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Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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Intergral of Brownian motion w.r.t. Brownian motion

I don't understand why $S$ (highlight on picture), I learned $$\int_0^t W(s) dW(s) = \left. \frac{1}{2} (W^2(s)-s) \right \vert_0^t$$ everyone please explain for me. Thank you
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For Ito Integrals with respect to a Brownian motion, why would the amount of stock held be a stochastic process?

Suppose that $B$ is a Wiener process and suppose $H$ is a right-continuous, adapted, and locally bounded process. Suppose $$\int_0^t H dB$$ is the Ito integral of $H$ with respect to the Wiener ...
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Find the brownian motion associated to a linear combination of dependant brownian motions

I have $N$ correlated standard one-dimensional Brownian motions $W_1,\ldots,W_N$ with correlation matrix $\rho$ and I consider the process $Z_t \equiv \sum_{i=1}^N \mu_i (t) W_t$ where the $\mu_i$ are ...
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Solution to a Geometric Ornstein Uhlenbeck Process $dX_t = \kappa(\theta - X_t)dt + \sigma X_t dW_t$

I've been searching for the solution to the modified Ornstein-Uhlenbeck process \begin{equation*} dX_t = \kappa(\theta - X_t)dt + \sigma X_t dW_t \end{equation*} but it surprisingly hard to find. The ...
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Differential of integral of Wiener process over time

I am trying to compute this quantity: $\frac{d}{dt}\int_{0}^{t} W_s ds$ Where $W_t$ is a Wiener process. Is there a theorem which tells how this can be computed? I have tried https://en.wikipedia....
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Statistical estimation vs Stochastic calibration of models

I have never been able to deduce the precise differences between model building from the statistical perspective and the stochastic processes/calibration perspective. I can only infer that these are ...
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What does $\int dS \phi (S - K)$ mean in Gatheral's book?

In Gatheral's book on stochastic volatility, he writes the price of an option as $$\int_K^\infty dS \phi (S - K)$$ where $\phi$ is a density. Where does this come from? I have multiple questions: ...
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How to derive an option price for an asset with these dynamics?

Assuming my underline asset price follows the process: $$d\ln (F_{t,T})=-(1/2)\sigma ^2e^{-2\lambda(T-t)}dt+\sigma e^{-\lambda(T-t)}dB_t$$ How should I derive an option price formula?
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Lookback option to find stock price

Consider the payoff equation for the lookback option $\psi(T)= max(S_t-S_T)$, where $t\in[0,T]$ and $S_t$ is modeled by the geometric Brownian motion with constant parameters. Find the price of stock ...
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Conditional expectation of a non stochastic process

In an example I was working through it was shown that $W_{t}^{2} - t$ was a martingale with respect to the Brownian motion filtration $\mathcal{F}_{s}^{W}$ with $t>s$. Everything was fine except a ...
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Bond Option Hedging

(My question) Please show me how to solve from (2) to (4) with computation processes. These are too difficult to solve. Thank you for your help in advance. (Cross-link) I have posted the same ...
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Derivation and expectation interchange

I would like to know when it is allowed to interchange derivation and expectation. Suppose $X$ is some r.v whose dynamic is controlled by some parameter $\sigma$ and suppose $h$ is some smooth ...
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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 ...
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stochastic calculus - brownian motion

I don't know how to prove this : let be $X_t = \int_{0}^{t}\sigma_{u}dW_{u}$ where $\sigma_{t}$ is a predictable process. If $|\sigma_{t}| = c$ a.s. how can I prove that $X_{t}=c*\beta_{t}$ (...
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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 $$\mathbb{P}(\tau_U\leq \tau_L) = \frac{\... 1answer 65 views prove the normality, with given moments, of this process: I have this process: dx_t = -\frac{k}{2}x_tdt + \frac{\beta}{2}dz_t and must prove it's normally distributed with first two moments: \mu = e^{-\frac{1}{2}kt}x_0 \sigma^2 = \frac{\beta^2}{4k}(... 3answers 291 views How to understand nonrandom/random process in Shreve book? [closed] I have been reading Chapter 4 of Shreve's Stochastic Calculus for Finance II. It is easy to understand the simple process, \Delta(t), defined on Page 126, which is just a constant inside a given ... 1answer 131 views How do one solve  \int_t^T \exp[\int_0^u-( r-\delta_s)ds] dW_u ? Double integral with general deterministic function \delta(t) How do one solve  \int_t^T \exp[\int_0^u-\left( r-\delta_s\right)ds] dW_u  ? \delta(t) is a general deterministic function. r is constant. 1answer 218 views Derivation using Ito's Lemma of price process Define q(t) as the log price minus a linear trend$$ q(t) = \ln P(t) - \mu t $$Assume the log price process = Equation 1:$$ dq(t) = - \Theta q(t) dt + \sigma dW(t)  Can you show that the ...
I have a question about the solution of the Black-Scholes PDE for the European call option when I read the book Stochastic Calculus for Finance II of Steven E.Shreve. Let $c(t,x)$ be the value of the ...