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Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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0answers
80 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
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1answer
244 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 $$E\left[ \int_0^t \sigma(s) e^{iuX(s)} dW(s)\right] = 0$$...
3
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2answers
166 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 $...
3
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1answer
427 views

Step by Step Guide to Learn Quantitative Finance [closed]

Can some one help in creating step by step guide to learn Quantitative Finance? The suggestions should be in the lines of 1- Which Maths topics needs to be learn 1st 2- Which Maths Books or ...
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1answer
204 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
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1answer
448 views

Mean Reverting to its own variance?

Good morning all, When trying to decipher some documentation I have come across this stochastic process which seems to me much like a Ornstein-Uhlenbeck (or Vasicek) process. $$dX_t=-\kappa(X_t-\...
0
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1answer
296 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
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1answer
358 views

Is this process of Brownian motion?

Background Information: The process $W = (W_t:t\geq 0)$ is a $\mathbb{P}$-Brownian motion if and only if i) $W_t$ is continuous, and $W_0 = 0$ ii) the value of $W_t$ is distributed, under $\mathbb{...
0
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2answers
216 views

How to compute the conditional variance of this jump process?

Let $N_t$ be a Poisson process with intensity $\lambda>0$ and $S_t$ follows a pure jump process $$dS_t=S_t(J_t-1)dN_t$$ where $J_t$ is the jump size variable if $N_t$ jumps at time $t$. Also, ...
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1answer
939 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'...
2
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1answer
231 views

Closed- solution for Convertible bond price two factor model

I am trying to find the closed- solution of convertible bond $V(s,r,t)$ under Vasicek model of two factor model of PDE shown in below link Ito lemma of Convertible Bond under Two-factor Model ...
2
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2answers
272 views

Ito lemma of Convertible Bond under Two-factor Model Interest Rate

@Behrouz Maleki has provided the PDE of two factor model in other post so could anyone please provide Ito lemma of this equation and how this PDE was derived from Vasicek model. as far as I know it ...
4
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1answer
503 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 $f(x,t)$ $$x_t=\frac{r_t}{\kappa}(1-e^{-\kappa(T-t)})+\int_0^tr_sds$$ $$f(x,t)=e^{a(T-t)-x_t}$$ ...
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0answers
82 views

Approximating an SDE for Volatility Estimation

Consider the SDE $$ dT(t) = ds(t) + a(s(t) - T(t))dt + \sigma dW(t) $$ where $s(t)$ is a deterministic function that turns out to be the long-term mean (this SDE is used to model daily temperature, so ...
2
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2answers
304 views

What is the strong solution for this SDE

I want to calculate $E_t[(X_T-K)^+]$ where $$dX_t=\frac{3}{X_t}dt+2X_t dW_t$$ and $X_0=x$. I don't know how extact the strong solution of this SDE. Indeed I used Ito's lemma but it was not usefule. ...
4
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1answer
400 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 ...
2
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1answer
191 views

Analytic ZCB call option under Vasicek

The call option with strike $X$ and maturity $T$ on a ZCB maturing at time $S$, where $T\le S$, is $$ZBO(t,T,S,X)=E_t[e^{-\int_t^Tr_sds}(P(T,S)-X)^+]$$ The ZCB price is denoted by $$P(t,T)=E_t[e^{-\...
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3answers
3k views

Variance of time integral of squared Brownian motion

I want to calculate the variance of $$I = \int_0^t W_s^2 ds$$ I was thinking I could define the function $f(t,W_t) = tW_t^2$ and then apply Ito's lemma so I get $$f(t,W_t)-f(0,0) = \int_0^t \frac{\...
2
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0answers
402 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
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1answer
133 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,...
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1answer
180 views

Variance of the Cox-Ingersoll-Ross short rate

Shreve II page 151, the Cox-Ingersoll-Ross model is defined as $$dr_t=(\alpha-\beta r_t)dt+\sigma\sqrt{r_t}dW_t$$ By applying Ito's Lemma, we obtain \begin{align} r_t&=r_0e^{-\beta t}+\frac{\alpha}...
7
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1answer
498 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
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1answer
663 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^{\...
2
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1answer
298 views

Differential of integral of a stochastic process

Let $Y_{t}$ be \begin{equation} Y_{t}=\int_{\Omega} g(X_{u}) du \end{equation} where $g(.)$ is a deterministic function and $\Omega=[t_{0},t]$ continuos partition of $\mathbb{R}$. Furthermore let $...
1
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0answers
57 views

Using malliavin derivative to find the worst Delta-positive hedge?

Background: I've heard that Malliavin Calculus can be used to show the explicit form of a delta-neutral hedge (given an SDE driven market model). For example, here is a sketch here on page 21 on how ...
1
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1answer
172 views

How do I find this Expectation?

I have an expectation given as: $\mathbb{E}\left(S_{T}\mathbb{1}_{S_{T}\geq K} \right)$ where $K$ is just an arbitrary number (i.e. the strike price, but that's unimportant) and $S$ can be modelled ...
0
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1answer
142 views

Integral with respect of $(dW_s)^n$

I know $$\int _0^t dW_s=W_t-W_0=W_t$$ Since $ dW_s dW_s=ds$ , so $$\int _0^t( dW_s)^2=\int_0^t ds=t-0=t$$ I Want to know why for $n\ge 3$ we have $$\int _0^t (dW_s)^n=0$$ My try $$(dW_s)^2 dW_s (...
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5answers
933 views

Free or Relatively Less Pricey Quant Finance courses online

I am trying to figure out what all online Quant Finance courses are out there which are free or relatively less pricey? CQF is not less pricey Financial Engineering course on Coursera - Not so great ...
0
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1answer
181 views

Discrete Time to Continuous Time and Summation of Two Geometric Brownian Motions

Could someone please suggest with detailed steps and/or a reference, 1) How to convert the below discrete time summation to continuous time form and write it as an integral? 2) Any methods to ...
2
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1answer
194 views

Computing Correlation between Forward Rates

I have the feeling this question has an extremely simple answer but I'll put it out to the group anyway. Imagine I have data for 3M and 6M forward rates following a lognormal process, and that I ...
4
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1answer
410 views

discounted price is a martingale under any measure?

Assume I have some dynamics for the stock price under 2 different measures: risk-neutral and forward measures: $$dS_t=r S_tdt+\sigma S_td\tilde{W_t}$$ $$dS_t=\alpha S_tdt+\sigma S_td\hat{W_t}$$ now ...
5
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1answer
446 views

Martingale representation theorem

Let $r_t, \theta_t$ denote some stochastic processes driven by a $N$ dimensional Brownian motion $W_t$ (they are of course assumed adapted to the natural filtration $\mathcal{F}_t$ of that Brownian ...
1
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1answer
144 views

Piecewise Ito formula

Usually Ito's lemma is stated for $C^{1,2}(\mathbb{R}^{d+1},\mathbb{R})$ functions. My question is does Ito still hold if the domain is restricted. That is if the semi-martingale $Z_t$ is only ...
4
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1answer
899 views

What's the variance of this Ito integral?

I am reading stochastic calculus and I have understood that the process $$X=\int_{0}^{1}\sqrt{\frac{\tan^{-1}t}{t}}dW_t$$ has normal distribution with mean zero. How can I find the variance of $X$?
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4answers
18k views

Integral of Brownian motion w.r.t. time

Let $$X_t = \int_0^t W_s \,\mathrm d s$$ where $W_s$ is our usual Brownian motion. My questions are the following: Expectation? Variance? Is it a martingale? Is it an Ito process or a Riemann ...
0
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1answer
3k views

Correlation coeffitiont between two stochastic processes

I want to find correlation coeffitiont between $W_t$ and $\int_{0}^{t}W_s ds$. I think that these are uncorrelated. But Why? So thanks
3
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1answer
349 views

Dupire's formula proof

I just have a question for the beginning of a proof: Suppose $\frac{dS_{t}}{S_{t}}=(r_{t}-q_{t})dt+\sigma(t,S_{t})dW_{t}$ with $r,q,S$ stochastic. In the book I read, it is written: We define the ...
4
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3answers
933 views

Variance of Brownian Motion

Can someone point me into the right direction to calculate this one: $E(B^4_t)=3t^2$ I had tried using the following property with no luck: $E(B^4_t)=E(B^2_tB^2_t)=E(\int B^2 dt )E(\int B^2 dt )=[E(\...
1
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1answer
396 views

Stratonovich Integral and Ito's lemma

Let $(\Omega, \mathcal{F},\mathbb{P},\{\mathcal{F}\}_t)$ be a filtered- probability space and $W_t$ be standard Wiener process. I want to show stratonovich integral of $W_t$, i.e $\int_{0}^{t} W_s ○ ...
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2answers
377 views

Close form solution for Geometric Brownian Motion

I have a very fundamental problem, please help me out. I am little confused with the derivation for the close form solution for the Geometric Brownian Motion, from the very fundamental stock model: $$\...
0
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1answer
113 views

Does the partition of time in a simple process depend on the omega in probability space?

In Steven Shreve's book "Stochastic Calculus for Finance 2", page 126, a simple process $\Delta(t)$ is a stochastic process such that there is a partition of time $0 < t_1 < ... < t_n \leq T$,...
2
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1answer
208 views

Mix of Arithmetic and Geometric Brownian Motion

Talking with some traders the other day, I found out that they were using a pricing model based on a mix between a geometric brownian motion and an arithmetic brownian motion to price certain ...
3
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1answer
583 views

What is the probability that a Brownian Bridge hits an upper barrier $U$ before a lower barrier $L$?

The probability that an arithmetic Brownian motion process $dt = \mu dt + \sigma dW$ hits an upper Barrier $U$ before it hits a lower barrier $L$ is given by $$ \mathbb{P}(\tau_U\leq \tau_L) = \frac{\...
5
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1answer
229 views

What is the probability that a OU process hits an upper barrier U before a lower barrier L?

What is the probability that the arithmetic OU process $dx_t= \theta(\mu-x_t)dt+\sigma dW_t$ hits barrier $U$ before hitting barrier $L$ when $L<x_0<U$ ?
1
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1answer
48 views

How to understand the following limits when kapa limits to Zero

The equation is quite simple, however it is not very obvious to me to have the following relationship: $$\begin{equation} \frac{1-exp(-\kappa(T-t))}{\kappa}\rightarrow(T-t) \quad \rm{when\space} \...
8
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1answer
638 views

Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative

The problem: Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \...
2
votes
1answer
138 views

stochastic interest rate $r_t=x_t+y_t$

Let $$dr_t=(\alpha(t)-\beta r_t)dt+\sigma dW_t$$ where $\alpha$ is non stochastic process and $\beta$ and $\sigma$ are constant. Can we write process $r_t$ in the form $$r_t=x_t+y_t$$ where the ...
1
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1answer
143 views

How to understand the following brownian integral using Fubini's method?

I am a little bit stucked with the following integral process, using Fubini's method, this is an intermediate step of short rate Merton Model. $\int_{t}^{T} W(s)ds=\int_{0}^{\hat {T}}ds\int_{0}^{s}...
7
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1answer
587 views

How to show that this process is “normally distributed”?

Say we have following SDE (Vasicek): $$dr(t) =(b-ar_t) dt + \sigma dW_t$$ I am able to reach an integral form of this SDE : $$r(t) = r(0) e^{-at} + \frac{b}{a}[1 - e^{-at}] + \sigma e^{-at}\int_0^t e^...
2
votes
1answer
83 views

Option price derivation with these dynamics

If my underlying follows a dynamics of the form \begin{align*} dF(t,T)/F(t,T)=\sigma_1(t,T)dW_1(t)+\sigma_2(t,T)dW_2(t), \end{align*} where $\sigma_1(t,T)=h_1e^{-\lambda(T-t)}+h_0$, and $\sigma_2(t,T)...