Utilités Progressives Dynamiques.

Abstract : In 2002, Marek Musiela Thaleia Zariphopoulo and introduced the concept of (\ em utility) forward, ie a dynamic utility, incremental, consistent with a given financial market. We can see this process as a random field $U (t, x)$ adapted to the available information, which every moment is a standard utility (hence in particular at the time $0$, compatible with a family of strategies, data $(X ^ (\ pi))$ in the sense that for all $t, h> 0$, $\ mathbb (E) (U (t + h, X ^ (\ pi) _ (t + h)) | \ mathcal (F) _t) \ leq U (t, X ^ (\ pi) _t)$ and there exists an optimal portfolio $X ^ *$ for which the inequality is an equality. \ \ The authors have done several articles on this, showing in particular how traditional utilities, power, exponential, etc need to be modified for dynamic progressive utilities. Limited attention has been paid to the investment universe. \ noindent In my thesis, I consider a much more general. Indeed, the contract is incomplete in the sense that an investor is forced, at every time $t \ ge 0$, choose those strategies eligible for closed convex cones, adapted $\ k_t (X_t)$ depend on the level of wealth X_t . I believe later that the random fields $U (t, x)$ evolve according to the dynamics \ begin (equation) \ label (eq: field) dU (t, x) = \ beta (t, x) + \ Gamma (t, x) dW_t, ~ U (0,.) = u (.) (\ text (given)) \ end (equation) As in the classical optimization, ( called retrograde rebuilt since the information from the end), I show that the% term% $\ beta (t, x)$ contains, necessarily contains a term like classical Hamiltonian% modified by the presence of volatility derived from% $\ Gamma (t, x)$ of the value gradually. And therefore all utility progressive% satisfied assumptions of regularities of Ito-lemma Ventzell% satisfied I propose to consider the equations of Hamilton-Jacobi-Bellman that satisfies a progressive utility $u (t, x)$. To conduct this study, I use a formula of generalized Itô formula call'd Ventzell-Friedlin, which allows for the decomposition Ito-type composed of a random field with an Ito process. I then show that the term $\ beta (t, x)$ contains necessarily a term of type classical Hamiltonian modified by the presence of derivative volatility $\ Gamma (t, x)$ of the value gradually. And therefore any incremental value that satisfies the assumptions of regularities of Ito-lemma Ventzell satisfy the stochastic differential equation \ begin (equation) \ EDPSU label () dU (t, x) = \ Big \ (-xU '_ (x) \, dt r_t + \ frac (1) () (xx 2U''_ (t, x)) \ | \ prod_ ( \ k_t (x) \ sigma_t) \ big ((x) U'_ (t, x) \ + eta_t \ Gamma'_x (t, x) \ big) \ | ^ 2 \ Big \) (t, x) \ , DT \> + \ Gamma (t, x) dW_t. \ end (equation) with $X as optimum portfolio process ^ *$ associated with the strategy $\ pi ^ *$ given by \ begin (equation) x \ pi ^ * (t, x) \ sigma_t =- \ frac (1) () (xx U''_ (t, x)) \ | \ prod_ (\ k_t (x) \ sigma_t) \ big (U ' _ (x) (t, x) \ + eta_t \ Gamma'_x (t, x) \ big) (t x) \ end (equation) \ noindent where $r$ is the short rate, $\ eta$ premium market, $\ sigma$ the variance-covariance matrix of assets and $\ prod_ (\ k_t (x) \)$ means sigma_t the projection operator on the cone $\ k_t (x) \$ sigma_t. \ \ This point of view to verify that the random field, if there is consistent with the investment universe. However, the question of convexity and monotony is a priori complex because there are no theorems comparison for progressive equations (which are (\ em forward)), unlike the case of backward equations. The question of interpretation and the role of volatility are found to be so central to this study. Contrary to the general framework that I considered here, and Mr. T. Musiela Zariphopoulo and C. Rogers et al were limited to cases where the volatility of utility is identically zero. The gradual process $u (t, x)$ is a deterministic function satisfying a nonlinear PDE, the authors have turned into space-time harmonic solution of the heat equation. \ \ My choice was étudire the issue of volatility in the technical change in cash, so I show stability of the utility concept gradual change of currency. The advantage of this technique compared with the conventional method,%% As with the classical, the problem is complicated by the fact that the space of n% Constrain is not invariant under change of numeraire. is that it can always be reduced to a market "martingale" ($r =$ 0 and $\ eta = 0$), which greatly simplifies the equations and calculations . The derivative of volatility appears to be a risk premium introduces instant market, which depends on the level of wealth of the investor. This new perspective can answer the question of the interpretation of the volatility utility. In the following, I study the dual problem and I show that the transform of (\ em Fenchel) $\$ tU concave function of $U (t, x)$ is also a Markov random field satisfying dynamics \ begin (eqnarray) \ label (EDPSDuale ') d \ tilde (U) (t, y) = \ left [\ frac (1) (2 \ (yy tU_ }''}( \ | \ tilde (\ Gamma) '\ | ^ 2 - \ | \ prod_ (\ k_t (- \ tU_y' (t, y)) \ sigma_t) (\ tilde (\ Gamma }^{'}_ y-y \ eta_t) \ | ^ 2 ) + y \ (y) tU_ 'r_t \ right] (t, y) dt + \ tilde (\ Gamma) (t, y) dW_t, ~ ~ \ tilde (\ Gamma) (t, y) = \ Gamma ( t \ tU_y '(t, y)). \ end (eqnarray) From this result I show that the dual problem admits a unique solution $Y ^ *$ in volatilté $\ nu ^ *$ is given by \ begin (equation) y \ nu ^ * (t, y) = - \ frac (1) (\ yy) (tU_'') \ Big (\ tilde (\ gamma) '+ y \-eta_t \ prod_ (\ k_t (- \ tU_y ') \ sigma_t) (\ tilde (\ Gamma }^{'}_ y-y \ eta_t) \ Big) (t, y). \ end (equation) \ noindent This will allow to establish the identities the following key: \ begin (eqnarray) & Y ^ * (t (U_x')^{- 1) (0, x)) = U'_x (t, X ^ * (t, x)) \-label (A) \ \ & (\ Gamma'_x U'_x + \ eta) (t, x) = (xU''(t, x) \ pi ^ * (t, x) \ sigma_t + \ nu ^ * (U_x '(t , x)) \ label (B). \ end (eqnarray)% Note that the term $(\ Gamma'_x U'_x + \ eta)$ is decomposed uniquely as% of its projection on the cone $\ K \ sigma$, which is the optimal strategy, and the projection on the dual cone $\ K ^ * \ sigma$,% which is the volatility of the optimal dual process. But our goal is two term project known as the projection% Á From the first identity we know that $U'_x (t, X ^ * (t, x))$ is simply the optimal dual process% Callback At this stage that the purpose of this study is carracteriser utilities progressive. The question then is: can we characterize the utility $U (t, x)$ for all $x> 0$ from the first identity? This may seem too much to ask as we seek to characterize the field $U$ knowing only its behavior along the unique optimal trajectory $X ^ *$. However, the answer turns out to be positive and relatively simple. Indeed, denote by $\ Y (t, x): = Y ^ * (t, (U_x')^{- 1) (0, x))$, and assume that the stochastic flow $X ^ *$ is invertible, $\ X$ denotes its inverse. So , reversing in (\ ref (A)), I deduce that $U_x '(t, x) = \ Y (t \ X (t, x))$. By integrating with respect to $x$, j' U get that $(t, x) = \ int_0 ^ x \ Y (t, \ X (t, z)) dz$, which proves the following theorem: \ begin (theo) Under assumptions of regularity and reverse flow $X ^ *$ $U$ processes defined by $U (t, x) = \ int_0 ^ x \ Y (t, \ X (t, z)) dz$ are the solutions of the incremental utility 'stochastic PDE (\ ref () EDPSU). \ end (theo)%% \ noindent Conversely, I show the theorem of stochastic PDE: \ begin (theo) Let $U$ be a random field solutions of the stochastic PDE (\ ref (EDPSU)). Using décompostion (\ ref (B)), if DHS following \ begin (eqnarray *) & dX ^ * _t (x) = X ^ * _t (x) (r_tdt + \ pi ^ * (t, X ^ * _t (x)) \ sigma_t (dW_t + \ eta_tdt)), X ^ * _0 (x) = x ~ \ \ & dY ^ * _t (y) = y ^ * _t (y) (-r_tdt + \ nu ^ * (t, Y ^ * _t (y)) dW_t), ~ Y ^ * _0 (y) = y \ end (eqnarray *) admit strong solutions unique and monotonic, then, noting $\ Y (t, x): = Y ^ * (t, (U_x')^{- 1) (0, x))$ and $\ X$ the reverse flow of $X$, we get that $U (t, x) = \ int_0 ^ x \ Y (t, \ X (t, z)) dz$. If in addition $X ^ *$ and $Y ^ *$ is increasing, $U$ is concave. \ end (theo) \ noindent% In this work, I still consider an incomplete market, in a second part of this work, I place myself in a much broader sense in which the assets are assumed to be locally-bounded cadlag and Therefore filtration is no longer a Brownian filtration. I replace the convex cone type constraints by constraints more general type convex set. The purpose of this section is to characterize all progressive utility with a minimum of assumptions, including with fewer assumptions on regularities of the random fields $U$. I no longer assume that $U$ is twice differentiable and therefore I can not apply the Ito-lemma Ventzell. The approach is so different: I first establish the optimality conditions optimal wealth process and the optimal dual process, and using the methods of analysis. Using these results I show, by elements of analysis, convexity and optimality conditions that all utilities generating progressive increasing wealth is of the form $\ int_0 ^ x \ Y (t, \ X (t, z)) dz$ with $\ Y$: $\ YX$ is supermartingale for all wealth $X$ and a martingale if $X = X ^ *$.
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Submitted on : Wednesday, July 21, 2010 - 9:32:07 AM
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Mohamed Mrad. Utilités Progressives Dynamiques.. Mathématiques [math]. Ecole Polytechnique X, 2009. Français. ⟨pastel-00005815⟩

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