Hi,
Just wondering if anyone would be able to help me with a question I'm stuck on:
Assume that initially, at time t_0 = 0 the particle is located at a point x_0 ∈ R.
At time t_1 = t_0 + δt for some δt > 0 the particle’s location is given by
X_1= x_0 + νδt + σW_1√(δt)
where W_1 is a standard normal variable and ν ∈ R is a constant drift term and σ > 0 is the diffusion constant that depends on the ambient temperature among other things. Notice that generally, the movement of the particle in equidistant time steps, i.e. at times t_1, . . . , t_n where t_n = t_(n−1) + δt, can be described recursively by
X_j =X_(j−1) + νδt + σW_j√(δt)
where W_1, W_2, . . . , W_n are independent standard normal random variables and X_(j−1) denotes the location of the tracked particle at time t_(j−1)
Based on the above info, how do I find an expression for X_j that is a function of W_1, . . . , W_j and the parameters but does not depend on the previous position X_(j−1)?
Thanks in advance!
Just wondering if anyone would be able to help me with a question I'm stuck on:
Assume that initially, at time t_0 = 0 the particle is located at a point x_0 ∈ R.
At time t_1 = t_0 + δt for some δt > 0 the particle’s location is given by
X_1= x_0 + νδt + σW_1√(δt)
where W_1 is a standard normal variable and ν ∈ R is a constant drift term and σ > 0 is the diffusion constant that depends on the ambient temperature among other things. Notice that generally, the movement of the particle in equidistant time steps, i.e. at times t_1, . . . , t_n where t_n = t_(n−1) + δt, can be described recursively by
X_j =X_(j−1) + νδt + σW_j√(δt)
where W_1, W_2, . . . , W_n are independent standard normal random variables and X_(j−1) denotes the location of the tracked particle at time t_(j−1)
Based on the above info, how do I find an expression for X_j that is a function of W_1, . . . , W_j and the parameters but does not depend on the previous position X_(j−1)?
Thanks in advance!
