simByMilstein2
Simulate diagonal diffusion Merton
sample paths by second
order Milstein approximation
Since R2023b
Syntax
Description
[
simulates Paths
,Times
,Z
,N
] = simByMilstein2(MDL
,NPeriods
)NTrials
sample paths of NVars
correlated state variables driven by NBrowns
Brownian motion
sources of risk and NJumps
compound Poisson processes
representing the arrivals of important events over NPeriods
consecutive observation periods. The simulation approximates the continuous-time
Merton jump diffusion process by the second order Milstein approximation.
simByMilstein2
is only valid for diagonal diffusion SDE
models.
[
specifies options using one or more name-value pair arguments in addition to the
input arguments in the previous syntax.Paths
,Times
,Z
,N
] = simByMilstein2(___,Name=Value
)
You can perform quasi-Monte Carlo simulations using the name-value arguments for
MonteCarloMethod
, QuasiSequence
, and
BrownianMotionMethod
. For more information, see Quasi-Monte Carlo Simulation.
Examples
Input Arguments
Output Arguments
More About
Algorithms
This function simulates any vector-valued SDE of the following form:
Here:
Xt is an
NVars
-by-1
state vector of process variables.B(t,Xt) is an
NVars
-by-NVars
matrix of generalized expected instantaneous rates of return.D(t,Xt)
is anNVars
-by-NVars
diagonal matrix in which each element along the main diagonal is the corresponding element of the state vector.V(t,Xt)
is anNVars
-by-NVars
matrix of instantaneous volatility rates.dWt is an
NBrowns
-by-1
Brownian motion vector.Y(t,Xt,Nt)
is anNVars
-by-NJumps
matrix-valued jump size function.dNt is an
NJumps
-by-1
counting process vector.
simByEuler
simulates NTrials
sample paths of
NVars
correlated state variables driven by
NBrowns
Brownian motion sources of risk over
NPeriods
consecutive observation periods, using the Euler
approach to approximate continuous-time stochastic processes.
Consider the process X satisfying a stochastic differential equation of the form.
The attempt of including a term of O(dt) in the drift refines the Euler scheme and results in the algorithm derived by Milstein [1].
Further refining of the Euler scheme gives out a metho with a weak order 2:
where dI is given by the area of the triangle with base dt and height dW.
References
[1] Milstein, G.N. "A Method of Second-Order Accuracy Integration of Stochastic Differential Equations."Theory of Probability and Its Applications, 23, 1978, pp. 396–401.
Version History
Introduced in R2023b