伊藤積分と利子の計算式

最近伊藤の定理と確率的の微分積分に関して悩んでいます。The solution to the stochastic differential equation for the Vasicek interest rate process, although it apparently conforms to Ito's Lemma, seems arbitrary.
Basically, given dr=a(b-r)dt + sigma*dz
the interest rate is r=exp(-at)*(x0+integral(0 to t)(ab*exp(as)ds)+integral(0 to t)(sigma*exp(as)dz)), where z is a Brownian motion that is a function of s.
この由来は線形微分方程式の答えです。The general form is, given dx = (ax+u)dt+sigma*dz,
X= phi*(x0+integral(0 to t)(phi^(-1)*u*ds)+integral(0 to t)(phi^(-1)sigma*dz))
This interestingly can be derived by solving for d(exp(ct)*x) using Ito's Lemma. Here, the phi is exp(at) and phi^(-1) is exp(-at) and function g=(phi^(-1))*x with dg =(-a*exp(-at)x+(ax+u)exp(-at))dt +sigma *exp(-at)dz = u*exp(-at)dt +sigma*exp(-at)dz. Note that the drift term is (ax+u). Integrating both sides results in the desired results. Slides from Prof. Kohn of NYU explain this derivation well for the case of the Ornstein-Uhlenbeck Process.
次は債券の値の偏微分方程式を証明します。
The PDE, which resembles the Black-Scholes PDE for equity options, is derived from the formula for the market price of risk for the bond. The equation is:
(mu-lambda*sigma)dF/dr + dF/dt + (1/2)(sigma^2)*(d^2F/(dr^2) = 0
In the case of the Vasicek model this translates to:
(a(b-r)-lambda*sigma)dF/dr + dF/dt + (1/2)(sigma^2)*(d^2F/(dr^2) = 0
or, letting gamma=b-lambda*sigma,
(a(gamma-r))dF/dr + dF/dt + (1/2)(sigma^2)*(d^2F/(dr^2) = 0
Here, gamma is the risk-neutral equivalent of b, and alpha is the equivalent of a. We rewrite the Q-measure interest rate process as
dr=alpha*(gamma-r)dt+sigma*dz
Deriving the closed-form expression for F (the bond price) from the PDE is the next problem. Interestingly, this can be done by guessing the solution and then verifying that the PDE is satisfied based on this guess. The guess is: p(t,T,r) = F = exp(a(t,T)-b(t,T)*r). If the functions for a(t,T) and b(t,T) can be found, then the expression is valid.
The differential equations resulting from the assumption are: da/dt = alpha*gamma*b-(1/2)*(sigma^2)*b^2
a(T,T)=0
db/dt = alpha*b-1
b(T,T)=0
The b expression, interestingly, can be solved by first transforming the equation to db/(alpha*b-1)=dt
Integrating both sides indefinitely and substituting for the terminal case to find the constant yields:
b(t,T)=(1-exp(-alpha*(T-t)))/alpha
The a expression involves the Fundamental Theorem of the Calculus.
a(T,T)-a(t,T) = (gamma*alpha)integral(t to T)(b(x,T)dx) -(1/2)(sigma^2)integral(t to T)(b(x,t)^2dx)
Since a(T,T)=0,
a(t,T) = -(gamma*alpha)integral(t to T)(b(x,T)dx) +(1/2)(sigma^2)integral(t to T)(b(x,t)^2dx)
These two integral a can be solved in terms of b and b^2 then, substituting for the integral values, we obtain:
a(t,T) = -(gamma-sigma^2/(2*alpha^2))(T-t-b)-sigma^2*b(t,T)^2/(4*alpha)
Having obtained expressions for b and a (after substituting for b) in terms only of the know. Parameters of the interest rate diffusion process, we can assert that the initial guess of a solution to the bond price pde is valid.