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You might want to set $a= \epsilon - d$ and write $\epsilon>0$ as a constraint. I guess $\textbf{lsqnonlin}$ is the suitable fonction for what you intend to do. I personnally like to use and play around with $\textbf{fmincon}$, which allows more flexibility and performs well, if you are willing to provide Jacobian and/or Hessian in algorithms options


If I have read the question correctly then I will assume that $a$, $b$, $c$, $d$, $T_i$, and $k_i$ are constants. If this is the case then the only term which we need to show is bounded is $$ \big(a + b(T_i - t)\big)\exp\big(-c(T_i-t)\big). $$ If we assume that we are only considering the temporal domain $0 \leq t \leq T_i$ such that $T_i - t \geq 0 $ then ...


The forward Libor rate at time $t$ is the forward rate over a certain accrual period $[T, T+\Delta]$, where $\Delta$, in years, can be 3 months or 6 months, and is defined by \begin{align*} L(t, T, T+\Delta) = \frac{1}{\Delta}\left(\frac{P(t, T)}{P(t, T+\Delta)}-1 \right), \end{align*} where $P(t, u)$ is the price at time $t$ of a zero coupon bond with unit ...

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