11Z")

</div>

Hey,

I wanted to simulate a geometric Brownian Motion of the form

GBB(t)=\exp\left(\beta t+\delta B(t)\right),

where B(t) is just some standard Brownian Motion.  
Or expressed as a SDE, I want to simulate the solution

dGBB(t)=\mu GBB(t)dt+\sigma GBB(t)dB(t)

with \mu=\beta+\frac{\delta^2}{2} and \sigma=\delta.  
(The latter formulation as an SDE might be more consistent to the parameters in the SciML implementation of a Geometric Brownian Motion.)  
However, I want to condition this on start and endpoint, so simulate a Geometric Brownian Bridge.

What I did at first was to simulate standard Brownian Bridges and apply the linear transform and exponential function to it, for example with the below code

```julia
using Plots
using DifferentialEquations
using Distributions
using Random

# package of SciML for Noise Processes including Brownian Bridges
using DiffEqNoiseProcess

# step size of the solver
dt = 0.01

# conditions for the Brownian bridges
t0 = 0.0
tend = 10.0
W0 = 0.0
Wend = 4.0

# parameters for the linear transformation
beta = 0.1
delta = 0.1

# first simulate a Brownian Bridge and apply linear transform and exponential by hand

function BB_to_GBB(beta, delta, BB)
    GBB = [exp(beta*(i-1)*dt+delta*BB.u[i]) for i in eachindex(BB.u)]    
    return GBB
end

SimpleBB = BrownianBridge(t0, tend, W0, Wend)
BBprob = NoiseProblem(SimpleBB, (t0, tend))

BBensemble_prob = EnsembleProblem(BBprob)
BBensemble_sol = solve(BBensemble_prob, trajectories = 1000, dt = dt)

TransBBmean = mean([BB_to_GBB(beta, delta, BBensemble_sol[i]) for i in eachindex(BBensemble_sol)]);
plot(t0:dt:tend+dt, TransBBmean)

```

This works nicely and the mean process looks as I would expect it, some kind of exponentially looking curve between beginning and endpoint.  
 ![Screenshot from 2024-01-16 15-56-19](https://global.discourse-cdn.com/julialang/original/3X/5/3/5384b5f9323dbc28afef1812a0dcdc7d6625d0b4.png)

I then found that the `DiffEqNoiseProcess.jl` package also has a function for directly simulating Brownian Bridges. So I tried this

```julia
GB0 = exp(beta*t0+delta*W0)
GBend = exp(beta*tend+delta*Wend)

# reformulate to get the parameters of the SDE expression
mu = beta+delta^2/2
GBB = GeometricBrownianBridge(mu, delta, t0, tend, GB0, GBend)
GBBprob = NoiseProblem(GBB, (t0, tend))

GBBensemble_prob = EnsembleProblem(GBBprob)
GBBensemble_sol = solve(GBBensemble_prob, trajectories = 1000, dt = dt)
GBBmean = mean([GBBensemble_sol[i].u for i in eachindex(GBBensemble_sol)]);

plt = plot(t0:dt:tend+dt, GBBmean)
plot!(plt, t0:dt:tend+dt, TransBBmean)

```

![Screenshot from 2024-01-16 15-58-37](https://global.discourse-cdn.com/julialang/original/3X/5/2/529f5ddba110c847bd6a0a0076d16e62e927b34f.png)

But as you can see from the plot, the mean is now just a linear interpolation between start and endpoint. For me this is what I would expect from a normal Brownian bridge but not the Geometric. Moreover, I would expect the two simulation approaches to be equivalent, so the plots should be the same.

Maybe someone can help me whether I got something wrong conceptually or in my implementation.  
Unfortunately, there is no documentation for the `GeometricBrownianBridge` function, so maybe I am just using it wrong.

Would appreciate any help on this.

Thanks  
Vincent

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<div class="post-metadata">

**Author:** ![ChrisRackauckas](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/chrisrackauckas/32/77_2.png) [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)\
**Post date:** [January 19, 2024, 3:32am UTC](https://discourse.julialang.org/t/unexpected-simulation-behaviour-of-geometric-brownian-motion-with-sciml/108870/2 "2024-01-19T03:32:24Z")

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This looks to be a bug. Fix discussion in [Unexpected simulation behaviour of `GeometricBrownianBridge` process. · Issue #186 · SciML/DiffEqNoiseProcess.jl · GitHub](https://github.com/SciML/DiffEqNoiseProcess.jl/issues/186)