Introduction of semimartingales and diffusions (5) 

This introduces the basic semimartingale identifier Smgl with associated basic stochastic differential identifier dSmgl:

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Attempts to reintroduce semimartingale or stochastic differential identifiers are reported as errors:

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This introduces and sets up a basic semimartingale identifier X with basic stochastic differential identifier dX and initial value expression X0, and satisfying the second- and first-order structure implied by the stochastic differential equation dX==sd:

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Introduce a single Brownian motion:

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Introduce independent Brownian motions:

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Altering the second- and first-order structure, and initial value (4) 

Change quadratic covariation Brktbydt[W1,W2] to rhodt/dt:

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This sets Drftbydt[dX]=DriftdX/dt:

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Now Drift[dX] yields DriftdX:

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Change initial value:

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Placeholders for structure (4) 

The expression Brktbydt is a placeholder for the formal quotient by dt of the bracket differential dX dY:

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Brktbydt has Orderless attribute:

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The expression Drftbydt is a placeholder for the formal quotient by dt of the drift of dX:

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The expression Fixed[t0,X] is a placeholder for the fixed value of the basic semimartingale X at time t0 (usually t0=0):

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Ito differential, expansion, drift, initial value (5) 

Use ItoD to compute the stochastic differential of a semimartingale expression f. ItoD assumes that f is a genuine semimartingale expression:

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Specify f:

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Use Drift to compute the drift of the stochastic differential expression df. Drift assumes that df is genuinely a stochastic differential expression:

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Use InitialValue to compute the value of the expression f at time t0:

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Since W1 is zero at time zero, f is also zero at time zero:

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Ito integral (4) 

Set up two Brownian motions and a semimartingale Z that depends on them:

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ItoIntegral[dX] represents the right-inverse of the stochastic differential ItoD[X]:

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Even without further properties, ItoIntegral can compute some simple stochastic integrals:

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Drift behaves well with ItoIntegral:

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See the Stochastic Integration Tech Note for ways to teach ItoIntegral more integration rules.

Random predicate (6) 

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RandomQ[x] returns True if x is an expression defined in terms of an existing semimartingale:

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Make x a function of a Brownian motion:

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Provide a list in the second argument of RandomQ to test if x is an expression in semimartingales or stochastic differentials excluding those given in the list:

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Use BSDQ[x] to test if x is a defined semimartingale differential:

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Can exclude differentials by using the second argument:

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List all defined semimartingales and their differentials:

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