Hi,I am trying to price a bermuda swaption (fixing dates are same as exercise dates) by PDE (Hull & White Model)And I am facing a problem with the fixing times :T_ j , j = 0,...,N-1 is the set of the fixing dates. They are separated by delta.and r : is the value of the interest rate (discretized : r_min, r_min+dr,....., r_max)I know the value of the Swaption at time T _ (N-1) which is given by v ( (T_ (N-1))- , r) = max [ delta*(K-r) ; 0]And theoretically, the value of the Swaption in T_ j (a given fixing date) is given by the maximum between the value just after T _ j (calculated by solving the PDE between T _ (j+1) and T _ j ) and the the sum of the expected value of the discounted value of the swaption : ==> v ( (T_ j)- , r) = max [ v ( (T_ j)+ , r) ; SUM(for j < i < N) E ( B(T_ j, T_ i, ) * v ( T_ i , r (T_i) ) ) | v(T _ j) = r ]the problem is : How to compute E ( B(T_ j, T_i ) * v ((T_ i), r (T_i)) ) | v(T _ j) = r ] in this case ?? (without Monte Carlo)Thank you very much for helping me !

Use a trinomial tree a full implementation of it is described in "Interest Rates Models Theory and Practice" by Brigo and Mercurio