Questions tagged [stochastic-calculus]
A branch of mathematics that operates on stochastic processes.
6
votes
1answer
66 views
Show that $(W_t, \int_0^t W_s ds)$ has a normal joint distribution
I have to show that, if $W_t$ is a 1-d Brownian motion then
$\biggl(W_t, \int_0^t W_s ds\biggr)$ has normal distribution.
Hint: apply Ito formula to this bivariate process.
Any idea or suggestion on ...
1
vote
1answer
44 views
How to check if $ E [\exp \{ \int_0^t \frac{Y_u^2}{1+Y_u^2}du \}]< \infty $
$dY_t=2Y_tdt+2\sqrt{1+Y_t^2}dW_t$ where $W_t$ is $P-$Brownian motion (Wiener process).
I have defined a new measure $Q$ where the Kernel density (In Girsanov theorem) is
$$ \phi_t = \frac{Y_t}{\sqrt{...
3
votes
2answers
96 views
Application of Ito's lemma
Let $X_t$ be some stochastic process driven by wiener process ($W_t)$ so it can be expressed as:
$$dX_t=(...)dt+(...)dW_t$$
Let $f(t,x)$ be some $C^2$ function. Define the process $Z_s=f(t-s,X_s)$ ...
4
votes
1answer
69 views
How to express a process using Itos formula
Let $F(t,x)$ be the solution to the PDE
$$
F_t(t,x)=aF_x(t,x)+\frac{1}{2}F_{xx}(t,x),t>0
$$ $$F(0,x)=g(x)$$ for some function $g$.
Let $X_t$ be a process defined by
$$dx_t=aX(t)dt+dW(t)$$
Now ...
5
votes
1answer
59 views
Distribution of time integral of Brownian motion squared (where the Brownian motion occurs in square root time)?
Let $I_t = \int_0^t W_{\sqrt{u}}^2du$. What is the distribution of $I$?
If I recall correctly, if the Brownian motion were instead $W_u$, then it would be $I_t \sim N\left(\frac{t^2}{2},\frac{t^4}{3}\...
4
votes
1answer
93 views
Compute distribution of a stochastic variable
$sign(x)=1$ if $x\geq0$
$sign(x)=-1$ if $x< 0$
Consider
$$
X_t = \int^t_0 sign(W_u)dW_u
$$
where $W_t$ is a wiener proces.
How can I determine the distribution of $X_t$ and compute $E[\exp(\...
0
votes
2answers
75 views
Integral of Wiener process over time
This should hopefully be an easy question to answer, but I am new to Stochastic Calculus and am gapping as to why the following is true, for a brownian motion $W_t$:
$$d(\int W_t dt ) = W_t dt$$
I ...
1
vote
1answer
88 views
Derive a mathematical equation for Eurodollar future rate
If we suppose that r(t) follows a Vasicek model, which is:
$$dr(t) = (\mu - \kappa r(t))dt + \sqrt\sigma dW(t)$$
How to derive an expression for Eurodollar future rate?
1
vote
1answer
37 views
Levy process and random measure
I am wondering if random measures are used under a Levy process and how this connects to finance (particularly pricing). Any paper or books for suggestions is welcomed.
3
votes
0answers
52 views
Solving BSDE in R
I was wondering how to implement a BSDE approximation in R. For example, if I have the toy BSDE
$$
dX_t = \mu dt + \sigma dW_t ; X_T\sim N(\mu_1,\sigma_1),
$$
for fixed real numbers $\mu,\mu_1,\sigma,...
1
vote
0answers
62 views
Question about Stochastic Calculus,(change of measure)?
Can any one give some hint for this question?
Let $\{S_t\}_{t=0}^\infty$ be an asset price process defined on the probability space $(\Omega,\mathcal{F},\mathbb{P})$. Assume that the log-return of $...
2
votes
1answer
47 views
Characteristic function and distribution of a random variable
This is exercise 4.3 in Bjork, Arbitrage Theory in Continous Time.
$$
X_t = \int^t_0 \sigma(s)dW_s
$$
$\sigma$ is a deterministic function and $W_t$ is brownian motion.
I am asked to find the ...
4
votes
1answer
164 views
For the Brownian motion integrate
I want to calculate
$$\operatorname{E} \left[ \int_0^1{W(t)dt \cdot \int_0^1{t^2W(t)dt}} \right].$$
I discovered that the first integral is $\operatorname{N}(0, \frac{1}{3})$ but I don't know how to ...
2
votes
2answers
79 views
Statistical estimation vs Stochastic calibration of models
I have never been able to deduce the precise differences between model building from the statistical perspective and the stochastic processes/calibration perspective. I can only infer that these are ...
1
vote
0answers
24 views
Variance of integrated dynamical system
Define time increment $\mu:=t_{k+1}-t_{k}$. Consider the signal $x(\mu)-\mathbb{E}[x(\mu)]$ defined as
$x(\mu)-\mathbb{E}[x(\mu)]=\frac{1}{\mu}\int_{t_{k}}^{t_{k+1}}\int_{0}^{\tau}e^{A(\tau-\delta)}...
0
votes
0answers
59 views
Black and Normal Model for Caplet using Python
I am able to Price Caplet using Black 76 model in Python. However, I am unable to price the same with Normal Model. Can anyone suggest what is missing ?
I am valuing caplet that caps interest rate on ...
2
votes
1answer
118 views
How to calculate the covariance between two stochastic integrals?
How to calculate the covariance between the integral of a Brownian motion at different times:
$$\text{Cov}\left(\int^{t_1}_0\sigma(t)dW_t,\int^{t_2}_0\sigma(t)dW_t\right)\ ?$$
I know the answer is:
$$\...
1
vote
0answers
32 views
Model of asset substitution/risk shifting in continuous time
Consider a firm with cash flows $X_t$, which under a risk-neutral probability measure, follows a geometric brownian motion:
$$dX_t = X_t[(r-\beta)dt + \sigma dZ_t]$$
where $r>0$ is the risk-free ...
3
votes
0answers
36 views
Euler discretization with jumps
There is a process
$B_t = B_0\prod_{i=1}^{N_t}(1-Z_n)$,
where $Z_n=e^{-ξ_n}$ for i.i.d exponentially distributed random variables $(ξn)_{n≥1}$ with rate $ρ=20$.
${N_t}$ is a counting process ...
1
vote
0answers
21 views
From one period to multi period risk neutral pricing
For a one period economy, we have the price of an asset as:
$ p_0 = E^Q [p_1 * \frac {B0}{B1}] $
where $B0 = e^{-r_0}$ = time 0 price of risk free bond maturing at time =1 and $r_0$ is known at t0. ...
0
votes
0answers
48 views
Correlated GBM and OU processes
I want to model two different stochastic processes, such that:
$X_t , V_t$ are correlated with coefficient $\rho$. Where:
$\frac{dX_t}{X_t}=\mu_1dt+\sigma_1 dW_{1,t}$ and $dV_t=\theta(\mu_2-V_t)dt+\...
1
vote
3answers
96 views
Need help to interpret the definition of a diffusion process
https://studentportalen.uu.se/uusp-filearea-tool/download.action?nodeId=1134155&toolAttachmentId=218130
In these lecture notes at page 15 and 16 I am looking at the definition of diffusion ...
3
votes
0answers
51 views
Price of a stochastic game between an agent and the market
In the article Pricing via utility maximization and entropy from Richard Rouge and Nicole El Karoui, they define the value function of the optimization problem as
\begin{align}
V(x,C) = \dfrac{1}{\...
0
votes
2answers
212 views
Integral of Brownian Motion w.r.t Time: what is wrong with this solution? [duplicate]
My question is about a stochastic integral of brownian motion w.r.t time.
Let $W(t)$ the Wiener process (or brownian motion). I want to calculate this:
\begin{eqnarray}
X(t)=\int_{0}^t dt' W(t').
\...
2
votes
1answer
79 views
The duality of the free energy and relative entropy used to deduce deduce the stochastic game between an agent and the market
I am reading the article Pricing via utility maximization and entropy by Richard Rouge and Nicole El Karoui. They talk about the relative entropy of a probability measure $Q$ with respect to the ...
2
votes
1answer
59 views
How to deduce the formula of the wealth process of a stochastic volatility model?
I am reading the paper Solution of the HJB Equations Involved
in Utility-Based Pricing from Daniel Hernandez and Shuenn Jyi Sheu.
The authors consider the utility function $U: \mathbb{R} \to \...
0
votes
1answer
43 views
standard brownian vs brownian motion
We say Xt with paramters (µ,σ) is brownian process if (Xt-s - X t) ~N (µs,σ2 s) AMONG other conditons .
Here we don't speak about any particular distribution for X t. We only say it is a brownian ...
2
votes
0answers
62 views
Pricing caplet with Bachelier (normal dynamic) using forward measure
I'm trying to price caplet with Bachelier under forward measure, but I can't find any solution.
Remind that Bachelier assumed rates follow a normal dynamic. So here what I was doing :
$C_t(T,T+d)$ ...
2
votes
2answers
128 views
Ho Lee model in Baxter&Rennie
I am currentyl reading Baxter&Rennie and I have a difficulty with understanding a derivation of formula for one function, $g(x,t,T)$ (this can be found on page 152 in the book). I know that there ...
1
vote
0answers
93 views
Compo Feature in Asian Option on Futures
I'm pricing an Asian option on futures using Turnbull–Wakeman (other suggestions welcome) where the average is defined as
$A _ { t _ { 1 } , t _ { n } } ^ { A , f } = \frac { 1 } { n } \sum _ { i = 1 }...
0
votes
0answers
31 views
If you have normally distributed returns, shouldn't you have the same adjustment factor as lognormally distributed?
We know that when using lognormal returns, the number you need to plug in is not the apparent return, but $\mu-\sigma^2/2$ because what you really have is, in essence, (1) a deterministic growth of $\...
2
votes
1answer
61 views
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:
...
1
vote
0answers
43 views
Extension of HJM to multiple factors
The HJM model calibrates the entire forward curve using the existing yield curve data and this results in the following expression for its instantaneous forward rate-
$$df(t,T)=\sigma(t,T)\int_0^T\...
3
votes
1answer
79 views
Application of Vibrato Montecarlo methods
Ciao,
I was studying Vibrato Montecarlo methods and I came up with a very simple question: what is an real application of this method? Let me explain.
In short the main idea of the method is the ...
2
votes
1answer
115 views
Discretizing a Continuous Time Stochastic Volatility Model
How does the discrete time stochastic volatility model arise from the continuous time one?
Also, forgive me for cross-posting.
I have the following continuous time SDE for a stochastic volatility ...
0
votes
0answers
32 views
Impact analysis of parameters in the 2 Factor Hull White Model
Through the 2-Factor-Hull White Model you can model the yield curve if you have the parameters $a, b, \sigma, \eta$ given.
Is there any way to measure the impact of these parameters on the yield ...
0
votes
0answers
81 views
Dynamics of an option on a future
I have trouble understanding why $$V_s=exp(\int_s^t r_u du) \int_s^t exp(−\int_t^v r_u du)\theta_v dW_v$$ is the solution to the following SDE
$dV_s=\theta_s dW_s+r_s V_s ds$.
I tried of course with ...
2
votes
0answers
46 views
Prove the given stochastic integral are equally distributed
Let $W^i_t$ and $W_t$ be pairwise independent Brownian motions for $i \in \{1, \dots , d\}$.
Let $X_t^i$ be $d$ independent Ornstein–Uhlenbeck processes for $i \in \{1, \dots , d\}$, i.e. each $X_t^i$...
1
vote
1answer
127 views
Two papers - two different solutions of the Ornstein-Uhlenbeck process
Bernal 2016 says that the solution of
$$ dr_{t}=\lambda*(\mu-r_{t})*dt+\sigma dW_{t} \qquad (eq.1) $$
equals
$$ r_{t}=r_0*exp(-\lambda t)+\mu(1-exp(-\lambda t))+\sigma \int_{0}^{t} exp(-\lambda t)...
1
vote
0answers
48 views
Bond prices at future times under Vasick one-factor model
In Vasicek one-factor model (and in other affine models), the price of a zero-coupon bond at time $t$ conditional on the information at this time is
$$P(t,T) = E[e^{-\int^T_tr(u)du}|F_t] = A(t,T)e^{-...
2
votes
1answer
141 views
Calculating the stochastic integral of $\exp(-rt)S_t$
I am currently reading lecture notes which aim to show that if
$$
S_t = S_0 \exp (\mu t + \sigma W_t)
$$
then, under the probability measure $\tilde{\mathbb{P}}$ with density
$$
\gamma_T = \exp (c W_T ...
2
votes
0answers
168 views
Applying Ito's formula to complex functions
Within my lecture notes, the following definition is given:
We say that the stochastic process $X_t$ has stochastic differential
$$
dX_t = b_t dt + \sigma_t dW_t
$$
if and only if
$$
X_t = ...
1
vote
0answers
41 views
autocorrelation function of the trending OU process
What is the autocorrelation function of the trending Ornstein-Uhlenbeck (OU) process?
First, the OU process
$dX_t = -\frac{1}{\mu} X_t + \sqrt{\frac{2\sigma^2}{\mu}} dW_t $
generates coloured noise ...
1
vote
1answer
68 views
Girsanov's Theorem for Multiple Risky Assets
Girsanov's theorem provides the measure transformation from probability measure P to Q such that-
$dW_t^Q=dW_t^P+\lambda dt\implies \xi_tW_t^Q$ is a martingale under the P measure where $\xi_t=e^{-\...
2
votes
1answer
129 views
Black Scholes in the case of dividends
Let's take the case where the underlying stock has the continuous dividend yield $\delta$. Then, in the risk-neutral world, $\frac{dS}{S}=(r-\delta)dt+\sigma dW^Q$. Suppose we want to price a ...
2
votes
1answer
182 views
Show that the Ito integral is Gaussian
Let $f(t), 0 \leq t \leq T$ be a deterministic function with $f(t) = \sum_{i=1}^na_{i-1}1_[t_{i=1}, t_i)(t)$ with $0 \leq t_0<t_1<...<t_{n-1} = T$. Show that the stochastic integral $I_t(f) ...
-2
votes
1answer
101 views
squaring stochastic calculus and other solutions [closed]
It is well-known that the solution to the stochastic SDE
$$ dS = S_0(\mu dt + \sigma dWt) $$
is
$$ S_t=S_0 e^{(\mu-\frac{\sigma^2}{2})t+W_t} $$
Were $\sigma=0$, this is simply the formula for ...
2
votes
1answer
89 views
Distribution in Heston
$$dV_t=-k(V_t-1)dt+ \epsilon\sqrt{V_t}dW_t$$
$W_t$ is wiener process and the rest is just some parameters.
For $T_{i+1}>T_{i}$ how do I find the expectation and variance of $V_{T_{i+1}}$ ...
1
vote
1answer
103 views
Finding the process of $X/Y$
This comes from Mark Joshi's concepts of mathematical finance exercise 4 chapter 11.
If
$$dX_t = \alpha X_t dt + \beta X_t dW_t$$
$$dY_t = \alpha Y_t dt + \gamma Y_t d\tilde{W}_t$$
with $W$ ...
3
votes
2answers
152 views
Isn't GBM the equivalent of adding infinitessimally small normally distributed returns?
The classic treatment of GBM for asset pricing leads to a point where eventually one gets a solution that is the same as assuming an underlying arithmetic Brownian motion, $X_t$, which has (over unit ...
