## stochastic calculus – Specifying general covariance to an ItoProcess?

I’m studying a system of stochastic equations and I would like to specify the covariance between the Wiener processes appearing in each. The Mathematica help’s page for `ItoProcess` indicates that this is possible, but does not make it clear how to indicate covariance.

For example, suppose I want to solve

$$qquad dx_1 = -gamma_1 x_1 dt + sigma_1 dW_1 qquad dx_2 = -gamma_2 x_2 dt + sigma_2 dW_2$$

where $$dW_1$$ and $$dW_2$$ are Wiener processes. Independently, this is easy since $$x(t)$$ and $$y(t)$$ are Ornstein Uhlenbeck processes, and Ito’s lemma gives $$dW_1^2 = dW_2^2 = dt$$. However, suppose they’re corelated such that $$dW_1dW_2 = xi dt$$ where $$-1le xi le 1$$?

How is this specified to `ItoProcess`?

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## general topology – \$(a, b] cup [c, d)\$ cannot be written as a union of open intervals

I know this looks like an obvious question, but I’m not exactly sure at the method of proof for this question and suspect it involves some topology (which I’ve never taken a formal course on).

Suppose $$I = (a, b) cup (c, d) subset mathbb{R}$$ satisfy $$a < b < c < d$$. I wish to show that it cannot be written as a union of open intervals.

Consider $$mathbb{R}$$ under the standard topology. Then $$I$$ is disconnected because it is not an interval. Because it is disconnected, it cannot be written as a union of open intervals in $$mathbb{R}$$.

Is my proof correct?

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## general topology – Supporting set in convex sets

I am traing to prove that: If $$f$$ is a continuous linear functional on a compact convex set $$C$$ then the set $$S={c in C : f(c)=maxf(C)}$$ is non empty convex and supporting set in $$C$$.where $$C$$ is convex set.

I’m using this definition of supporting set : $$S$$ is
called a supporting set in $$C$$ if S is convex and has the property that if an “inside
point” of a line segment from $$C$$ belongs to $$S$$, then the entire line segment must
belong to $$S$$.That is, if $$c_{1},c_{2} in C$$ and $$lambda c_{1} +(1- lambda)c_{2} in S$$ for some $$0 < lambda < 1$$, then both $$c_{1}$$ and $$c_{2}$$ must be in $$S$$.

Any idea?

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## general topology – Theorem that \$A Leftrightarrow B\$ where \$A = \$”\$f\$ is continuous, maps saturated open sets to open sets”, \$B=\$”\$f\$ is a quotient map” is false.

Suppose $$f:X to Y.$$ Let $$A$$ be the assertion $$f$$ is continuous and maps saturated open sets to open sets, let $$B$$ be the assertion $$f$$ is a quotient map. I am struggling to prove $$A Leftrightarrow B,$$ which was supposedly “””proven””” here.

My proof of $$B Rightarrow A$$ is the same as the given proofs, and my proof of $$A Rightarrow B$$ proceeds along the same lines until I get to the statement $$f(f^{-1}(U)) = U.$$ This is false because we do not know $$f$$ is surjective. In fact, I believe $$A Rightarrow B$$ is false and the book has a critical mistake of forgetting to mention surjectivity in the “is equivalent to…” discussion; are my suspicions correct?

## Define Function(s) to Retrieve System of Equations

``````Clear(P,V,n,R,T);
Rval=QuantityMagnitude@UnitConvert@Quantity(1, "MolarGasConstant");
idealGasEqn := Module({R=Rval,eqns}, eqns = {P*V == n*R*T})
``````

## Known Variables

### Case 1: P, V, and n are knowns (Solve for T)

``````Pval1 = Quantity(1.5, "Atmospheres");
Vval1 = Quantity(3, "Liters");
nval1 = Quantity(1, "Moles");
``````

### Case 2: V, T, and n are knowns (Solve for P)

``````Vval2 = Quantity(3, "Liters");
nval2 = Quantity(1, "Moles");
Tval2 = Quantity(55,"Kelvins");
``````

## Procedure

### Setup

Equations, solve variables, and inputs

• get system of equations based on input argument (e.g. type = “IdealGas”) using a Switch statement.
• define list of solve variables (Symbols that are left unassigned)
• define list of input variables (mixture of unassigned and assigned)

Units

• Get output unit and SI unit Quantities both with magnitude 1
• find positions in solve variable and input lists based on variable type (Symbol or Quantity) using Position
• replace quantities with magnitude of SI-converted quantities

### Solve

• solve for unknowns using SI magnitudes, no units in output (i.e. unitless) using Solve
• attach SI magnitudes to unitless solution and convert to output units

### Output

• output a unitless or unit-containing solution

## Module

``````idealGasSolver(P1_,V1_,n1_,T1_,type_:"IdealGas",unitlessQ_:False) :=
Module(
{eqns,vars},
(*get system of equations*)
eqns = Switch(type,"IdealGas",idealGasEqn);

vars = {P,V,n,T}; (*Symbols for solve, keep unassigned throughout*)
valsTmp = {P1,V1,n1,T1}; (*input values, some are Symbols, some are Quantities*)

(*units with magnitude 1*)
outUnits = Quantity(1,#)&/@{"Atmospheres","Liters","Moles","DegreesCelsius"};
SIunits = Quantity(1,#)&/@QuantityUnit@UnitConvert@outUnits;

(*find positions based on variable type*)
quantityIDs = getIDs(Quantity);
symbolIDs = getIDs(Symbol);

(*replace quantities with magnitude of SI - converted quantities*)
{quantityIDs,QuantityMagnitude@UnitConvert@valsTmp((quantityIDs))});
vals = ReplacePart(valsTmp,rules1);

(*solve for unknowns using SI magnitudes, no units in output*)
unitlessSoln = Solve(eqns/.rules2,vars((symbolIDs)))((1));

(*convert solution to output units and include units*)
rules3 = MapThread(#1 -> #2 &, {vars((symbolIDs)),
vals((symbolIDs))*SIunits((symbolIDs))});
{vars((symbolIDs))/.rules3/.unitlessSoln,outUnits((symbolIDs))});

(*output a solution based on unitlessQ argument*)
outsoln = If(unitlessQ,unitlessSoln,unitSoln)
)
``````

## Case 1

Make sure that temperature input is unassigned and specifically use the variable “T”.

``````Clear(T);
idealGasSolver(Pval1, Vval1, nval1, T) (*output in units based on outUnits (deg C)*)
idealGasSolver(Pval1, Vval1, nval1, T, "IdealGas", True) (*output temperature SI unit (K) magnitude*)
``````

`{T -> Quantity(-218.31031631383098, "DegreesCelsius")}`
`{T -> 54.83968368616898}`

We get units with the first output, and an SI magnitude with the second.

## Case 2

Make sure that pressure input is unassigned and specifically use the variable “P”.

``````Clear(P);
idealGasSolver(P, Vval2, nval2, Tval2) (*output in units based on outUnits (atm)*)
idealGasSolver(P, Vval2, nval2, Tval2, "IdealGas", True) (*output pressure SI unit (Pa) magnitude*)
``````

`{P -> Quantity(2286477219992141/1519875000000000, "Atmospheres")}`
`{P -> 2286477219992141/15000000000}`

Exact arithmetic is preserved in this case.

## Case 3 (additional case, underdetermined system of equations)

Make sure that all inputs except temperature are unassigned and specifically use the variables “P”, “V”, and “n”.

``````Clear(P, V, n)
idealGasSolver(P, V, n, Tval2) // N
idealGasSolver(P, V, n, Tval2, "IdealGas", True) // N (*output SI magnitude*)
``````

`{P -> UnitConvert(P*Quantity(1., "Kilograms"/("Meters"*"Seconds"^2)), Quantity(1., "Atmospheres")), V -> UnitConvert(V*Quantity(1., "Meters"^3), Quantity(1., "Liters")), n -> UnitConvert(P*V*Quantity(0.002186770091685928, "Moles"), Quantity(1., "Moles"))}`
`{n -> 0.002186770091685928*P*V}`

The second output (SI magnitude) is more parsable and less subject to issues if you were to apply this process successively (i.e. use the outputs as inputs to the next system of equations).