# Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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### Transformation into Martingale

If $f$ is some function of BV on $\mathbb{R}$ and $dZ_t = f(W_t)dW_t + \mu_t dt$ ($W_t$ is a $1$-dimensional standard Brownian Motion), then what choice of real valued function $F$ makes: \begin{...
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### Bachelier model: number of stocks in replicating strategy

Given: Consider a two-asset, continuous time model (B,S) where \begin{equation} dB_t = B_t r dt, \quad dS_t = \mu dt + \sigma dW_t. \end{equation} The question is: Show that there exists a trading ...
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### Option on $\left( \int_0^T dW_t \right)^2$

Silly question, but how would you actually price $$E_0 \left( \left( \int_0^T dW_t \right)^2 - K \right)_+$$ where $dW_t$ are standard Brownian motions. Is there a closed form analytical solution?...
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### Building up an Economic Scenario Generator [closed]

I am trying to build an Economic Scenario Generator in VBA or Python. Can anyone please help me with some good resources which I can follow or some basic procedures which explains how to go about in ...
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### Is it meaningful to look at $\int f(W_t, t) \,dt$?

CONTEXT (can skip): My textbook looks at two things - 1) Ito integrals for deterministic functions—i.e. $\int f(t) \,dW_t$. We are able to say that they are normally distributed, with a mean of 0 ...
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### Martingale representation of European option

Let stock price $S$ satisfy $$S(t)=S(0)e^{(\int_0^t\sigma(s)dB_s-\frac{1}{2}\int_0^t\sigma(s)^2ds)}$$ I want to calculate the Martingale representation $V(t)=E(F|F_t)$ of European option with strike ...
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### Interchange Expectation and Supremum in Snell Envelope/American Options

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,...
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### Computing Malliavin Derivative for European Call Payoff

Let $X_t$ be a continuous local-martingale modeling the stock price given by $$X_t = \int_0^t \sigma_t(T,K)dW_t ,$$ and $\sigma_t(T,K)$ is an $L^2$-measurable process not adapted to $W_t$'s ...
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### 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 ...
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### Idea of using logarithm for solving SDE in Black-Scholes model

In the Black-Scholes model they consider that the stock follows this stochastic differential equation: $$dS = \mu S dt + \sigma S\ dW$$ I was wondering, was it common at the time they work on this ...
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### Levy process and random measure

I am wondering if random measures are used under a Levy process and how this connects to finance (particularly pricing). Any paper or books for suggestions is welcomed.
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### Is a wiener proces measurable? (exercise from Bjork)

I will claim $$E[W(T) \vert F_t] = 0$$ for $t<T$. Anyway, in an exercise in Bjork the results requires that $$E[W(t) \vert F_t] = 0$$ But why? Isn't $W(t)$ measurable at time $t$ and hence not ...
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### Trading over a Ornstein/AR process

For a OU/AR(1) process is there anyway to analytically calculated most probable period of time the process is likely to diverge from the average, before turning to converge. Basically I am looking ...
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### Proof that $f$ is continuous if and only if it has 0 quadratic variation?

I understand that $f$ continuous $\Rightarrow Q(f) = 0$ where this is defined over a bounded interval [0,T] as then we may use uniform continuity and the mean value theorem. But I am not sure how the ...
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### Derivation of stock price formula John C. Hull 9th Ed p309

It says assuming a no-uncertainty Weiner process that models stock price: $$\Delta S = \mu S\Delta t$$ Can be rearranged to (after taking the limit of $\Delta t \to 0$... $$\frac{dS}{S}=\mu dt$$ ...
A question based from Springer's Stochastic Calculus for Finance II book - I've tried working this out, but keep ending up in circles. Let $S(t)$ be given by the usual formula for an asset price ...
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