I want to solve the two-factor Black-Scholes PDE,
q = 0;
\[Sigma] = 0.3;
T = 1;
r = 0.06;
NDSolve[{ D[V[Subscript[S, 1],Subscript[S, 2],t],t] + r Subscript[S, 1] D[V[Subscript[S, 1],Subscript[S, 2],t],Subscript[S, 1]]+ r Subscript[S, 2] D[V[Subscript[S, 1],Subscript[S, 2],t],Subscript[S, 2]] + 1/2 \[Sigma]^2 Subscript[S, 1]^2 D[V[Subscript[S, 1],Subscript[S, 2],t],Subscript[S, 1],Subscript[S, 1]] + 1/2 \[Sigma]^2 Subscript[S, 2]^2 D[V[Subscript[S, 1],Subscript[S, 2],t],Subscript[S, 2],Subscript[S, 2]] - r V[Subscript[S, 1],Subscript[S, 2],t] == 0,
V[Subscript[S, 1],Subscript[S, 2],T] == Max[Subscript[S, 1],Subscript[S, 2]],
V[0,Subscript[S, 2],t] == Subscript[S, 2],
V[Subscript[S, 1],0,t] == Subscript[S, 1],
V[100,Subscript[S, 2],t]== 100,
V[Subscript[S, 1],100,t] == 100
},
V,
{Subscript[S, 1],0,100},{Subscript[S, 2],0,100},{t,0,T}
]
I got two warning
Warning: boundary and initial conditions are inconsistent
and
The initial conditions did not evaluate to an array of numbers of depth 2 on the spatial grid. Initial conditions for partial differential equations should be specified as scalar functions of the spatial variables
If I open up Wolfram Cloud Link in my browser and I patiently wait for the "Type your Wolfram Language Input..." to appear and I left mouse click in the empty box below that and I tap the "x" key and wait for that to appear and I tap the backspace key and wait for that to disappear and then I scrape-n-paste exactly this
q=0; \[Sigma]=0.3; T=1; r=0.06;
sol=V/.NDSolve[{ D[V[s1,s2,t],t] + r s1 D[V[s1,s2,t],s1]+ r s2 D[V[s1,s2,t],s2] +
1/2 \[Sigma]^2 s1^2 D[V[s1,s2,t],s1,s1] + 1/2 \[Sigma]^2 s2^2 D[V[s1,s2,t],s2,s2] -
r V[s1,s2,t]==0, V[s1,s2,T]==Max[s1,s2], V[0,s2,t]==s2, V[s1,0,t]==s1,
V[100,s2,t]==100, V[s1,100,t]==100}, V, {s1,0,100},{s2,0,100},{t,0,T}][[1]];
Table[{s1,s2,t,sol[s1,s2,t]},{s1,0,100,25},{s2,0,100,25},{t,0,1,1/4}]
into that empty box and patiently wait for that to appear and I tap shift+enter and wait for the result to appear
then I get three warnings about grid points, but those seem safe to ignore, and after another delay I see the table of {s1,s2,t,sol[s1,s2,t]} appear.
I do not see either of the error messages you report.
Can you reproduce exactly the steps that I have shown here and get exactly the same results?
Perhaps this is a version problem. What version were you using when you got the error? Perhaps this is a Subscript problem. Perhaps this is a cache issue using prior assignments in your notebook.