Questions tagged [integral]
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16 questions
3
votes
2
answers
299
views
Show that $E [ \int_s^t W_u \, du \,|\, \mathcal{F}_s ] = (t - s) W_s$
Show that
$$E \left[ \int_s^t W_u \, du \,|\, \mathcal{F}_s \right] = (t - s) W_s$$
where $W_u$ is a standard Brownian motion and $\mathcal{F}_s$ is the filtration up to time $s $.
- 175
1
vote
1
answer
142
views
Step by step integration of the Hull-White SDE
I'm struggling to understand the integration process of the Hull-White equation:
\begin{equation}
dr(t)=[\nu(t)-ar(t)]dt+\sigma dW(t)
\end{equation}
In the majority of the references that I have ...
- 61
1
vote
0
answers
37
views
Inverse differencing in continuous time
I want to fit a continuous time ARMA (CARMA) model to traffic data $T_t$. After removing trend and seasonality I need first order differencing to obtain stationarity. Then I fit a CARMA model (yuima ...
- 135
0
votes
2
answers
97
views
Obtain B-S-M from a binomial tree as n goes to infinty using Lebesgue integral
My question is simple, consider a European call with payoff max(S_T-K, 0), Let's suppose that the underlying stock follows a binomial tree with up and down factors I know as we take n goes to infinity ...
- 1
2
votes
0
answers
104
views
Path integral approach to price call option on zero coupon bonds
I am given the following identities:
$$
Z[J,t_1,t_2]=\int D W e^{\int_{t_1}^{t_2}dtJ(t)W(t)}e^{S}=e^{\frac{1}{2}\int_{t_1}^{t_2}dtJ(t)^2}
$$
$$
\int_t^Tdx\alpha(t,x)=\frac{1}{2}\left[\int_t^Tdx\sigma(...
- 133
2
votes
2
answers
295
views
Calculate value of Integral of Wiener process $\int_{0}^t e^{\lambda u } dZ_u$
I am not quite sure how to solve this integral to be able to do numerical calculations with it. $\lambda$ is a constant, $u$ is time, and $Z_u$ is a wiener process. Can anyone provide some direction ...
- 315
1
vote
0
answers
102
views
Choice of grid for numerical integration
I have to compute an integral involving the characteristic function for pricing options in a model and it so happens that accurate approximation seems to be mostly about putting lots of points in ...
- 2,536
0
votes
1
answer
114
views
How to evaluate the following integral?
I stumbled across an expression and I wonder how to evaluate this: $-\int_ {0} ^ {+\infty} {v(x)} dw^{+} (1-p(x))$ where $v(x)$ is some utility function and $w(p(x)) $ is a decision weighting function,...
- 600
2
votes
0
answers
34
views
How to derive Parameter Derivative within an FFT integral
I have the following function (Carr-Madan) of which I am trying to take the derivative wrt $\theta$:
$c(k)=\int_0^\infty \frac{e^{-iuk}}{\alpha^2 + \alpha - u^2 + i(2\alpha+1)u} e^{\phi_T(u-(\alpha+1)...
- 21
6
votes
1
answer
701
views
Is there a closed-form solution for the following integral?
The integral under consideration is as follows:
$$
F=\int_{a}^{1} \exp\Big\{c\Phi^{-1}(x+b) + d\Big\}\; \mathrm dx,
$$
where $0<a, b<1$, and $c>0, d\in\mathbb{R}$ are constants, and the ...
- 419
5
votes
2
answers
505
views
Expectation of integral where one of limits of integration is a random variable
Is it correct to write
\begin{equation}
E_t \int_0^{X_T} f(z) dz = \int_0^\infty \left(\int_0^x f(z) dz \right) p(x)dx \,\,?
\end{equation}
Here $X_T$ is a positive random variable with density $p(x)...
user34971
1
vote
1
answer
121
views
Regression of stochastic integral on Wiener process
This question is a follow-up from the following: conditional expectation of stochastic integral
so I won't repeat myself regarding assumptions and notation.
Using Brownian bridge approach, we know ...
- 700
1
vote
2
answers
284
views
Show that $\int_0^T(T-t)dW_t=\int_0^TW_t \ dt$
I want to show that $$\int_0^T(T-t)dW_t=\int_0^TW_t \ dt \tag1$$
By Ito's lemma we can write $$d((T-t)W_t)=(T-t)dW_t-W_t \ dt.\tag{2}$$
Now If I integrate this expression and use that $W_0=0$ I should ...
- 221
6
votes
1
answer
118
views
Esscher Premium: Integral Transform Proof
I have some difficulty understanding the following proof and I hope someone can help me with that.
Claim: I want to show that
$E_\alpha(S)=\frac{d}{dr} \log M_S(r)|_{r=\alpha} $, where $M_S(r)=E(\exp(...
- 181
4
votes
1
answer
534
views
Ornstein–Uhlenbeck process – integration by parts
While deriving the solution for the stochastic differential equation that models the Ornstein–Uhlenbeck process, Paul Wilmott (Paul Wilmott on Quantitative Finance, chapter 4, page 87) performs the ...
10
votes
2
answers
1k
views
conditional expectation of stochastic integral
let $M_t$ be the following stochastic integral
$$
M_t = \int_0^t \sigma_s dW_s
$$
where $\sigma_t$ is a sufficiently regular deterministic function and $W_t$ is a standard Wiener process (that is $...
- 700