5
votes
References for Stochastic Control for finance
Peter Forsyth of UWaterloo is my favourite author on this topic (one of my top three in MathFin!)
Personal Homepage with Lots of Papers
Optimal allocation under wealth goals, optimal decumulation ...
5
votes
Code examples of solving Stochastic Optimal Control problems
FWIW, I implemented one such control solution for my course project. See here
- 514
4
votes
Accepted
example Hamilton-Jacobi-Bellman Equation - clarification of $dX_t$ derivation using $\pi_t$, $\Pi_t$
We assume that
\begin{align*}
\frac{dX_t}{X_t} &= (r+\pi Y_t)dt + \pi\sigma dW_t,\tag{1}\\
dY_t &= -\lambda Y_t + dB_t.\tag{2}
\end{align*}
From $(2)$,
\begin{align*}
Y_t = Y_0 e^{-\lambda t}...
- 21k
3
votes
Accepted
Question on derivation step in portfolio replication under different borrowing and lending rates
Noting that
$$ B= V -\alpha S = V - (\alpha S)^+ + (\alpha S)^- $$
$$ = (V - (\alpha S)^+)^+ - (V - (\alpha S)^+)^- + (\alpha S)^-,$$
a clearer way to write the dynamics of the funding costs (funding ...
- 5,008
3
votes
How do your solve for trader's optimal demand in market similar to Kyle's model?
Generic knowledge about this kind of models
Let me try to get your model close to elements that are known:
Time continuous Kyle's model is something that is solved in Çetin, Umut, and Albina Danilova....
- 11.5k
3
votes
References for Stochastic Control for finance
Look into Huyên Pham, Continuous-time Stochastic Control and Optimization with Financial Applications; Salvatore Federico, Giorgio Ferrari, Luca Regis (Editors). Applications of Stochastic Optimal ...
- 11.9k
3
votes
How to, from various hypotheses on the P&L, get known models (BS, Heston etc ...)
You can characterize local volatility (LV) models by the existence of a delta-hedging strategy which reduces the variance of P&L to zero, together with an assumption that P&L is a continuous ...
- 1,875
2
votes
Accepted
Merton portfolio allocation problem proportions/weights >1 or <0?
Your statement should be correct, the weights into the risky asset are not bounded between $0$ and $1$. Essentially, by setting $r=0$ you omit the term which shows that your weights always sum up to ...
- 1,456
2
votes
Closed-form solution to optimal single assset position sizing with predicted returns
As you have defined the Sharpe ratio, it is independent of your position. You have
$$
\mathrm{E}[rf(z_1)] = \alpha z_1 f(z_1)
$$
and
$$
\mathrm{Var}[rf(z_1)] = \sigma_2^2f(z_1)^2
$$
and hence
$$
\frac{...
- 5,891
2
votes
Game theory and stochastic calculus
The field you have in mind is covered with differential game theory, and it game birth to Mean Field Games (MFG), the book posted in a comment is certainly the reference: Probabilistic Theory of Mean ...
- 11.5k
1
vote
Beta Weighting Deltas: What happens to the non-correlation part?
"At Beta=1 the underlying is expected to be as volatile as the index as well as move (more or less) together with the index." is not right.
Beta has nothing to do with volatility, at-least ...
- 1,875
1
vote
Accepted
American option pricing formulation
In explicitly wording my own question yesterday and naming my doubts, I think I may have stumbled upon the explanation:
On the one hand, indeed we have
$$
{\text{ess}\sup}_{s\in[0,T]}\mathbb{E}\left[...
- 51
1
vote
non-Markovian/path-dependent optimal log utility and HJB-PDE
This answer will provide somewhat of an educated guess, but is by no means rigorous or exact. It is based on my preliminary readings/understanding of subsections 5.4.1 and 5.4.2 of Applied Stochastic ...
- 225
1
vote
The duality of the free energy and relative entropy used to deduce deduce the stochastic game between an agent and the market
Such relationships are commonly covered in statistical mechanics, so any decent statsictal mechanics book should help. Here is an article that gives a very nice summary of the key concepts:
https://...
- 6,204
1
vote
Ansatz and HJB equation
derivatives had a wrong sign (had to be γ(-f) instead of γf for example)
- 121
1
vote
Stochastic optimization and mean field games : textbooks
I answer my own question. A starting point would be :
the summer school on mean field games, provided by the University of Chicago.
summer school
- 111
1
vote
Understanding the HJM drift condition's dimensions
Your issue is that you misinterpreted the NA criterion, it reads:
$$
a_t(x) \triangleq \sum_{i=1}^{\infty} \left(b^i_t(x) \int_0^x (b_t^i(u))^T du\right)e_i,
$$
where $b^i_t$ denotes the $i^{th}$ ...
- 353
1
vote
Accepted
optimal strategy problem (using Jensen's inequality)
I assume that the problem is $$\max_{\pi} E\left(\ln Z_T^{\Pi} \right).$$
Note that $\ln Z_t^{\Pi} = \ln X_t^{\Pi} -\ln X_t^{\rho}$. Moreover,
\begin{align*}
d\ln Z_t^{\Pi} &= d\ln X_t^{\Pi} -d\ln ...
- 21k
1
vote
Accepted
trading strategy problem - initial capital x buys S over time [0,T] at the constant rate of x/T euros per unit of time
Equations (1) to (3) are correct. Your investment strategy is then, $\forall t > 0$
$$ X_t = \theta _ t S_t $$
Provided you use this strategy as part of self-financing portfolio you can write the P&...
- 14.5k
1
vote
Accepted
investor terminal value of portfolio with two risky assets 1) correlated 2)not correlated $\phi_t^1=S^{2}_{t}, \ \phi_t^2=S^{1}_{t}$
Let $Y_t := 2 S_t^1 S_t^2 $. Applying (multivariate) Itô to the function $f(t,S_t^1,S_t^2)=2 S_t^1 S_t^2$ yields a stochastic differential equation for $Y_t$
$$ \frac{dY_t}{Y_t} = \frac{dS_t^1}{S_t^1}...
- 14.5k
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