# Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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### If $W_T$ is standard Brownian motion, what is $\int_0^T W_T \ln(W_T) dW_t$?

If $W_T$ is standard Brownian motion, what is meant by $\int_0^T W_T dW_t$ in finance? Furthermore, what then is the meaning of $\int_0^T W_T \ln(W_T) dW_t$?
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### Anticipating stochastic integral $\int_0^T W_T dW_t$

Using basic techniques from Malliavin calculus it can be shown that $$\int_0^T W_T dW_t = W_T^2 - T$$ As can be seen the above integral is a non-adapted stochastic integral. We also know using Ito ...
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### Explicit expression for option prices in SABR?

I am trying to get a grip of the current state of research regarding option pricing in the SABR model. Am I correct in that, so far, there is no known general formula for the option price in the SABR ...
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### How can the increments of a CIR process be derived?

For a CIR process, which has SDE $$dr_t = \alpha (\mu - r_t) dt + \sigma \sqrt{r_t} dW_t$$ how can I derive the increments over the discrete time-interval from $r_t$ to $r_{t+1}$?
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### Black Scholes to Heat Equation

Equation (2) was derived by setting r=0 in the Black-Scholes equation for the Bachelier model (1). Can someone please help me understand all the steps for how we get from the heat equation under time ...
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### Infinitesimal generator - Is it obtained from a stochastic process or It can construct the process

We can see here that the generator is an operator which can be determined for a stochastic process. But, in the answers and comments here we can see that the brownian motion on sphere can be ...
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### Option that never expires

I have been struggling with the problem below for quite some time now. I really don't know how to approach it. All I could think of is to use the Black-Scholes formula with $T \rightarrow \infty$, ...
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### Ito Integral of functions of Brownian motion

How does one show that: $$\mathbb{E}\left[ \int f(W_s)dWs \right] = 0$$ For all $f()$ that are powers of $W(s)$?? I assume that one would have to go via the definition of Ito integral and express ...
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### Why is the Schöbel-Zhu model affine?

In the Schöbel-Zhu model, the stochastic volatility process is $dv_t=\kappa(\theta-v_t)dt+\sigma dW_t$. The characteristic function of the stock process can be found by arguing that the model is ...
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### Statistical test for comparing two different speed of mean reversion parameters for CIR model

I am trying to compare two different values of speed of mean reversion parameter for CIR model. I would like to know if there exists a statistical test for comparing these two parameters. the estimate ...
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### Dynamic programming and Bellman equation to obtain the maximum

This is the problem of Marhsall (1992) "Inflation and Asset Returns in a Monetary Economy" and Balvers and Huang (2009) "Money and the C-CAPM" Suppose an endowment economy where the representative ...
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### How to replicate the future instantaneous short rate?

Suppose we have an interest rate model $R(t)=\alpha(t)d(t)+\sigma d\tilde{W}(t)$, where the brownian motion is under the risk neutral measure. Suppose $S(t)$ is the price at time $t$ for a contract ...