Questions tagged [stochastic-calculus]
A branch of mathematics that operates on stochastic processes.
732
questions
2
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Hagan's explanation of the Local Volatility model
Long story cut short: I am asking why the Local Volatility function can be thought of as a function of the underlying, when in fact it appears to be the function of the strike.
Long story:
The well-...
- 5,366
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0
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32
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Reference request: Approximate mapping of a multi-factor stochastic volatility model to single-factor stochastic volatility model
I am looking for approaches to transform a more complicated stochastic volatility model such as the one shown in Section 2.2 of Smile Dynamics II to a single-factor model such as the one shown in ...
- 540
2
votes
0
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44
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Feymann Kac pde with correlated process
I have to solve the following PDE:
\begin{equation}
\begin{cases}
\dfrac{\partial F}{\partial t}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial x^2}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial y^2}+\dfrac{1}{...
- 153
0
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1
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49
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Ito's lemma in stochastic volatility models [closed]
I couldn't help but notice that in all stochastic volatility models articles I consulted, whenever Ito lema is applied with a process of the sort
$$\frac{d S_t}{S_t} = \sigma_t d W_t $$
With $(\...
- 237
1
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1
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Dynamics of discounted prices (multi-dimensional)
My objective is to find the dynamics of the discounted prices, given by $\mathbf{y}_{t} = \mathbf{P}_{t}\mathrm{e}^{-\int^{t}_{0} r_{s} ds}$. I know the dynamics should be $d\mathbf{y}_{t} = \mathrm{...
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1
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1
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115
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Conditional Expectation of Integral of Squared Brownian Motion - PDE Approach
I am looking to compute the following using Ito's formula.
$$u(t,\beta_t) = \mathbb{E}(\int_t^T\beta_s^2ds|\beta_t)$$
Knowing the properties of brownian motion, it is rather easy to show that the ...
0
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0
answers
54
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Pricing of Zero Coupon bond under Risk-neutral pricing measure
Pg 242 Topic 5.6.2: Futures contract
Risk-neutral pricing of a zero-coupon bond is given by the below formulae:
$$ B(t,T) \, = \,\frac{1}{D(t)}. \tilde E~[D(T)\mid F(t)], 0\,\leq \,t\,\leq\,T\,\leq\,\...
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2
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0
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Munk (2011) exercise 3.6
I'm trying to solve the exercise in Munk (2011). The exercise reads:
"Find the dynamics of the process: $\xi^{\lambda}_{t} = \exp\left\{-\int^{t}_{0} \lambda_{s} dz_{s} - \frac{1}{2}\int^{t}_{0} \...
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1
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0
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86
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Analytical expression for SDE
I'm trying to find an analytical expression for the following. Suppose $X$ is a geometric Brownian motion, such that: $dX_{t} = \mu X_{t} dt + \sigma X_{t} dW_{t}$. Suppose furthermore, that the ...
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65
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How to derive the HJM drift condition?
I'm trying to derive the Heath Jarrow Morton drift condition (from Björk, page 298) and this equation is the part that I'm not able to derive:
$$ A(t,T) + \frac{1}{2} ||S(t,T)||^2 = \sum_{i=0}^d S_i(t,...
- 51
1
vote
1
answer
112
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Put price characterisation
I am reading Shreve's Stochastic Calculus for Finance II: Continuous-Time Models.
I am trying to understand the below two concepts:
Topic 8.3.3 Analytical Characterization of the Put price on Page ...
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2
votes
1
answer
100
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Integral of Function of Brownian Motion w.r.t Time (Context: Computing Quadratic Variation)
I am looking to compute the quadratic variation of $$S_t = S_0e^{\sigma B_t}$$ where $B_t$ is Brownian Motion. Applying Itô's lemma, I having the following
$$(dS_t)^2 = S_0^2\sigma^2e^{2\sigma B_t}dt$$...
0
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0
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60
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Backset LIBOR contract
Below is an extract from Steven Shreve’s BK 2, Chapt 10: Term Structure models. LINK
I am trying to understand Stochastic Calculus from the above book with the help of a Pure Math PhD student.
Despite ...
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0
votes
1
answer
61
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Construction of Itos integral
I am trying to understand the below:
Question 1 how can [W(t1) - W(t0)] = [W(t1) - W(0)] =[W(t1) - 0] =some positive number be a profit or loss?
In this calculation the purchase price is not taken ...
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2
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If $\Delta \log(V_{t})$ behaves like the increments of fractional Brownian motion, why do we model the rough volatility as follows
From Gatheral's paper, Volatility is rough and empirical evidence, it is clear that $\big\{\log(V_{t+1})-\log(V_{t})\big\}_{t}$ behaves like the increments of fractional Brownian motion $B^{H}$ with ...
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60
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Locally riskless
Most derivations of the Black-Scholes formula end up with the following dynamics of some (hedged) portfolio:
$$
\int_{t=0}^{T} \left(\frac{\partial f}{\partial \tau}(S(t),t)+\frac{1}{2}\cdot\frac{\...
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4
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0
answers
131
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optimal stopping time problem
I'm currently reading a paper (The Optimal Stopping Time for Selling an Asset When It Is Uncertain Whether the Price Process Is Increasing or Decreasing, American Journal of Operations Research, March ...
2
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2
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164
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Black-Scholes PDE derivation gap
Most derivations of the Black-Scholes formula end up with the following dynamics of some (hedged) portfolio:
$$
\int_{t=0}^{T} \left(\frac{\partial f}{\partial \tau}(S(t),t)+\frac{1}{2}\cdot\frac{\...
- 33
2
votes
1
answer
100
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Deriving the variance of G2++ Model
I'm studying G2++ Model in Brigo(2007)'s book.
The model constructed as follows,
$$
r(t) = x(t) + y(t) + φ(t), \quad r(0) = r_0\\
$$
with the dynamics of $dx(t)$ and $dy(t)$ described by:
\begin{align}...
- 303
1
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1
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190
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Euler Discretization python code
Write the Euler discretization of the 1-dimensional stochastic equation
$dXt = b (t, X_t) \space dt + \sigma (t, X_t) \space dW_t$
For this part I would say all right because it is a purely ...
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2
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0
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139
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Local volatility implied spot vol correlation
I have a question about local volatility models.
In a lot of articles it is stated that the implied spot vol correlation of this model is -1 and we usually compare this with stochastic volatility ...
- 21
4
votes
2
answers
400
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Transformation of local volatility model
Assume we have an SDE
$$dX_t=\mu(X_t)dt + \sigma(X_t)dW_t$$
where $\sigma>0$ and $W_t$ is a Wiener process. Is there a transformation $y(X_t)$ that will make the dynamics of the transformed process ...
- 43
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0
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35
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Maximum entropy probability distribution for $S_T$ implied from discrete market quotes
Consider a maturity $T$, for this maturity I have some implied volatility from market denoted $\sigma^{0}_{i}$. I want to interpolate these volatility using Entropy approach, by using $\sigma^{0}_{i}$...
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0
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73
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How to calculate mean and variance in Vasicek Model
In the Vasicek model, the short rate of interest under the risk-neutral probability measure is
given by:
where k, θ, σ > 0 and W is a standard Brownian motion. Consider the related process
where ...
- 1
3
votes
1
answer
186
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Numerically stable method for estimating $\partial_t \mathbb{E}[f(X_t)]$ where $X_t$ is an n-dim Ito process and $f:\mathbb{R}^n\rightarrow\mathbb{R}$
Assume $(X_t)_{t\geq 0}$ follows an SDE of the form:
$$dX_t = a(t, X_t) dt + b(t, X_t) dW_t$$
where $W$ is a standard $n$-dimensional Brownian motion, $a$ and $b$ are mappings from $\mathbb{R}_+\times\...
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1
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I want to know stochastic derivation of zero coupon bond formula
I'm elementary level of stochastic calculus.
In the above picture, from equation (11) to (12) I don't know what is the clue of $μ(t)$ is the expectation of $r(t)$ and how from this identity we can get ...
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2
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0
answers
67
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Relation between SABR parameters and Taylor expansion parameters
Suppose a SABR model framework (with $\beta=1$)
$$dF_t=\sigma_t S_t dW^{S}_{t}$$
$$d\sigma_t=\alpha \sigma_t dW^{\sigma}_{t}$$
$$dW^{S}_{t}dW^{\sigma}_{t}=\rho dt$$
I know that the Implied Volatility ...
- 805
0
votes
1
answer
75
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Integration of exponential raised with Brownian Motion wrt the Brownian Motion
I have to derive several things for my thesis, however, I have the following expression:
$$
\int^{t}_{0} \exp\{\sigma W_{t}\}.dW_{t}
$$
Does anyone know what the solution for this is?
Kind regards.
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93
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Implied Volatility is the harmonic average of Local Volatility
I am trying to demonstrate the famous result that states that when $T \rightarrow 0$, the Implied Volatility is the harmonic average of Local Volatility.
I am st the final stage, and I have the ...
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5
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0
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105
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Closed formula for computing Implied Volatility from Local Volatility function
The main result of this paper (Asymptotics and Calibration in Local Volatility Models, Berestycki, Busca, and Florent. Quantitative Finance, 2002) is equation (16) on page 63, that states that:
In the ...
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2
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158
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The meaning of filteration ( coin toss example )
Reference book is 'Steven Shreve: Stochastic Calculus and Finance'
What I don't understand is $F_3$ below picture
I understand that 'filteration' have accumulative information.
So when we tossed the ...
- 303
0
votes
1
answer
123
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Black-Scholes differential equation rewritten [closed]
I have seen that the Black-Scholes equation
$$\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2}+
rS\frac{\partial V}{\partial S}-rV=0$$
can also be written in the ...
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1
vote
1
answer
126
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Calculating Expectation of Stochastic Volatility
I have a question while reading THE NELSON–SIEGEL MODEL OF THE TERM
STRUCTURE OF OPTION IMPLIED VOLATILITY
AND VOLATILITY COMPONENTS by Guo, Han, and Zhao.
I don't understand why the above equations ...
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1
vote
0
answers
67
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Simultaneous Stochastic Differential Equations
I was thinking about cointegrated time series and came up with the following simultaneous equations model:
$dY_t = \alpha (Y_t - \gamma X_t)dt + \sigma dB_t$
$dX_t = \beta (Y_t - \delta X_t)dt + \tau ...
1
vote
0
answers
90
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Differential vs. derivative in the Vasicek model [closed]
Can anyone help me in understanding how we get the line I have marked with a red arrow?
I guess I have trouble in understanding the difference between differentials and derivatives, i.e. what is the ...
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3
votes
1
answer
154
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Where does the term $\gamma$ come from when moving from measure $\mathbb Q^{N}$ to $\mathbb Q^{M}$?
Consider two measures $\mathbb Q^{M}$ and $\mathbb Q^{N}$, as well as the two numéraires $M$ and $N$, furthermore assume that $X\frac{N}{M}$ is a $\mathbb Q^{M}$-martingale. Furthermore, the ...
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1
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246
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How am I supposed to understand the following statement on the convexity adjusted rate
Given, a numéraire $(N(t))_{0\leq t \leq T}$ and an index $(X(t))_{0\leq t\leq T}$ that is a $\mathbb Q^{N}$-martingale, we consider the natural payoff $V_{N}(T)$, where it pays
$$V_{N}(T):=X(T)N(T) \...
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0
answers
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Finding the distribution of $I(T_{1},T_{n})$ under an appropriate measure if the forwards are lognormal? [duplicate]
My question follows beneath the "lengthy" setting I describe:
Given a tenor discretization $0 = T_{0}< ... < T_{n} =T$,
and under the assumption that under $\mathbb P$, for all $i = 1,....
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votes
3
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679
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Why does the diffusion term remain the same when we change pricing measure?
Consider some Itô process $dS(t)=\mu(t)dt+\sigma(t)dW^{\mathbb P}_{t}$ under the measure $\mathbb P$, where $W^{\mathbb P}$ is a $\mathbb P$-Brownian motion
In plenty of interest rate examples, I have ...
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votes
1
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617
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Where does 1/2 in Fourier Transform method of pricing options come from?
I am reading Jianwe Zhu's Applications of Fourier Transform to Smile Modeling. On page 26, the author is describing how to use the Fourier tranform to price vanilla European call options. If $f_j$ is ...
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1
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Obtaining the dynamics of the Vasicek model using Itô
Consider the following expression for the short-term interest rate
$$r_t=r_0 e^{\beta t}+\frac{b}{\beta}\left(e^{\beta t}-1\right)+\sigma e^{\beta t}\int_0^te^{-\beta s}dW_s \tag{1},$$
which is ...
- 221
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0
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68
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SDE of a Geometric Levy process with compound Poisson process
Suppose that a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ is given. A geometric Levy process is defined in the form of $S_t=S_0 exp(X_t)$ where $S_0$, let's say, is the initial price and $...
- 409
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0
answers
101
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Largest class of real world probability models admitting explicit risk-neutral change of measure
Assume we have two assets, a random asset $A_t$ and deterministic risk-free bond $B_t = e^{rt}$. Let $P$ be a model of the real-world probabilities of $S$ and $Q$ the unique associated risk-neutral ...
- 21
4
votes
1
answer
236
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Pricing a contract
I'm currently trying to price some different kinds of contracts. I'm stuck on this following exercise, which I can't seems to find a good solution for. The following is assumed:
We are in a standard ...
- 65
2
votes
3
answers
270
views
Integral of brownian increments
I'm stuck at a problem and I'm not sure on how to proceed. My question is how would one go about and integrate the following
$$\sigma\int_{t}^{T}\mathrm{e}^{a\cdot u}\cdot (W_{u}-W_{t})du.$$
I've been ...
- 65
1
vote
2
answers
408
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Why would exchange rates follow a geometric brownian motion?
I'm reading Shreve's Stochastic Calculus for Finance.
On page 382, he begins talking about exchange rates:
Finally, there is an exchange rate $Q(t)$, which gives units of domestic currency per unit ...
- 41
1
vote
0
answers
34
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From the perspective of a company, when is the right time to start paying dividends?
I am trying to understand geometric Brownian motion as it relates to the present discounted value of future dividend payments.
I am supposing that a company has a revenue stream $f(t)$. This is just $...
- 41
1
vote
1
answer
241
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Jump Diffusion Process question
I have a European call option with time maturity $T=3$ years,$K=50$, and given that $S(t)$ refers to the derivative is being described by the geometric Brownian motion with $S_{0}=100$ and $r = 0.04$....
user57440
0
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0
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194
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what is $\int t dW$ and $\int W dt$? [duplicate]
More explicitly, if $W(t)$ is Brownian motion, what would be
$$f(t) := \int_0^t u dW(u)$$
and $$g(t) := \int_0^t W(u) du$$?
- 2,109
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votes
0
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132
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Are Stochastic Differential Equation diffusion terms always invariant under a change of measure?
I'm struggling with learning change of numeraire, and stochastic differential equations. I'm reading the beginning of Brigo and Mercurio's Interest Rate Models- Theory and Practice, and I'm on the ...
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