# Questions tagged [girsanov]

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### Girsanov Theorem and Quadratic Variation

Girsanov theorem seems to have many different forms. I've got a problem matching the form in wiki to the one in Shreve's book, due to the difficulty of quadratic variation calculation. Below is the ...
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### Radon-Nikodym derivative and risk natural measure

I need help with my understanding of changing probability measure. Im not a mathematician so I hope for answers that are not too technical. As shown in this Wikipedia article http://en.wikipedia.org/...
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### How to compute the Radon-Nikodym derivative?

Suppose $B(t)$ is a standard Brownian motion, and $B_{1}(t)$ is given by $dB_{1}(t)=\mu dt+dB(t)$. Suppose $P$ is the Wiener measure induced by $B(t)$ on the $C[0,\infty)$, and $P_{1}$ is the Law ...
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### SDE simulation: P or Q?

Let's take a GBM under $P$: $dS=\mu dt+\sigma dW_{t}^{P}$ and then under $Q$ $dS=r dt+\sigma dW_{t}^{Q}$, where $dW_{t}^{Q} = dW_{t}^{P} + (\mu - r)/\sigma dt$ Now, let's say that I have ...
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### Uniqueness of equivalent martingale measure in Black Scholes-Model

Let's consider standard Black-Scholes model with price process $S_t$ satisfying SDE $$dS_t = S_t(bdt + \sigma dB_t)$$, where $B_t$ is standard Brownian Motion for probability $\mathbb{P}$. I ...
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### Bayes Theorem with change of measure

Tomas bjork- arbitrage theory in continuous time. Appendix B, proposition B41 says: The proof is not clear to me. Thanks to Gordon's comment below of $E^Q (X/G)$ being $G$ measurable, I think the ...
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### Are all change of measure operations between equivalent probability measures Doléans-Dade exponentials?

Let $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ be a filtered probability space, where $\mathbb{F}=\left(\mathcal{F}\right)_{t\in[0;T]}$ and $\mathcal{F}=\mathcal{F}_T$. Let $(W_t)_{t\in[0;T]}$ be ...
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### Understanding Girsanov's theorem in Bjork's book

In Bjork's arbitrage theory in continuous time, he writes S, essentially we define $Q$ using $h_t$, and then pick $h_t$ so $Q$ is a martingale measure. But, $Q$ needs to be an equivalent measure. ...
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### Uniqueness of Risk-neutral measure: Probabilistic view

Suppose we are working on the Black and Scholes Framework. There are only two assets, the risk-less bank account and a stock. The discounted process is a GBM under the physical measure with drift term ...
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### Change of measure from physical to risk-neutral under Radon-Nikodym and Girsanov Theorem

Given a stochastic process, how do we prove and generate the change-of-measure? I have been trying to prove the change-of-measure as under the Radon-Nikodym theorem and Girsanov Theorem, but ...
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### Why do we need $dS_t=r S_tdt+\sigma S_tdW_t^Q$?

Suppose $S_t$ is the stock price and follows the dynamics $$dS_t=\mu S_tdt+\sigma S_tdW_t$$. According to Girsanov, we can apply change of measure and obtain $dS_t=r S_tdt+\sigma S_tdW_t^Q$, this ...
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### Change-of-measure: Dynamics of $\log(S_t)$ with $S_t$ as numeraire [duplicate]

Let $S$ be a GBM with dynamics $dS_t/S_t=rdt+\sigma dW_t$. We want to compute the following expected value: \begin{align*} \mathbb{E}(S_T\log(S_T)). \end{align*} Using a change of measure we can write ...
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### Black Scholes model without using Girsanov's theorem? It might happen?

We can calculate the stock price by the equation: $\frac{dS_t}{dt} = \mu dt + \sigma dB_t$,where $B_t$ is a Brownian motion. First i create a portfolio that consists of $\Phi$ units of stock share ...
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### Change of Numeraire to price European swaptions

In the pricing of a European swaption, it is common to use the annuity factor $A(t)$ as the Numeraire. I was trying to write down the pricing formula via the bank account as numeraire to see if they ...
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### How do we derive the Radon-Nikodym derivative for T-forward measures?

Let $Q^{T_e}$ denote the $T_e$-forward measure and let $Q^{T_p}$ denote the $T_p$-forward measure. I have seen the following Radon-Nikodym derivative being used in derivations. For $0 \le t \le T_p$, ...
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