Let [$]R_{t}[$] be the CIR process at time [$]t[$]. I want to calculate the expectation $$E(R_{t}|R_{T})$$ conditional on [$]R_{T}[$] and [$]R_{0}[$] , where [$]t<T[$]. Then, is there a good way to calculate it?

generally speaking Brownian bridge it seems is defined as w ( s ) - w ( t ) - ( s - t ) ( T - t )^ -1 [ w ( T ) - w ( t ) ] where T and t are fixed and s is a variable on [ t , T ]. As far as I remember R t Is F t measurable random process for each t > 0. Here F t is the filtration generated by Wiener process increments on [ 0 , t ]. Then F t belongs to F T. Hence it looks like that E ( Rt | RT ) = Rt. If Ro = x is nonrandom then E ( Rt | Ro ) = E R. If Ro is random then E ( Rt | Ro ) = [ E R ] | x= Ro. Though I have not dealt with such math for along time and can be wrong

Well, it is a bridge, but not Brownian. Maybe a Feller bridge would do? The question is well specified, but I suspect not all that easy to solve analytically. And to list -- it is not the expectation conditional on [$]F_T[$] but rather conditional on the sigma-field generated by [$]R_T[$], which does not know about [$]R_t[$].

If diffusion coefficient is non degenerated then filtration generated by Wiener process and filtration generated by the solution of the corresponding SDE coincide. I suspect that as far as cir process does not reach 0 over a finite period therefor diffusion of the cir process can be considered as non degenerative. Though I could missing here something.

You are mixing up the filtration generated by observing [$] R_t \forall t \in [0,T] [$] with the sigma-field generated by observing [$]R_T[$] only.

If we write R T as a solution of stochastic integral equation we can see that R T for any T > 0 is defined by wiener process increments. ie R T is F T measurable random variable

QuoteOriginally posted by: fmfreshmanLet [$]R_{t}[$] be the CIR process at time [$]t[$]. I want to calculate the expectation $$E(R_{t}|R_{T})$$ conditional on [$]R_{T}[$] and [$]R_{0}[$] , where [$]t<T[$]. Then, is there a good way to calculate it?As pointed out, your [$]R_t[$] is a diffusion bridge process. It has a known time-inhomogeneous SDE, say eqn (68) here.The bridge SDE drift is known in terms of the standard (unbridged) SDE coefs and the standard (CIR) transition density, [$]p(t, R_t|R_0)[$], which is known. So,(*) [$]dR_t = \gamma(t,R_t; T, R_T) dt + \sigma \sqrt{R_t} dW_t [$],where [$]\gamma(\cdot)[$] is the coefficient of dt at (68).Integrating,(**) [$]E[R_t] - R_0 = E[\int_0^t \gamma(s,R_s;T, R_T) \, ds | R_0] = \int_0^{\infty} \int_0^t \gamma(s,y;T, R_T) \, p(s,y|R_0) \, ds \, dy[$].Note the last term of the integrand is, again, the known CIR transition density.So, unless I have made a mistake, that's it. [The expectation in the l.h.s. of (**) is what you seek].In Mathematica, for example, just give that double integral to NIntegrate and you are done.You could check my formula by doing a Monte Carlo with the bridged SDE process (*).=================================================================================Correction.Thinking about it some more, the first equality in (**) seems correct, but not the second.In the second, the last term of the integral should really be the transition density for the bridged process.Since we don't know that one, that seems to leave either a Monte Carlo or numerical PDE solution for [$]E[R_t][$].Alternatively, what I posted should be approximately correct if [$]t \ll T[$], since in that case the motion shouldbe practically unrestricted. Finally, googling "simulating diffusion bridges" will turn up some things.

QuoteOriginally posted by: list1If we write R T as a solution of stochastic integral equation we can see that R T for any T > 0 is defined by wiener process increments. ie R T is F T measurable random variableAlan is much better than I am at this stuff, so I will just try to tidy up in the corners here. Yes, [$]R_T[$] is [$]F_T[$] measurable, but that does not mean that [$]F_T[$] is equal to the sigma-field generated by [$]R_T[$] (it is, in fact, far bigger). Why don't you go off and look up the distribution of a Brownian bridge? It is a simpler version of the problem at hand, and contains all the potential for filtration related confusion.

QuoteOriginally posted by: AlanQuoteOriginally posted by: fmfreshmanLet [$]R_{t}[$] be the CIR process at time [$]t[$]. I want to calculate the expectation $$E(R_{t}|R_{T})$$ conditional on [$]R_{T}[$] and [$]R_{0}[$] , where [$]t<T[$]. Then, is there a good way to calculate it?As pointed out, your [$]R_t[$] is a diffusion bridge process. It has a known time-inhomogeneous SDE, say eqn (68) here,in terms of the standard CIR transition density, [$]p(R_t|R_0)[$], which is known. So,(*) [$]dR_t = b(t,R_t; T, R_T) dt + \sigma \sqrt{R_t} dW_t [$],where [$]b(\cdot)[$] is given at (68).Integrating,(**) [$]E[R_t] - R_0 = E[\int_0^t b(s,R_s;T, R_T) \, ds | R_0] = \int_0^t b(s,R_s;T, R_T) \, p(R_s|R_0) \, ds [$].So, unless I have made a mistake, that's it. [The expectation in the l.h.s. of (**) is what you seek].You could check my formula by doing a Monte Carlo with the SDE process (*).I thought that cir process is defined ordinary SDE. If R t is defined by equation (*) the stochastic integral on the right hand side (*) does not Ito integral and it should be defined separately.

list, read the link I posted. Also you copied in my post before it was done, so see my edits.

QuoteOriginally posted by: bearishWell, it is a bridge, but not Brownian. Maybe a Feller bridge would do?Almost good CIR is a Bessel process, so is called the bridge - Bessel bridge. There're quite many papers and the formulas are ugly.

Alan, eq (68) and eq (8) are of the different types. Function p ( Y , T | Xt , t ) is Xt measurable while R T is a rv that depends on future moment T. It is known fact that conditional density is a solution of the SPDE but all stochastic imbedded in Ito scheme. With the help of orthogonal series one can define expanded stochastic integrals but it is a different topic. About conditional density SPDE a number of papers were published by Liptser&Shiryaev and Rozovsky in 70-ies and more details can be found in their books translated in English. Another people derived SPDE for conditional density before them.

Eqn (68) is just an ordinary SDE in which (y,T) are simply fixed parameters for the drift function at [$](t,X_t)[$].For example, for the Brownian bridge, it reduces to[$]dX_t = \frac{y-X_t}{T-t} \, dt + dW_t[$], which is quite an ordinary SDE as long as [$]t \in [0,T)[$]. So, I don't see the need to introduce anything more exotic.

QuoteOriginally posted by: AlanEqn (68) is just an ordinary SDE in which (y,T) are simply fixed parameters for the drift function at (t,X_t).For example, for the Brownian bridge, it reduces to[$]dX_t = \frac{y-X_t}{T-t} \, dt + dW_t[$], which is quite an ordinary SDE as long as [$]t \in [0,T)[$].Yes I agree. In eq (*) R T can not be in the drift b ( )