Questions tagged [itos-lemma]
The itos-lemma tag has no usage guidance.
2
votes
0answers
38 views
Ito Diffusion with Change of Measure
Let $(X_t)$ be an Ito diffusion with speed $(V_t)$, under a probability measure P. Could there exist a change of measure to a probability measure Q, with Q ~ P, under which $(X_t)$ is an Ito diffusion ...
3
votes
1answer
90 views
Expectation in a stochastic differential equation
I'm new to stochastic calculus, I want to find the mean of $X_2$ with $X_t = \exp(W_t)$, with $W_t$ a Wiener process.
I used Ito's Lemma is arrive at the SDE:
\begin{align}
d(X_t) = \frac{1}{2}X_t dt ...
2
votes
0answers
53 views
For an Ito Process, $d\ln{X} \neq \frac{dX}{X}$ and $(d\ln{X})^2 = (\frac{dX}{X})^2$, but $d\ln{X} \neq \pm \frac{dX}{X}$
In normal calculus we can write $d\ln{x} = \frac{dx}{x}$ since there is no quadratic variation to deal with. This isn't true for stochastic processes, and Ito's Lemma is used to calculate $d\ln{X}$. ...
4
votes
1answer
134 views
Ito's Lemma for this problem
I'm attempting to prove a lemma from a paper, in the context of optimal contracts.
$r,\rho,\gamma,\alpha,\sigma$ are all known constants.
$dR_t = (\alpha + r)dt + \sigma dZ_t$ where $Z_t$ is a ...
3
votes
1answer
98 views
HJM model Baxter Rennie: differentiating the discounted asset price using Ito
From Baxter and Rennie Page 145:
$Z(t,T) = exp(\int_{0}^{t}\Sigma(s,T)dW_s - \int_{0}^{T}f(o,u)du - \int_{0}^{t}\int_{s}^{T}\alpha(s,u)duds)$
where $\Sigma(t,T) = \int_{t}^{T}\sigma(t,u)du$
How ...
6
votes
0answers
92 views
Random variable minus Integral of Ito Generator is a Martingale under what conditions?
I am reading about american option pricing and the variational inequality, and the book I am reading states, in the derivation of the variational inequality, the following is a martingale: $$M_s = U(s,...
4
votes
1answer
111 views
Expected payoff at future time
Let $a$, $b$, $c$, and $e$ be constants, $W_1$ and $W_2$ be Brownian motions with correlation $\rho$, and $f(t)$ and $g(t)$ be deterministic functions of time. Let $X$ satisfy $$d(X(t))=(aX(t)+ef(t)g(...
3
votes
2answers
208 views
Application of Ito's lemma
Let $X_t$ be some stochastic process driven by wiener process ($W_t)$ so it can be expressed as:
$$dX_t=(...)dt+(...)dW_t$$
Let $f(t,x)$ be some $C^2$ function. Define the process $Z_s=f(t-s,X_s)$ ...
4
votes
1answer
83 views
How to express a process using Itos formula
Let $F(t,x)$ be the solution to the PDE
$$
F_t(t,x)=aF_x(t,x)+\frac{1}{2}F_{xx}(t,x),t>0
$$ $$F(0,x)=g(x)$$ for some function $g$.
Let $X_t$ be a process defined by
$$dx_t=aX(t)dt+dW(t)$$
Now ...
5
votes
1answer
127 views
Distribution of time integral of Brownian motion squared (where the Brownian motion occurs in square root time)?
Let $I_t = \int_0^t W_{\sqrt{u}}^2du$. What is the distribution of $I$?
If I recall correctly, if the Brownian motion were instead $W_u$, then it would be $I_t \sim N\left(\frac{t^2}{2},\frac{t^4}{3}\...
3
votes
2answers
96 views
Random Walk with normal increments and n time periods why is the increment $\sqrt{(t/n)}$?
Question is basically in the title. I have found several sources stating that $R_i = \sqrt{\frac{t}{n}}$, but I couldn't find the intuition behind taking the square root. And it seems to be crucial ...
-1
votes
1answer
66 views
Differential product Correlated processes
I am trying to derive the differential of the product of two processes, but I got stuck. This is what I have until now:
We have the following two stochastic processes:
$dX_t= \mu_t dt +\sigma_t dW_t$...
2
votes
1answer
117 views
Differential of integrating factor $d(e^{at}r_t)$ in Vasicek model
I am attempting to solve the Vasicek model SDE (using Wikipedia parametrisation):
$$
dr_t = a(b-r_t)dt + \sigma dW_t
$$
Every solution is proceeding to multiply both sides of the equation by the ...
3
votes
1answer
160 views
Why is Ito applied this way?
Given the price of a call option :
$$C = \mathbb{E}\left[ D_{0,T} (s-K)1_{s>K} |\mathcal{F_0}\right] $$
with $D_{0,T}=e^{-\int_0^Tr(u)du}$
I read somewhere that applying Itô gives :
$$dC = \...
-2
votes
2answers
65 views
2
votes
0answers
88 views
Pricing caplet with Bachelier (normal dynamic) using forward measure
I'm trying to price caplet with Bachelier under forward measure, but I can't find any solution.
Remind that Bachelier assumed rates follow a normal dynamic. So here what I was doing :
$C_t(T,T+d)$ ...
0
votes
0answers
84 views
Dynamics of an option on a future
I have trouble understanding why $$V_s=exp(\int_s^t r_u du) \int_s^t exp(−\int_t^v r_u du)\theta_v dW_v$$ is the solution to the following SDE
$dV_s=\theta_s dW_s+r_s V_s ds$.
I tried of course with ...
5
votes
1answer
133 views
Basic question on Ito integrals
$Let \space X(t) =\begin{cases}
2, \qquad\text{if} \space 0\le t \le 1 \\
3, \qquad\text{if} \space 1 < t \le 3 \\
-5, \qquad\text{if}\space 3 < t \le 4
\end{cases}
$
or in one forumala $...
2
votes
0answers
210 views
Applying Ito's formula to complex functions
Within my lecture notes, the following definition is given:
We say that the stochastic process $X_t$ has stochastic differential
$$
dX_t = b_t dt + \sigma_t dW_t
$$
if and only if
$$
X_t = ...
2
votes
1answer
168 views
Black Scholes in the case of dividends
Let's take the case where the underlying stock has the continuous dividend yield $\delta$. Then, in the risk-neutral world, $\frac{dS}{S}=(r-\delta)dt+\sigma dW^Q$. Suppose we want to price a ...
2
votes
0answers
64 views
$\int_{0}^1W_x(t)dW_y(t)/(\int_{0}^1W_x^2(t)dt)^{1/2}$ normally-distributed?
I have came across the following stochastic integrals:
$$\frac{\int_{0}^1W_x(t)dW_y(t)}{(\int_{0}^1W_x^2(t)dt)^{1/2}}$$
which was claimed to be standard normally distributed ($W_x$ and $W_y$ are ...
2
votes
1answer
84 views
How to derive the dynamic of the log forward price?
I have the following Schwartz model:
$$dS_t=a(\mu-\ln S_t)S_tdt+\sigma S_tdW_t$$
$$X_t=\ln S_t$$
$$dX_t=a(\hat{\mu}-X_t)dt+\sigma dW_t$$
with $\hat{\mu}=\mu-\frac{\sigma^2}{2a}\sigma$
$$F_t(T)= \exp\...
0
votes
1answer
116 views
Self finance conditions - proof check
Find expressions for the process $\psi=(\psi(t),\ 0\leq t\leq T)$ , so the portfolio $(\phi,\ \psi)$ is self-financing when:
(1) $\phi(t)= \int_{0}^{t}S_{s}ds $
(2) $\phi(t)=S_{t}$
where $\phi(t)$ ...
3
votes
2answers
132 views
Show that the two solutions of the SDE are equivalent
I have a process:
$$dr_t = (W_t^1 - ar_t)dt +\sigma dW_t^2$$
where $W_t^1$ and $W_t^2$ are brownian motions with instantaneous correlation coefficient $\rho$.
I want to show that the solution of this ...
0
votes
1answer
260 views
exercise on multivariate Ito's lemma + jumps (Poisson)
Given the two jump-diffusions:
\begin{equation}
\begin{aligned}
dX_{1,t} &= a_1 dt + b_1 dW_t + c_1 dN_t(\lambda) \\
dX_{2,t} &= a_2 dt + b_2 dW'_t + c_2 dN_t(\lambda) \\
corr(dW,dW') &= \...
1
vote
1answer
122 views
Mark Joshi, Chapter 5 Problem 2 of The concepts and practice of mathematical finance
If $$dX_t = \mu(t,X_t)dt + \sigma(X_t)dW_t$$
with $\sigma$ positive, show there exists a function $f$ such that $$d\left(f(X_t)\right) = v(t,X_t)dt + V dW_t$$ where $V$ is constant. How unique is $f$...
2
votes
0answers
64 views
How to calculate the product of forward rates with different reset times using Ito's lemma?
I am curious about a calculation I saw in this question.
Specifically in this equation:
\begin{align*}
&\ L(T_s, T_p, T_e) L(T_s, T_s, T_e) \\
=&\ L(t_0, T_p, T_e) L(t_0, T_s, T_e) e^{-\...
4
votes
1answer
537 views
Ito's Lemma: Multiplication Rule
I have a conceptual question about Ito's lemma, in particular, the multiplication.
Ito's multiplication rule states, that multiplying dt by itself or by dx (the stochastic differential) equals zero. ...
5
votes
1answer
161 views
Ito representation unique up to indistinguishability? Proof?
Given an Ito-process $X(t)$, $t\in[0,T]$
$$X(t)=X_{0}+\int_{0}^{t}F(s)ds + \int_{0}^{t}G(s)dW(s)$$
with $F\in \mathbb{L}^{1}(0,T)$ and $G\in\mathbb{L}^{2}(0,T)$.
It is now often claimed that this ...
1
vote
1answer
275 views
Intuition behind Ln transformation of stock price when applying Ito lemma [closed]
I am able to replicate steps and arrive to the option price using Black Scholes framework. Here however I am more interested to understand, at least intuitively, why the ln transformation of price ...
2
votes
1answer
248 views
Quantile normal and lognormal
Let's assume we have a normal distribution $X\sim \mathcal{N}(\mu,\sigma^2)$. In a normal distribution the quantile can be calculated as follows:
\begin{equation}
\Phi_X ^{-1}(p)=\mu +\sigma {\sqrt {...
-3
votes
1answer
626 views
On the application of Itos lemma to Geometric Brownian motion [closed]
I recently read this from a book:
The canonical SDE in financial math, the geometric Brownian motion,
${{d{S_t}} \over {{S_t}}} = \mu dt + \sigma d{W_t}$ has solution
$${S_t} = {S_0}{e^{(\mu -...
0
votes
1answer
194 views
Self-Financing Portfolio
Why when we are using self-financing portfolios to replicate some external payoff we do not consider the quadratic variation of the portfolio weights? Say, in Black-Scholes world, when we are using $...
1
vote
1answer
75 views
Is it possible to approach finding the risk premium of this derivative using Ito's Lemma?
I understand the author's intended solution to the below problem, but I thought I would see if I could solve this using first principles and Ito's Lemma instead for practice.
Let $V(S(t), t) = e^{rt}\...
-1
votes
1answer
2k views
Given $S$ is a Geometric Brownian Motion, how to show that $S^n$ is also a Geometric Brownian Motion?
Suppose that a stock price $S$ follows Geometric Brownian Motion with expected return $\mu$ and volatility $\sigma:$
$$dS = \mu S dt +\sigma S dz$$
How to find out the process followed by variable $...
2
votes
0answers
76 views
Transformation of coupled forward-backward stochastic differential equations in 3 dimensions with Ito formula
Maybe this is the right place for my question:
I have a system of coupled FBSDEs in 3 dimensions as follows (in cartesian coordinates):
$$ \mathrm{d}\vec{r}(t) = \vec{u}(\vec{r}(t))\mathrm{d}t + \...
2
votes
1answer
237 views
How to show that $E\left[ \int_0^t \sigma(s) e^{iuX(s)} dW(s)\right] = 0$?
Let $\sigma(t)$ be a given deterministic function of time and define the
process $X_t$ by
$$X(t) = \int_0^t \sigma(s)dW(s)$$
I want to show $$E\left[ \int_0^t \sigma(s) e^{iuX(s)} dW(s)\right] = 0$$...
-1
votes
1answer
466 views
Bond price and its process
Suppose that x is the yield to maturity with continuous compounding on a discount bond that pays off $1 at time T. Assume that the x follows the process
$dx=a(x_0-x)dt + sxdz$
where $a, x_0$ and $s$ ...
4
votes
2answers
162 views
Moment Ito's Process Proof
I have a following stochastic integral - related problem that I have difficulty to solve:
Given
\begin{equation}
dX_t = -\alpha X_tdt+\sigma\sqrt{X_t}dW_t
\end{equation}
and the second moment of $...
-1
votes
1answer
179 views
Why does the partial derivative, $X_t$, of an ABM $X(t)$ not involve standard Brownian motion $Z(t)$, even though $Z(t)$ varies with $t$?
Consider the arithmetic Brownian motion $X(t) = \alpha t + \sigma Z(t)$ and evaluating $dX(t)$ using Ito's lemma.
We have $\frac{\partial X}{\partial t} = \alpha$, which does not involve $Z(t)$, even ...
0
votes
1answer
231 views
How to define the $f$ function to apply Ito's lemma?
\begin{equation}
Z(t) = \exp (a W(t))
\end{equation}
I am asked to find $dZ$. I am pretty sure it can be done using Ito's lemma. But in all my textbook (Bjork) examples Ito's lemma is giving from a $...
1
vote
1answer
281 views
Stochastic differential equation of a Brownian Motion
I have two questions about Ito's Lemma with respect to calculating SDEs. The examples are simple enough, but I haven't found an answer yet.
Take $W_t$ as a standard Brownian motion and $g(s)$ as some ...
-2
votes
1answer
756 views
How is the Wiener integral $\int{WdW}$ calculated?
I want to calculate $\int ^t _0 W_tdW_t$
I know that the reasoning is the following:
Let $x(t)=W(t)$ with $a=0$ and $b=1$ in the definition of an Ito Process, and $f(t,x)=x^2$.
Then, applying Ito'...
5
votes
1answer
479 views
Application of Ito's Lemma, finding the condition for the martingale
The Vasicek short rate model is
$$dr_t=\kappa(\theta-r_t)dt+\sigma dW_t$$
Define the processes $x_t$ and $f(x,t)$
$$x_t=\frac{r_t}{\kappa}(1-e^{-\kappa(T-t)})+\int_0^tr_sds$$
$$f(x,t)=e^{a(T-t)-x_t}$$
...
4
votes
1answer
371 views
Chain rule for Ito's Lemma
The CIR short rate model is
$$dr_t=k(\theta-r_t)dt+\sigma\sqrt{r_t}dW_t$$
under the risk-neutral measure.
The bond price is of the form
$$P(t,T)=A(t,T)e^{-B(t,T)r_t}$$
where the continuously ...
1
vote
0answers
319 views
Stochastic Leibniz rule
We have the following single-factor HJM model
$$d_tf(t,T)=\sigma(t,T)dW_t+\alpha(t,T)dt$$
$$f(t,T)=f(0,T)+\int_0^t\sigma(s,T)dW_s+\int_0^t\alpha(s,T)ds$$
The discounted T bond is then
\begin{align}
Z(...
1
vote
1answer
119 views
Simple HJM model, differentiating the bond price
We have the following simple HJM model
$$f(t,T)=f(0,T)+\int_0^t\alpha(s,T)ds+\sigma W_t$$
$$r_t=f(0,t)+\int_0^t\alpha(s,t)ds+\sigma W_t$$
$$P(t,T)=\exp-\bigg(\int_t^Tf(0,u)du+\int_0^t\int_t^T\alpha(s,...
1
vote
1answer
856 views
Partial derivative of an integral
Suppose I have a model for the short rate $r$ as ($W(t)$ is standard Brownian motion)
$r(t) = c+ \int_0^t \sigma (s) ^2 (t-s) ds+ \int_0^t \sigma (s) dW(s)$
I then want to find the dynamics of $r$, ...
8
votes
1answer
437 views
Baxter & Rennie HJM: differentiating Ito integral
From Baxter and Rennie, page 138:
$$f(t,T)=\sigma W_t+f(0,T)+\int_0^t\alpha(s,T)ds$$
$$Z_t=\exp-\bigg(\sigma(T-t)W_t+\sigma\int_0^tW_sds+\int_0^Tf(0,u)du+\int_0^t\int_s^T\alpha(s,u)ds\bigg)$$
$$dZ_t=...
2
votes
1answer
550 views
Ito's Lemma, differentiating an integral with Brownian motion
In How were these SDE derived? I don't understand one part of Gordon's answer, specifically:
$$\ln S_t=\ln F_{0,t}-\frac{\sigma^2}{4\lambda}(1-e^{-2\lambda t})+\sigma e^{-\lambda t}\int_0^t e^{\...
