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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