# Tag Info

### 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 ...

### Code examples of solving Stochastic Optimal Control problems

FWIW, I implemented one such control solution for my course project. See here
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### 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}...
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### 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....
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $i^{th}$ ...
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### 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 ...
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### How do you actually solve a stochastic HJB equation in practice?

An HJB is made of two components: a "core component" that corresponds to applying an optimal control the "dynamics" of the value function (that is surrounding this optimizattion). ...
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### 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 ...
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### References for Stochastic Control for finance

it is a bit late but the 2007 book "Applied Stochastic Control of Jump Diffusions" from Oksendal and Sulem is quite good too.
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### non-Markovian/path-dependent optimal log utility and HJB-PDE

"Has anyone used this functional calculus to generalize the HJB-PDE from Markovian SDEs to non-Markovian SDEs? Can we simply write down the HJB-PDE with his new notions of derivatives and obtain ...
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### 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 ...
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### 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
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### 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://...
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### 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 ...
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### 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&...
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### 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}...
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