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Stoch_Process_Functions

Function Initial Value within Stoch Process returns the initial values of the state variables.

Function Drift within
Stoch Process returns the drift part of the SDE at a given observation time t1 when the observed value of the state variable is x1.
More specifically, if we write the SDE as dx = μ(x,t)dt + σ(x,t)dw, the drift part is the function μ(x,t) and this function returns the array of numbers μ(x1,t1)
Note that the dimensionality of the array μ(x1,t1) is the same with the dimensionality of the state variable x.

Function Diffusion within
Stoch Process returns the diffusion part of the SDE at a given observation time t1 when the observed value of the state variable is x1.
More specifically, if we write the SDE as dx = μ(x,t)dt + σ(x,t)dw, the diffusion part is the function σ(x,t) and this function returns the matrix of numbers σ(x1,t1)
Note that the dimensionality of the matrix σ(x1,t1) is NxN where N is the dimensionality of the state variable x.

Function Expectation within
Stoch Process returns the expectation of the state variable x at time t1 + dt, given the observed value x(t1) at time t1.
The interval dt is treated as small, so that if we write the SDE as dx = μ(x,t)dt + σ(x,t)dw, the coefficients μ(x,t) and σ(x,t) are kept constant during the interval dt.
Note that the dimensionality of the returned array is the same with the dimensionality of the state variable x.

Function Std Deviation within
Stoch Process returns the standard deviation of the state variable x at time t1 + dt, given the observed value x(t1) at time t1.
The interval dt is treated as small, so that if we write the SDE as dx = μ(x,t)dt + σ(x,t)dw, the coefficients μ(x,t) and σ(x,t) are kept constant during the interval dt.
Note that the dimensionality of the returned array is the same with the dimensionality of the state variable x.

Function Evolve within
Stoch Process returns the value x(t0 + dt) that the state variable x has reached at time t1 + dt when both its initial value x(t1) at time t1 and the stochastic shock dw are known.
The interval dt is treated as small, so that if we write the SDE as dx = μ(x,t)dt + σ(x,t)dw, the coefficients μ(x,t) and σ(x,t) are kept constant during the interval dt.
Note that the dimensionality of both the stochastic shock dw and the returned array is the same with the dimensionality of the state variable x.

Function Simulated Values within
Stoch Process returns the values attained by a given array of N stochastic processes after some specified time interval, as produced by simulation.
If N > 1, all processes must be one-dimensional, so that each process describes the time evolution of one asset.
If N = 1, the single process may be multi-dimensional, as for example the Heston process that carries two dimensions, one for the asset and one for its volatility.
For a specified simulation sample size S, the returned values are in the form of a matrix consisting of S rows and D columns, where D is the total number of dimensions.
If N > 1 then D = N.
Each row contains the simulated values of the stochastic processes.
This function also allows the optional specification of the usage of the antithetic technique.

Function Run Simulation within
Stoch Process simulates the evolution a given array of N stochastic processes over a specified time interval and returns the corresponding values mapped by a given real function of N variables.
It also returns the average and standard deviation of the mapped real value.
The returned output is an object of type
Simulation Report