Lemma

Market practitioners armed with a term-structure model next need to determine how this relates to the fluctuation of security prices that are related to interest rates. Most commonly, this means determining how the price T of a zero-coupon bond moves as the short rate r varies over time. The formula used for this determination is known as Itos lemma. It transforms the equation describing the dynamics of the bond price P into the stochastic process (4.5). 

The subscripts rr in (4.5) indicate partial derivatives—the derivative with respect to one variable of a function involving several variables. The terms dr and drY are dependent on the stochastic process that is selected for the short rate r. If this is the Ornstein-Uhlenbeck process represented in (4.4), the dynamics of P can be expressed as (4.6), which gives these dynamics in terms of the drift and volatility of the short rate. 

Building a term-structure model involves these steps  

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The following lemma is needed for our result. Consider a matrix Hamiltonian h coupling two states, whose energy difference is 2ra... 

Thus, we still relate to the same sub-space but it is now defined for P-states that are weakly coupled to <2"States. We shall prove the following lemma. If the interaction between any P- and Q-state measures like 0(e), the resultant P-diabatic potentials, the P-adiabatic-to-diabatic bansfomiation maOix elements and the P-curl t equation are all fulfilled up to 0(s ). 

For more than two almost invariant sets one has to consider all eigenmea-sures corresponding to eigenmodes for eigenvalues close to one. In this case, the following lemma will be helpful. 

Lemma 3. Let p M be a probability measure and let X and Y be disjoint sets which are dx- resp. dy-almost invariant with respect to p. Moreover suppose that f (X) n Y = 0 and f Y) n A = 0. Then X UY is 6xuy-almost invariant with respect to p where... 

This approximation is a special case of the Baker-Campbell-Hausdorff lemma for additional discu.ssion and extensions to more general classes of methods. 

Lemma 1.1. For every convex and differentiable functional J the function... 

The right-hand side is nonnegative here in view of (1.61), therefore (A) > 0. Lemma 1.1 is proved. 

Taking into account Lemma 1.1, we conclude that the left-hand side of this inequality converges from above to J(,(uo — u)- Thus... 

By (1.97), the right-hand side is nonnegative, which proves the assertion. Lemma 1.4. The following estimate takes place ... 

The proof of this theorem is based on the following lemmas. 

By making use of the strong monotonicity of A, Holder s inequality, and Lemma 1.2, we obtain the estimate... 

Applying Holder s inequality and using the strong monotonicity of A, the monotonicity of / , Lemma 1.2 and the estimates for the norms, we obtain... 

This inequality reduces to the estimate (1.122) and completes the proof. From Lemma 1.9 the following assertion is immediately deduced. 

Take the norm of both parts of this equality and use the Lipschitz continuity of P (see Lemma 1.2). By the linearity of I, it provides... 

Lemma 1.12. If the boundary dflc belongs to the class and a function u G iF (f2c) is given, then the normal derivatives at the boundary dflc are uniquely defined. 

Proof. By utilizing the local coordinate systems (1.135), the assertion of Lemma 1.13 reduces to the case... 

By Lemmas 1.12, 1.13 and property (1.137), from Theorem 1.24 the next statement follows. 

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