# Questions tagged [itos-lemma]

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### Deriving the stochastic process for a dividend-yielding stock (under Black-Scholes assumptions)

In order to derive the Black-Scholes equation for a stock $S(t)$ yielding dividends at the continuous rate $d$ $$S(t) = S_0 e^{(\mu - d - \frac{\sigma^2}{2})t + \sigma \sqrt{t} N(0,1)} \text{,}$$ M. ...
1 vote
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### Integral of brownian motion wrt. time over [t;T]

From the post Integral of Brownian motion w.r.t. time we have an argument for $$\int_0^t W_sds \sim N\left(0,\frac{1}{3}t^3\right).$$ However, how does this generalise for the interval $[t;T]$? I.e. ...
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### Integration of exponential raised with Brownian Motion wrt the Brownian Motion

I have to derive several things for my thesis, however, I have the following expression: $$\int^{t}_{0} \exp\{\sigma W_{t}\}.dW_{t}$$ Does anyone know what the solution for this is? Kind regards.
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A trading strategy is defined as follows: starting capital $v_0 = 5$ and 1 risky asset holdings $\varphi_t = 3W_t^2-3t$ where $W$ is a Wiener process. The problem is to find the probability of the ...
1 vote
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### Show that the solution to a SDE is strong

I have the following SDE \begin{equation} dX_t = - \frac{1}{1+t}X_t dt + \frac{1}{1+t}dB_t \end{equation} that has the solution: \begin{equation} \begin{aligned} X_t = \frac{X_0 + B_t}{1+t} = \frac{...
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### Obtaining the dynamics of the Vasicek model using Itô

Consider the following expression for the short-term interest rate $$r_t=r_0 e^{\beta t}+\frac{b}{\beta}\left(e^{\beta t}-1\right)+\sigma e^{\beta t}\int_0^te^{-\beta s}dW_s \tag{1},$$ which is ...
1 vote
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### Ito's lemma for option pricing with Levy-alpha stable drift

Consider $$dS=\omega\left(\Lambda-S\right)dt+\sigma_S S dW_t,$$ such that such that $W_t$ is a Wiener process, $\sigma_S$ is constant, $\omega: t\rightarrow\mathbb{R}$ represents anticipated drift and ...
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### Stochastic process as integral over window function

Consider the following stochastic integral of a deterministic function $f(t,s)$ with respect to the Wiener process $W_s$: $$\int_0^\infty f(t,s) d W_s$$ My questions are: Is such an integral ...
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### Clarification on Paul Wilmott's derivation of Ito's Lemma

I'm currently self-studying to be quant and have been thoroughly enjoying PW's book. I have some questions regarding his derivation of Ito's lemma. Specifically, I can see that the first line in his ...
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### Ito's Lemma in option pricing for a stock satisfying $dS=\frac{P-S}{\omega}dt+SdW_t$

Suppose a stock follows the stochastic differential equation $$dS=\frac{P-S}{\omega}dt+SdW_t,$$ such that $W_t$ is a wiener process, $\omega\in\mathbb{R}^+$, and $P_t,S_t\in\mathbb{R}$. If the value ...
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### Volatility and drift of the instantaneous forward rate under risk neutral measure using the zero coupon bond

I have question about this problem. I believe I have derived $f(t,T)$ correctly using the zero-coupon bond. But I am unsure about how to go forward with the question and how to use the second part. ...
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### Parametric Stochastic Integral

I need help. Defining the parametric stochastic integral $$F_t = \int_t^T\xi(t,s)g(s)ds$$ $\\\\$ with $\xi$ a generic stochastic process such that $d\xi(t,s) = \mu(t,s)dt + \sigma(t,s)dW_t$, I'm ...
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### Covariance of the product of log normal process and normal procces

I tried to compute the following covariance : $$Cov(e^{\int_{t}^{T}W^1_sds},\int_{t}^{t+1}W^2_sds)$$ where $W^1_t$ and $W^2_t$ are Brownian motions such that $dW_t^1dW_t^2=\rho dt$ My idea was to ...
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### Question on Ito's lemma involving $\mathrm{d}W(t)$

I am new to Ito-calculus, so please forgive me if the question is stupid. Let $W(t)$ be a Brownian-Motion and $f(W(t))=W(t)^2$. If I want to calculate the differential $\mathrm{d}f(W(t))$, Ito's lemma ...
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### Hermite polynomials as martingales [closed]

Let $\left\{W_{t}: t \geq 0\right\}$ be a standard B.M. on the filtered probability space $\left(\Omega, \mathcal{F},\left\{\mathcal{F}_{t}\right\}_{t \geq 0}, \mathbb{P}\right)$. Define the Hermite ...
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### Ito calculus is Gaussian (using method of characteristic function)

Let $h$ be a deterministic function and define $X_{t}=\int_{0}^{t} h(s) d W_{s} .$ Show that $$\mathbb{E} \exp \left(i u X_{t}\right)=\exp \left(-\frac{u^{2}}{2} \int_{0}^{t} h^{2}(s) d s\right),$$ ...
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### Mutual variation of Brownian motions

Let $\{W^1\}_{t\geq0}$ and $\{W^2\}_{t\geq0}$ be two Brownian motions with correlation coefficient $\rho \in [0, 1]$, i.e., $\mathbb{E}[(W^1(t)-W^1(s))(W^2(t)-W^2(s))]=\rho(t-s)$ for all $t,s \geq 0$. ...
1 vote
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### What does it mean to "compute" an Itô integral?

I'm reading Shreve's Stochastic Calculus for Finance II. On page 191, Exercise 4.6, we are given the problem Exercise 4.6. Let $S(t)=S(0)\exp\Big \{\sigma W(t)+(\alpha-\frac{1}{2}\sigma^2)t\Big\}$ be ...
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### Proving that a stochastic process is a martingale using Ito's Lemma

Assume a Wiener process W and a bounded F-adjusted stochastic process a. Show that the following process is a martingale on F $$X(t)=(\int_{0}^{t}a(s)dW(s))^{2}-\int_{0}^{t}a^{2}(s)ds,\ t\geq0$$ Can ...
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### Summary of Stochastic Derivatives, Integrals, Expectations, and Variances

I wanted to make a summary table of stochastic functions to improve my understanding. Maybe the following should be a wiki page on this site so others can add functions and examples? Does the ...
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### Differentiability of solutions of a stochastic differential equation

I would like to clarify a confusion I have. It is well known that a Wiener process (Brownian motion) is nowhere differentiable. I have no difficulty in understanding that. But I am wondering about the ...
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### Trouble With Applying Ito's Lemma

I am having trouble applying Ito's Formula to the following: Let $Z_t = W_{1t}^2 e^{W_{1t}+ \int_0^t W_{3s}dW_{2s}}$. Find $dZ_t$. $W_1,W_2,W_3$ are independent Brownian motions. I know the formula ...
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### Is it possibile to use Ito Formula here?

I have this process: $dY_s^y=\alpha(s,Y_s^y)ds + \frac{1}{2}\beta^2(Y_s^y)^2dW_s$ with inital value $Y_s^y=y$. Moreover $\alpha(s,y)$ is a linear function in $y$ and bounded is $s$. I was wondering if ...
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### Volatility of a function of an asset

Suppose that $G$ is a function of the underlying asset $S$, which follows a geometric Brownian motion. Suppose that $\sigma_{S}$ and $\sigma_{G}$ are the volatilities of $S$ and $G$, ...
1 vote