Questions tagged [itos-lemma]
The itos-lemma tag has no usage guidance.
118
questions
0
votes
0answers
11 views
Independence of increments of the stochastic process $\frac{1}{t}\int_0^t u dW_u $
I am wondering about the stochastic process
$$X_{t} =\frac{1}{t}\int_0^t u dW_u $$
I know that for
$$Y_{t} =\int_0^t u dW_u$$
Y_t-Y_s is independent of Y_s where t>s
But is this also true for $X_t$...
1
vote
0answers
10 views
Probability distribution of the stochastic process $\int_{0} ^{t}\frac{u}{t}dW_{u}$
I am wondering about the probability distribution of the stochastic process
$$X_t=\int_0^t \frac{u} {t} dW_{u}$$
I thought of using the Kolmogorov equation but after converting this into An SDE
$$...
0
votes
0answers
45 views
How does this follow from Ito's formula?
Let $S_t$ be a geometric brownian motion and let $M_t = \sup_{0 \le u \le t} S_t$ be the maximum process.
Define $X_t = \frac{M_t}{S_t}$.
A book I am reading states the following:
How does this ...
5
votes
2answers
202 views
More questions about integral of Brownian Motion w.r.t time
A similar question have been posted earlier but one part has remained unanswered. Let us define:
$$X_t = \int_0^t W_s ds,$$
where $W_t$ is a standard Brownian Motion. Is $X_t$ an Itô process or a ...
1
vote
1answer
77 views
Are the Ito's Lemma given in Mark Joshi's Concept and Practice in Mathematical Finance same as what I learn?
In Joshi's Concepts and Practice in Mathematical Finance, page $110,$ he stated the Ito's Lemma:
Theorem $5.1$ (Ito's Lemma) Let $X_t$ be an Ito process satisfying
$$dX_t = \mu(X_t,t)dt + \sigma(...
2
votes
1answer
76 views
Stochastic Processes (Applying Ito's Lemma on Ho-Lee Model )
I seek a basic form (SDE) to understand the Ho-Lee model.
I already understand the models from Vasicek, Merton and Cox-Ingereoll-Ross, etc.. For example,
\begin{align*}
dX_t &= -1/2 \alpha X_t ...
1
vote
1answer
59 views
How to derive the expression for the forward rate?
The following RN dynamics of a ZCB maturing at time is given:
$$\frac{dZ(t,T)}{Z(t,T)} = r_tdt + \sigma_Z(t,T)dX_t$$
and the forward rate is given:
$$f(t,T,T+\delta) = \frac{ln(Z(t,T)) - ln(Z(t,T,...
1
vote
1answer
88 views
Integration and expectation of geometric Brownian motion
Let the stock price S follows the geometric brownian motion:
$$dS=\mu Sdt+\sigma Sdz$$
$$\frac{dS}S=\mu dt+\sigma dz$$
where $dz$ is a wiener process.
Naively integrating the second equation above ...
2
votes
1answer
77 views
Zero-coupon bond pricing equation derivation
I'm trying to understand how in Chawla's paper that I've linked below, how he obtains equation (2.5) for the zero coupon bond pricing equation?
The equation is:
$\frac{\partial B}{\partial t} + \...
3
votes
1answer
128 views
List: Behavioural characteristics of key Ito processes used in finance
My hope from this question is to become a repository of the behavioural characteristics, use-cases and interesting features of the key Ito processes used in quantitative finance - examples being GBM, ...
2
votes
0answers
89 views
Ito's lemma for special case
Assume a HJM framework with the same Brownian motion driving the dynamics for every tenor.
$$
df(t,T) = \alpha(t, T)dt + \sigma(t,T) dw_t \,,
$$
with $\alpha(t, T) = \sigma(t,T)\int_t^T \sigma(t,s)ds$....
3
votes
1answer
130 views
What is the easiest way to learn Option pricing with PDE?
I was reading about Ito's formula and Girsanov theorem, but I am still struggling to grasp how in reality these are combined to compute the price of an option. What are the main source to understand ...
1
vote
1answer
110 views
Brownian Motions theorems
I know that if $W$ and $W′$ are two independent brownian motions, then $dWt \ dWt′$ = 0.
How can I prove/demonstrate this theorem?
Additionaly, how can we prove that if $W$ and $W′$ are dependent, ...
2
votes
1answer
181 views
Ito formula (lemma) problem
I am trying to solve this problem
Consider the following one-dim. stochastic process
$$dX_t = b_t dt + \sigma_t dW_t$$
where $W$ is a one-dim. Brownian motion. The above SDE is well-defined.
...
1
vote
1answer
80 views
Stochastic solution (mean, variance) to lognormal drift and normal volatility
I have trouble deriving the state equations for a mixture of normal/lognormal stochastic differential, namely for its
a) expected mean, (b) variance, and (c) drift adjustment for LMM - libor model
I ...
-1
votes
1answer
71 views
Different Forms of Geometric Brownian Motion [closed]
If the stock price S follows the geometric brownian motion:
$$dS=\mu Sdt+\sigma Sdz$$
$$\frac{dS}S=\mu dt+\sigma dz$$
Where $dz=\epsilon\sqrt{dt}$ is a wiener process.
Integrating this to get $S_T$ ...
4
votes
1answer
460 views
Ito`s Lemma problem
Can someone help me with calculus for this problem.
I have these 3 equations and with Ito`s Lemma I have to find $dXt$.
\begin{cases} dY= μYdt+σYdB
\\ X=\frac{1}{2}cY\\ dc =-aαcdt\end{cases}
3
votes
4answers
688 views
Log of square of Geometric Brownian Motion
Which of the two calculations below, is wrong?
Why?
$dF = \sigma F dW$
First:
$dF^2 = (F^2)' dF + \frac{1}{2}(F^2)''dF.dF$
$dF^2 = 2F dF + dF.dF$
$dF^2 = 2 \sigma F^2 dW + \sigma^2 F^2 dt$
$\...
3
votes
0answers
54 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
130 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
104 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
182 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
115 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 ...
7
votes
0answers
117 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
116 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 $$d(X(t))=(aX(t)+ef(t)g(...
4
votes
2answers
331 views
Application of Ito's lemma
Let $X_t$ be some stochastic process driven by wiener process ($W_t)$ so it can be expressed as:
$$dX_t=(...)dt+(...)dW_t$$
Let $f(t,x)$ be some $C^2$ function. Define the process $Z_s=f(t-s,X_s)$ ...
5
votes
1answer
90 views
How to express a process using Itos formula
Let $F(t,x)$ be the solution to the PDE
$$
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)$$
Now ...
5
votes
1answer
304 views
Distribution of time integral of Brownian motion squared (where the Brownian motion occurs in square root time)?
Let $I_t = \int_0^t W_{\sqrt{u}}^2du$. What is the distribution of $I$?
If I recall correctly, if the Brownian motion were instead $W_u$, then it would be $I_t \sim N\left(\frac{t^2}{2},\frac{t^4}{3}\...
3
votes
2answers
109 views
Random Walk with normal increments and n time periods why is the increment $\sqrt{(t/n)}$?
Question is basically in the title. I have found several sources stating that $R_i = \sqrt{\frac{t}{n}}$, but I couldn't find the intuition behind taking the square root. And it seems to be crucial ...
-1
votes
1answer
70 views
Differential product Correlated processes
I am trying to derive the differential of the product of two processes, but I got stuck. This is what I have until now:
We have the following two stochastic processes:
$dX_t= \mu_t dt +\sigma_t dW_t$...
2
votes
1answer
210 views
Differential of integrating factor $d(e^{at}r_t)$ in Vasicek model
I am attempting to solve the Vasicek model SDE (using Wikipedia parametrisation):
$$
dr_t = a(b-r_t)dt + \sigma dW_t
$$
Every solution is proceeding to multiply both sides of the equation by the ...
3
votes
1answer
168 views
Why is Ito applied this way?
Given the price of a call option :
$$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 = \...
-1
votes
2answers
77 views
2
votes
0answers
143 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 :
$C_t(T,T+d)$ ...
7
votes
1answer
167 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 $...
3
votes
0answers
284 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
$$
X_t = ...
2
votes
1answer
221 views
Black Scholes in the case of dividends
Let's take the case where the underlying stock has the continuous dividend yield $\delta$. Then, in the risk-neutral world, $\frac{dS}{S}=(r-\delta)dt+\sigma dW^Q$. Suppose we want to price a ...
2
votes
0answers
64 views
$\int_{0}^1W_x(t)dW_y(t)/(\int_{0}^1W_x^2(t)dt)^{1/2}$ normally-distributed?
I have came across the following stochastic integrals:
$$\frac{\int_{0}^1W_x(t)dW_y(t)}{(\int_{0}^1W_x^2(t)dt)^{1/2}}$$
which was claimed to be standard normally distributed ($W_x$ and $W_y$ are ...
3
votes
1answer
94 views
How to derive the dynamic of the log forward price?
I have the following Schwartz model:
$$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
140 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 $\phi(t)$ ...
4
votes
2answers
144 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 ...
1
vote
1answer
353 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
148 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
67 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
637 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
172 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 ...
2
votes
1answer
388 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
274 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 {...
-2
votes
1answer
1k 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
222 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 $...
