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Alan
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Finite difference - CFD technique

May 12th, 2009, 3:06 pm

What are Assumptions 2 and 3?
 
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Cuchulainn
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May 12th, 2009, 3:46 pm

dbl
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May 12th, 2009, 3:46 pm

QuoteOriginally posted by: AlanWhat are Assumptions 2 and 3?1. Ideally, book or journal article treatments of this question.2. Problems or counter-examples or issues with my proposal.3. Alternative proposals. Maybe they are 'Actions'.
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Alan
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Finite difference - CFD technique

May 12th, 2009, 4:56 pm

Ok, we'll go to two factors. I copied and pasted my earlier post and made the necessary modifications:-----------------------------------------------------------------------------------------------------------------------------------Well, for general 2-factor, we are talking about a PDE(A) f_t = a_ij(t,x) f_ij + b_i(t,x) f_i - c(t,x) fwith f(0,x) a given (non-negative) function.My notation: x = (x1,x2) is a two-vector. f_i = Partial f/Partial xiThere is an implied summation on repated indices.If I write b( ) with no subscript, I mean the 2-vector (b1,b2).If write a( ) with no subscripts, I mean the 2 x 2 matrix [a_ij] Let x=B denote any point on a degenerate boundary -- such points are defined by det[a(t,B)] = 0 there. Again B is a 2-vector. I have in mind x = (S,V) for stochastic vol. models, so state space is the positive real first quadrant: x > 0For Example Cases 1,2,3 (see bottom of the post), there are two degenerate boundariesBoundary 1: x1 = 0 (S=0) and Boundary 2: x2 = 0 (V=0 or S2=0 for Example Case 3)So, x=B means x is a point on Boundary 1 or x is a point on Boundary 2. Assume the other boundaries are well-understood and not an issue.So, the question becomes, under this premise, when does it make sense to simply employ(*) f_t = a_ij(t,B) f_ij + b_i(t,B) f_i - c(t,B) fon a degenerate boundary, where certain components of [a_ij(t,B)] may vanish (or none vanish, but the det. does) I want the killing term c(t,x) to not be singular, so,for simplicity, let's assume (i) 0 <= c(t,x) < infinity for all (t,x).Take Boundary 1 and assume the associated SDE for the first component of X, at the boundary, reduces todX_1 = b_1(t,0,X2) dt Similarly, take Boundary 2 and assume the associated SDE for the second component of X, at the boundary, reduces todX_2 = b_2(t,X1,0) dt This is not the most general thing that can happen when det(a) = 0, but it is a good start. Then, the process can "keep running" when Boundary i is hit if and only if b_i > 0 there, where i = 1,2Why? Because this means that once the boundaryis hit, the particle is simply returned to the interior of the state space x > 0. The probability ofbeing killed there is not an issue because c(t,B) is bounded. In addition, even if theboundary is never hit (entrance boundary, say), the process can still be started there.So, from this point of view, everything proceeds nicely.This suggests to me that ======================================================A sufficient set of conditions for a well-posed numerical PDE problem (A), using the BC (*) at a degenerate boundary x=B is that:(i) det[a(t,B)] = 0, and(ii) n . b(t,B) > 0, and(iii) 0 < = c(t,B) < infinityHere n is the inward-pointing normal to the boundary at x=B and "." is the dot product.In other words, the drift will return the particle to the interior of the state space(Of course, there are additional standard regularity conditions, like a(t,x) > 0 for x > 0, etc.)==================================================================Example Cases: 1. Heston: Boundary1. (S=0): b_1(t,S=0,V) = 0, so the sufficiency condition (ii) is violated.Boundary2. (V=0): b_2(t,S,V=0) > 0, so the sufficiency condition (ii) is OK, and (*) may be applied there. 2. SABR: Boundary1. (S=0): b_1(t,S=0,V) = 0, so the sufficiency condition (ii) is violated. Boundary2. (V=0): b_2(t,S,V=0) = 0, so the sufficiency condition (ii) is violated. 3. Multi-asset (n = 2) PDE with correlation Boundary1. (S1=0): b_1(t,S1=0,S2) = 0, so the sufficiency condition (ii) is violated. Boundary2. (S2=0): b_2(t,S1,S2=0) = 0, so the sufficiency condition (ii) is violated. 4. Convertible bonds -- what is the PDE?
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Cuchulainn
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May 12th, 2009, 5:29 pm

Quote4. Convertible bonds -- what is the PDE?We can take this pde
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Alan
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May 12th, 2009, 5:44 pm

Ok, I looked at the link you gave.Since the second factor diffusion coef is strictly positive in that model, there is only one degenerate boundary and we have 4. Convertible bonds (Daniel's linked-to model)Boundary1. (S=0): b_1(t,S=0,r) = 0, so the sufficiency condition (ii) is violated.Regarding your other question (which I see just got erased), the payoff function (value of f(t=0,x)) doesn't matter to what I said.
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Finite difference - CFD technique

May 12th, 2009, 9:21 pm

Here are two references that treat some of the questions you are discussing. The first one deals with the term structure equation for 1-factor short rate models with nonnegative rates (including CIR), and the second paper with the pricing equation for stochastic volatility models.http://www.math.uu.se/~ekstrom/bux.pdfh ... /sv.pdfFor the term structure equation the results are as follows. Assume that the diffusion coefficient is 0 and the drift is nonnegative at r=0. Then the bond price B(r,t) is continuously differentiable up to the boundary r=0, and at the boundary the derivatives satisfy B_t(r=0,t) = b(r=0,t) B_r(r=0,t) (*)(i.e. the equation obtained by formally plugging in r=0 into the pricing PDE; b is the drift). Note that this boundary behaviour is satisfied regardless if the process can reach zero or not.Now, if the process can reach zero, then the boundary condition (*) is needed to ensure uniqueness of solutions to the PDE. (In fact, the PDE can be solved with any boundary condition, but only the boundary condition (*) will give the solution given by stochastic representation.)On the other hand, if the process cannot reach zero, then MATHEMATICALLY no boundary condition is needed (to ensure uniqueness). However, the relation (*) remains true also in this case, and may of course be employed when determining the bond price NUMERICALLY.The results are true under some conditions. The main assumption is that the diffusion coefficient sigma^2(r) should be C^1, i.e. sigma(r) is Hölder(1/2), and that the drift should be Lipschitz. In fact, if I remember correctly, the boundary behaviour (*) is not true if these (seemingly harmless) assumptions are removed.The second reference provides the corresponding results for stochastic volatility models (including Heston).
 
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May 13th, 2009, 2:41 am

QuoteThe results are true under some conditions. The main assumption is that the diffusion coefficient sigma^2(r) should be C^1, i.e. sigma(r) is Hölder(1/2), and that the drift should be Lipschitz. In fact, if I remember correctly, the boundary behaviour (*) is not true if these (seemingly harmless) assumptions are removed.Thanks for joining the thread. I have seen some of your papers and am in general agreement. Your last statementis interesting -- can you give an example of the case you have in mind where (*) fails.?Also, we are struggling here for the most general statement that could be made with N spatial variables, wherethe diffusion coefficient degenerates at some boundary. Any thoughts on that general case?
 
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May 13th, 2009, 3:05 am

Estrom,Welcome to the thread and thanks for the links.As further example, I also suggest we investigate:5. Asian PDEas well because it is an exceptional case (no diffusion in I/A). What do you think?
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May 13th, 2009, 5:15 am

QuoteOn the other hand, if the process cannot reach zero, then MATHEMATICALLY no boundary condition is needed (to ensure uniqueness). However, the relation (*) remains true also in this case, and may of course be employed when determining the bond price NUMERICALLY.In the discussion to follow, do we take far-fields into consideration or just look at 'zero boundaries'? The former is better imo but is more work and it means I don't have to worrry about BC at the far field (numerically, that is).
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ekstrom
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Finite difference - CFD technique

May 22nd, 2009, 1:30 pm

QuoteThanks for joining the thread. I have seen some of your papers and am in general agreement. Your last statementis interesting -- can you give an example of the case you have in mind where (*) fails.?Also, we are struggling here for the most general statement that could be made with N spatial variables, wherethe diffusion coefficient degenerates at some boundary. Any thoughts on that general case?Unfortunately, I do not remember an example where (*) breaks down (if no regularity of the coefficients is assumed). I will have to think more about that.Second, I do not know anything about the general case with N spatial dimensions, but it would be surprising ifthe analogue of (*) did not hold (again, under some regularity assumptions on the coefficients).
 
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August 12th, 2009, 5:06 pm

Update on the CIR pde:1. I have transformed it to one on the interval [0,1], let's say (1) U_t = LU where L is the new elliptic operator LU = A(y)d/dy((1-y)^2dU/dy) + B(y)dU/dy.2. Now I multiply each side of (1) by U, integate from 0 to 1, integration by parts 3 or 4 times, use Gronwall's inequality to prove well-posedness in the L2 norm but there is a term "getting in the way" as it were. 3. The term is [sig^2/2 - a] X U^2(0,t)Now, the first term is negative in Feller case, so no BC needed. No surprises.But if the first term is positive, the inequality will contain a term like M * U^2(0,t). Now to get rid of it I need to say 1) U^2(0,t) = 0 (Dirichlet) or get an estimate for U^2(0,t) in some way,in which case no BC are needed for any values of [sig^2/2 - a]. We need something like value on the boundary is less than the L2 norm in the full region..Anyways, I have reconstructed Fichera conditions using energy estimates when the Feller condition holds. Now see if it works when 'Feller condition is not satisfied'. In my analysis it hinges on what U^2(0,t) is.So, can I get U^2(0,t) <= Integral(U^2) on interval (0,1) or even if it is bounded I am happy.
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November 22nd, 2009, 9:42 am

On a related topic, how to model the economy using CFD techniques and water integrator:Phillips Hydraulic Computer The Moniac QuoteThe MONIAC (Monetary National Income Analogue Computer) also known as the Phillips Hydraulic Computer and the Financephalograph, was created in 1949 by the New Zealand economist Bill Phillips to model the national economic processes of the United Kingdom, while Phillips was a student at the London School of Economics (LSE), The MONIAC was an analogue computer which used fluidic logic to model the workings of an economy. The MONIAC name may have been suggested by an association of money and ENIAC, an early electronic digital computer. QuoteThe actual flow of the water was automatically controlled through a series of floats, counterweights, electrodes and cords. When the level of water reached a certain level in a tank, pumps and drains would be activated. To their surprise, Phillips and his associate Walter Newlyn found that MONIAC could be calibrated to an accuracy of ±2 %.
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July 9th, 2010, 10:04 am

More CFD models. Probably no diffusion, just drift?
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July 24th, 2011, 3:48 pm

For those who split PDE, here is Marchuk's 1-2-2-1 scheme
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