Martingale
Martingale represents a discrete-time adapted process whose conditional future expectation equals its present value under the supplied filtration.
For a martingale \((M_n)\),
\[\mathbb E[M_{n+1}\mid\mathcal F_n]=M_n.\]
The same object supports the submartingale and supermartingale tests used by the package.
Constructor
Martingale(process, filtration)
Properties
| Property | Meaning |
|---|---|
process |
Underlying discrete-time process. |
filtration |
Filtration with respect to which the process is analyzed. |
n_steps |
Number of available time steps. |
Methods
value_at(n)— value/process state at timen.conditional_next(n)— conditional next-step expectation.martingale_residual(n)— difference defining the martingale condition.is_martingale()— test the martingale property.is_submartingale()— test the submartingale property.is_supermartingale()— test the supermartingale property.conditional_future(n, k)— conditional future value at a later step.expectations()— expectations across the time grid.transform(function)— transform the process pointwise while preserving the appropriate public process representation.doob(random_variable, filtration)— construct the Doob martingale supported by the implementation.stopped(stopping_time)— construct the stopped process.
Example
M = Martingale(process, filtration)
print(M.is_martingale())
print(M.martingale_residual(2))