<r>A theorem along these lines is Lemma A.1 in Kurtz's paper "The Yamada-Watanabe-Engelbert theorem for general stochastic equations and inequalities" downloadable for free at <URL url="http://ejp.ejpecp.org/article/view/431">http://ejp.ejpecp.org/article/view/431</URL>. It says that you can generat...

<t>The strict inequality in the original post would make sense if variance was really a conditional variance, i.e., the random variable Var[X|Y] := E[(X-E[X|Y])^2|Y] (this is a measurable function of the random variable Y), as long as Var[X|Y] was a non-constant random variable (Jensen's inequality ...

EBal got the answer, here's the reason: X has the same distribution as F^{-1}(U), where U is uniformly distributed over [0,1], so E[F(X)] = E[F(F^{-1}(U))] = E = 1/2, since F is continuous by assumption.

We might try using the reflection principle as Karatzas and Shreve do in their book for finding P(\tau^H \in dt) for a single BM.

<t>A libor-in-arrears swap means the floating payment made at time S is F(S,S,T) = F(S,T) (denoted L(S,T) in Lesniewski's notes). Any cashflow C made at time S has time 0 value of P(0,S)E^{Q_S}[C], which is P(0,S)E^{Q_S}[L(S,T)] in this case.The general formula for time t arbitrage-free price of cas...

<t>Denote:W = event of T winningA = event of A played on TB = event of B played on T~A = event of A did not play on T~B = event of B did not play on TThen the given info says:.4 = P(W|A).5 = P(W|~A).7 = P(W|B).45 = P(W|~B)The 4 "quadrants" correspond to:i) P(~A~B) = P(W~A~B) + P(~W~A~B)ii) P(A~B) = ...

<t>I'm not sure what you mean by using a dummy variable, but if it caused your sample matrix X (rows of X are sample vectors) to have more than 1 constant columns, then it would ruin the positive-definiteness of the scatter matrix X'X (the estimator of the covariance) because then the column rank wo...

<t>But x is not r, it is r - \alpha.Adding alpha(t) to both sides of (1.3), I get:x(t)+\alpha(t) = x(s)\exp(-a(t-s)) + \alpha(t) + \sigma\int_s^t \exp(-a(t-u))dW(u)The left side is now r(t) (i.e., the left side of (1.1)), and the right hand side is:(r(s)-\alpha(s))\exp(-a(t-s)) + \alpha(t) + \sigma\...

<t>Shifted lognormal and lognormal both have the problem of a lower limit. You might want to consider a Black-normal model; in this model, rates are normally distributed. The SABR model with beta=0 corresponds to this situation, and Hagan et. al. explicitly cover this example in their SABR papers. W...

<r>There are many ways to calculate CVAs on a portfolio of interest rate instruments; with varying degrees of sophistication. How accurate do you need it to be? Be aware that any method you chose will be very sensitive to the credit curve (containing the probabilities of default during each piece of...

<t>This is a standard problem in linear filtering theory. See, e.g., Jazwinski's book "Stochastic Processes and Filtering Theory", or Bar Shalom and Li's "Estimation with Applications to Tracking and Navigation", or Anderson and Moore's "Optimal Filtering" (esp. Ch 10).I can't remeber all the detail...

<t>All probability measures are finite (i.e., they have finite measure on the whole space \Omega since P(\Omega) = 1 < \infty), and are therefore sigma-finite; a sigma-finite measure can be decomposed as a countable (hence the "sigma") sum of finite measures. The Lebesgue-Radon-Nikodym decomposition...

<t>But is it guaranteed that you will not lose money using that trading strategy (assuming you have correctly deduced the "true" volatility surface)? Maybe I'm being too theoretical. It seems that such a trade would be likely to produce profit, but not guaranteed. It seems possible that the implied ...

After looking at it again, I see (5) follows from rearranging the equation in the line preceding (3) and then applying the same logic as that in deriving (3). I believe this paper now.

<t>As Frank mentioned, the third line from (11) follows from the second line in (11) directly from the definition of E^T_t: for any random variable X, E^T_t[X] := E_t[\exp(-\int_t^T r_C(u)du)X]/P_C(t,T). In the case of moving from the second to the third line of (11), X=\exp(-\int_t^T s_F(u)du)S(T)....