# Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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### Process with negative quadratic variation

Today seems to be question day for me, sorry. The complex process $$dX = i\sigma dW$$ where $i = \sqrt{-1}$ and $dW$ is a standard (real-valued) Brownian motion will have a negative variance ...
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### Steven Shreve: Stochastic Calculus and Finance

The lecture notes have the following theorem: Let $\theta\in \mathbb{R}$ be given and $B(t)$ stands for the Brownian motion which is a martingale, then $Z(t)=exp\{-\theta B(t)-\dfrac{1}{2}\theta^2t\}$...
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### What is the SDE of this equation? [closed]

I am new and struggling to understand how to solve this using Ito lemma. Can someone please explain it to me: $$dS_t=-\frac{1}{2}\sigma^2 S_t dW_t$$ what is the solution with explanation please
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### What is Variance of delta of brownian motion [closed]

I am new to this. If variance of Brownian motion b is t, what is the variance of db? db is delta of b
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### Covariance of logarithms of geometric Brownian motion

Suppose I have a Geometric Brownian Motion process, $$dX_t=\mu X_t dt + \sigma X_t dW_t$$ I'd like to find the covariance of $\log(X_t)$ and $\log(X_s)$ where $s<t$. We can write $\log(X_t)$ in ...
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### Solve the following SDE: $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$

Let $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$ be a stochastic differential equation where $a$, $b$, and $c$ are positive constants, so I tried to solve it but I got stuck in ...
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### Realized Variance as an approximation of the Integrated Variance

Realized Variance is written as $RV_{[0,T]}^{n} = \sum_{j = 1}^{n} r_{j,n}^2$, where $r_{j,n}$ is the log return for the $j$th increment, and $n$ is the total number of sample points in the time ...
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I'm reading an interview book called A Practical Guide to Quantitative Finance Interview and I have some doubts regarding part of its solution and highlighted them in bold: Question: What are the ...
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### Intuition behind Scaling Symmetric Random Walk

I am reading a section in Shreve (2008) where we are scaling down the step size but speeding up the time a symmetric random walk, so that in the limit, we produce a Brownian motion. I understand the ...
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### Independence of increments of the stochastic process $\frac{1}{t}\int_0^t u dW_u$

Let $X_t$ be a stochastic process such that $$X_{t} =\frac{1}{t}\int_0^t u dW_u$$ 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 ...
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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 $$... 0answers 26 views ### Call Option on the Square of a Log-Normal: Process of Underlying under Stock Measure and Risk Neutral Measure I'm working on some quant interview questions from the book called Quant Job Interview Questions And Answers (by Mark Joshi and other authors). Here are the questions from the bookd, and the answers ... 1answer 39 views ### Accumulation Rate of Variance in Random Walk I am slightly confused with the terminology Shreve (2008), he states: "The variance of the symmetric random walk accumulates at rate one per unit time, so that the variance of the increment over ... 1answer 86 views ### Justify a backward differential equation Regards of 4.5.1, how we get 4.5.5? 1answer 63 views ### Arithmetic Asian Option Assume the risk-free bond Bt and the stock St follow the dynamics of the Black & Scholes model without dividends (with interest rate r, stock drift μ and volatility σ). Let A_T:=\frac{1}{T}... 1answer 108 views ### Asian Options-Change of Numeraire Assume the risk-free bond B_t and the stock S_t follow the dynamics of the Black & Scholes model without dividends (with interest rate r, stock drift \mu and volatility \sigma). Show that ... 2answers 290 views ### Stochastic Integral Graph As we can represent the integration of f(x) on [a,b] with the graph below, I was wondering how to represent the following integral with X(t) a Brownian motion, f(t) any function and t_j = ... 2answers 530 views ### Finding price of the power option Let's assume a market with d=1 and X=X^1 satisfying dX_t=\sigma X_t\,dW_t,\: \: X_0=1, where (W_t) is a standard Brownian motion. Assume that \mathbb{F} is the natural filtration of X ... 1answer 121 views ### How do we calcualte E[W_sW_t|W_s] W_t is a Brownian motion. How do we calculate this expectation? there are two cases: s < t t < s Do we have to distinguish the two cases or there is a unified way of calculating it 1answer 125 views ### stochastic dominance displaced diffusions Suppose I have two processes both satisfying a displace lognormal diffusion:$$ dX(t) = \alpha(t)[X(t) - a] dW(t)  dY(t) = \beta(t)[Y(t) - b] dW(t) $$Note that the processes are perfectly ... 2answers 91 views ### Instantaneous change in value of portfolio I am trying to figure out an intuitive explanation for the instantaneous change for the value of a portfolio (essentially I'm creating a self-financing portfolio to replicate a derivative payoff). ... 1answer 102 views ### What the expectation of S^2 is from GBM? [closed] I was at an interview and was asked to write down the SDE for GBM.$$ dS = S\mu dt + S\sigma dX $$Then I was asked how I would compute the expectation of S^2. I didn't know where to start. Any ... 1answer 171 views ### Evaluating the SDE dX_t = t\,dS_t The process S is a geometric Brownian motion with an SDE: dS_t = S_t(\sigma\, dB_t + \mu\, dt). I'm stuck evaluating E(X_t) and V(X_t), where dX_t = t\,dS_t. 1answer 68 views ### Stochastic Vol Mathematical derivation [closed] I want to understand the mathematical steps done. Can someone please simplify the derivation of d(pi) from Pi? Thanks in advance. 1answer 203 views ### Solving dX_{t} = \mu X_{t} dt + \sigma dW_{t} I want to solve the following SDE:$$ dX_{t} = \mu X_{t} dt + \sigma dW_{t} \quad X_{0} = x_{0}$$Integrating, I get:$$ X_{t} - x_{0}= \mu \int_{0}^{t} X_{s} ds + \sigma \int_{0}^{T} dW_{t}  ...
I had a question about the properties of a snell envelope, $\sup_{t\le\tau\le T} \Bbb E\left(Z_\tau\mid \mathcal F_t\right)$, which came to me while studying American options. I know that in general,...
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(...