Questions tagged [stochastic-calculus]
A branch of mathematics that operates on stochastic processes.
0
votes
1answer
123 views
When $E[f(\alpha,X)] = f(\alpha, E[X])$
When $E[f(\alpha,X)] = f(\alpha,E[X])$, where $f$ is some convex function of the first and second variables, except when the first variable takes the value $\alpha$ in which case the equality holds, ...
3
votes
2answers
98 views
Find the brownian motion associated to a linear combination of dependant brownian motions
I have $N$ correlated standard one-dimensional Brownian motions $W_1,\ldots,W_N$ with correlation matrix $\rho$ and I consider the process $Z_t \equiv \sum_{i=1}^N \mu_i (t) W_t$ where the $\mu_i$ are ...
3
votes
1answer
71 views
Derivation and expectation interchange
I would like to know when it is allowed to interchange derivation and expectation.
Suppose $X$ is some r.v whose dynamic is controlled by some parameter $\sigma$ and suppose $h$ is some smooth ...
3
votes
1answer
71 views
Solution to a Geometric Ornstein Uhlenbeck Process $dX_t = \kappa(\theta - X_t)dt + \sigma X_t dW_t$
I've been searching for the solution to the modified Ornstein-Uhlenbeck process
\begin{equation*}
dX_t = \kappa(\theta - X_t)dt + \sigma X_t dW_t
\end{equation*}
but it surprisingly hard to find. The ...
1
vote
0answers
31 views
Milstein discretization of the CIR process
Given the CIR process $\ dX_t = (a − bX_t ) dt +
\sigma \sqrt{X_t}dW_t$ - I want to show that its Milstein scheme is $\ X_{i+1} - X_i = ((a − bX_i) - 0.25\sigma^2)\Delta + \sigma\sqrt{X_i}\sqrt{\...
2
votes
0answers
39 views
Ito Diffusion with Change of Measure
Let $(X_t)$ be an Ito diffusion with speed $(V_t)$, under a probability measure P. Could there exist a change of measure to a probability measure Q, with Q ~ P, under which $(X_t)$ is an Ito diffusion ...
4
votes
0answers
63 views
Feynman-Kac to derive stochastic representation
$u_t + \frac{1}{2}\sigma^2x^2u_{xx} - \alpha + \lambda((K_d - x)^+ - u) = 0$ with terminal condition $u(T, X) = (K_m - X(T))^+$
$dX = \sigma X(t)dW_t$
$\alpha$ and $\lambda$ are constants
Ok so ...
1
vote
0answers
34 views
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 ...
2
votes
0answers
53 views
For an Ito Process, $d\ln{X} \neq \frac{dX}{X}$ and $(d\ln{X})^2 = (\frac{dX}{X})^2$, but $d\ln{X} \neq \pm \frac{dX}{X}$
In normal calculus we can write $d\ln{x} = \frac{dx}{x}$ since there is no quadratic variation to deal with. This isn't true for stochastic processes, and Ito's Lemma is used to calculate $d\ln{X}$. ...
1
vote
0answers
53 views
Proving Flow Property of Stochastic Differential Equation
I am trying to show that $X_t^{s,x} = X_t^{r, X_r^{s,x}}$ for $0 \leq s \leq r \leq t$, $x \in \mathbb{R}^n$ is a given initial condition for time $s$, for some SDE:
\begin{equation*}
d X(u)=b(X(u))d ...
2
votes
1answer
57 views
If S(t) is geometric Brownian motion, what is the distribution of S(t+h)-S(t)?
Suppose we have a geometric Brownian $S(t)$ which follows a lognormal process. Say
$$
\begin{equation}
dS_t = \mu S_t dt + \sigma S_tdW_t
\end{equation}
$$
My question is what is the distribution of $...
1
vote
2answers
95 views
How to numerically simulate exponential stochastic integral
For example given an integral
$$
\int^t_0 \exp(aW(t'))\,dt', t\in\mathbb R_+
$$
where $W(t')$ is a standard Wiener process.
I've been very confused about stochastic integrals like $\int^t_0 W(t')\,...
2
votes
0answers
57 views
SDE of futures price under non-constant interest rate and volatility process
I'm trying to figure out the form of the SDE of futures price under the risk neutral measure, when stock price follows GBM:
&...
1
vote
0answers
99 views
On quadratic covariation
I ran through an equality in a paper I was reading but couldn't check if it is correct.
Let $W^1_t$, $W^2_t$ and $W^3_t$ be three brownian motions, not necessarily independent, is it true that the ...
3
votes
1answer
98 views
HJM model Baxter Rennie: differentiating the discounted asset price using Ito
From Baxter and Rennie Page 145:
$Z(t,T) = exp(\int_{0}^{t}\Sigma(s,T)dW_s - \int_{0}^{T}f(o,u)du - \int_{0}^{t}\int_{s}^{T}\alpha(s,u)duds)$
where $\Sigma(t,T) = \int_{t}^{T}\sigma(t,u)du$
How ...
3
votes
1answer
128 views
Differential of integral of Wiener process over time
I am trying to compute this quantity:
$\frac{d}{dt}\int_{0}^{t} W_s ds $
Where $W_t$ is a Wiener process. Is there a theorem which tells how this can be computed?
I have tried https://en.wikipedia....
3
votes
1answer
119 views
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 ...
2
votes
1answer
143 views
How to compute the dynamic of stock using Geometric Brownian Motion?
I have been given the following question:
Given that $S_t$ follows Geometric Brownian Motion, write down the dynamic of $S_t$ and then compute the dynamic of $f(t,S_t) = e^{tS^{2}}$
For the first ...
2
votes
4answers
164 views
Basic book on stochastic calculus, Itô and jump processes and Brownian Motion
I was looking for a good book that explains at beginner-level the basic of stochastic calculus, probability and random variables, Itô and jump processes as well as Brownian Motion.
At university we ...
6
votes
0answers
92 views
Random variable minus Integral of Ito Generator is a Martingale under what conditions?
I am reading about american option pricing and the variational inequality, and the book I am reading states, in the derivation of the variational inequality, the following is a martingale: $$M_s = U(s,...
0
votes
1answer
139 views
Why don't I get this right $\frac{d}{dt}\mathop{\mathbb{E}}\left[ e^{-\int_t^Tr(s)ds}|\mathscr{F}_t \right]$
Let $r$ a random process defined by :
$$dr_t=\theta(t)dt + \sigma dW_t$$
$\theta$ is deterministic in $t$ and $W$ a brownian motion.
I don't know where my calculation below is going wrong :
Let $...
1
vote
0answers
97 views
Ultra Powerfull Vibrato Montecarlo for delta sensitivities of a not regular payoff
Ciao,
I am working on a derivative with the following payoff at time $T$:
$$
\sqrt{(S_T - K)^+}
$$
where $S_T$ is the value of the stock at the expiring date.
As usual we will assume $S_t$ to be a ...
1
vote
0answers
58 views
kolmogorov backward equation intuition
The kolmogorov backward equation equation states that the probability density of a random variable $x$ which follows $dx= \mu dt + \sigma dw$
is given by
$-p_t = \mu p_x + 0.5\sigma^2 p_{xx} $
...
19
votes
1answer
426 views
What is the trickiest thing to get right in Rates Quant recently (2019)?
What are the biggest challenges for Rates Quants in 2019? Most quants have been through a lot over the past years-shifting their SABR models in JPY swaptions, fixing the FVA models for negative rates, ...
3
votes
2answers
73 views
Conditional Expectation with Indicator Functions for Poisson Process First Jump Time (Option Pricing PDE)
This is supposed to be for the derivation of a PDE for pricing a specific type of option, from the book 'Nonlinear Option Pricing' (Guyon).
The option delivers $g(\tau, X_{\tau})$ at time $\tau$ if $\...
0
votes
1answer
117 views
negative values in geometric brownian motion
A GBM
$ \frac{dx}{x} = \mu dx + \sigma dW $
solves to
$x_t = x_o e^{(\mu - \sigma^2)t + \sigma W_t}$
From the solution, it is clear that $x_t$ cannot become negative. However, it is not so clear ...
3
votes
1answer
90 views
Limits of integration when applying stochastic Fubini theorem to Brownian motion
I'm looking at the solution below from Quantuple, it's a nice, succinct solution but I'm confused about how the limits of the integrals in the second line come from. Could someone please elaborate on ...
3
votes
0answers
61 views
Stochastic Differential equation: CAPM
Let $R=(R_1, \dots ,R_M)$′ denote a vector of excess returns of M assets observed at $n$ time points, $0<t_1<t_2< \cdots <t_n<T$, within a time span $T>0$.
We wish to explain the ...
2
votes
0answers
77 views
Pre-requisites for Finance Mathematics
I would like to pursue research in the areas of Financial Mathematics. Hoping to look into Operations Research, Risk Management and Stochastic Modeling. Anyone got some suggestions on useful resources ...
4
votes
3answers
153 views
How to prove that $X_s=\int^s_0 f(u)dW_u$ is independant from $X_t-X_s$
I am asked to prove that $X_s$ and $X_t-X_s$ are independant for $s<t$ then $$X_t=\int^t_0f(u)dW_u$$ for a deterministic function $f$ and brownian motion $W_t$. For the proof I am giving a hint to ...
3
votes
0answers
31 views
Stochastic integral representation of $F(T-s,X_s)$-type equations
For $T\in R$ given and fixed consider:
$$
{\rm d}F(T-t,X_t)=g(T-t,X_t)\,{\rm d}W_t.
$$
where $g(t,x)$ is a given functions and $X_t$ is a given process driven by a brownian motion ($dX_t=(...)dt+(...)...
6
votes
1answer
140 views
Show that $(W_t, \int_0^t W_s ds)$ has a normal joint distribution
I have to show that, if $W_t$ is a 1-d Brownian motion then
$\biggl(W_t, \int_0^t W_s ds\biggr)$ has normal distribution.
Hint: apply Ito formula to this bivariate process.
Any idea or suggestion on ...
1
vote
1answer
57 views
How to check if $ E [\exp \{ \int_0^t \frac{Y_u^2}{1+Y_u^2}du \}]< \infty $
$dY_t=2Y_tdt+2\sqrt{1+Y_t^2}dW_t$ where $W_t$ is $P-$Brownian motion (Wiener process).
I have defined a new measure $Q$ where the Kernel density (In Girsanov theorem) is
$$ \phi_t = \frac{Y_t}{\sqrt{...
3
votes
2answers
208 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
83 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
127 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
112 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
119 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
103 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
48 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
59 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
72 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 $...
3
votes
1answer
55 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
244 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
95 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
26 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)}...
2
votes
1answer
187 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 ...
3
votes
1answer
133 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:
$$\...
1
vote
0answers
38 views
Model of asset substitution/risk shifting in continuous time
Consider a firm with cash flows $X_t$, which under a risk-neutral probability measure, follows a geometric brownian motion:
$$dX_t = X_t[(r-\beta)dt + \sigma dZ_t]$$
where $r>0$ is the risk-free ...
3
votes
0answers
50 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 ...
