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# 数学代写|随机分析代写Stochastic Calculus代考|IMSE760 Intra-Personal Equilibria with Fixed Initial Data

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## 数学代写|随机分析代写Stochastic Calculus代考|Intra-Personal Equilibria with Fixed Initial Data

Consider an agent at time 0 with a fixed state $x_0$ who correctly anticipates that her self at each future time $t$ faces the problem (2) and who has no control of future selves at any time. A strategy $\hat{\mathbf{u}}$ can be consistently implemented by the agent throughout the entire horizon $[0, T]$ if the agent has no incentive to deviate from it at any time along the state path. Actions that the agent might be taking were she not on the state path are irrelevant. To be more precise, for any fixed initial data $\left(0, x_0\right)$, we define $\hat{\mathbf{u}}$ to be an intra-personal equilibrium starting from $\left(0, x_0\right)$ if (5) holds for any $\mathbf{a} \in \mathbf{U}, t \in[0, T)$, and $x \in \mathbb{X}_t^{0, x_0, \hat{\mathbf{u}}}$, where $\mathbb{X}_t^{0, x_0, \mathbf{u}}$ denotes the set of all possible states at time $t$ along the state path starting from $x_0$ at the initial time and under the strategy $\hat{\mathbf{u}}$.

It is evident that the intra-personal equilibrium defined in Definition 2 is universal in that it is an equilibrium starting from any initial data $\left(0, x_0\right)$. On the other hand, starting from a fixed state $x_0$ at time 0 , the state process in the future might not be able to visit the whole state space; so an equilibrium starting from $\left(0, x_0\right)$ is not necessarily universal, i.e., it is not necessarily an equilibrium when the agent starts from other initial data. For example, He et al. (2020) consider a continuoustime portfolio selection problem in which an agent maximizes the median of her terminal wealth. With a fixed initial wealth of the agent, the authors derive a set of intra-personal equilibrium strategies starting from this particular initial wealth level. They show that these strategies are no longer equilibria if the agent starts from some other initial wealth levels, and in particular not universal equilibria in the sense of Definition 2.

## 数学代写|随机分析代写Stochastic Calculus代考|Set of Alternative Strategies

In Definition 2, the set of strategies that the agent can choose at time $t$ to implement for the period $[t, t+\varepsilon)$, denoted as $\mathbf{D}$, is set to be the entire set of feasible strategies U. This definition is used in Björk et al. (2017), Ekeland and Pirvu (2008), and Ekeland et al. (2012). In some other works, however, $\mathbf{D}$ is set to be the set of constant strategies $\mathbb{U}$; see for instance Ekeland and Lazrak (2006, 2008, 2010), Björk and Murgoci (2010), and Basak and Chabakauri (2010). He and Jiang (2019) show that the choice of $\mathbf{D}$ is irrelevant as long as it at least contains $\mathbb{U}$. Indeed, this can be seen from the observation in Theorem 1 that $\Gamma^{\tau, y, \mathbf{u}}(t, x ; \mathbf{a})=\Gamma^{\tau, y, \mathbf{u}}(t, x ; \mathbf{a}(t, x))$ for any $\mathbf{a} \in \mathbf{U}$. He and Jiang (2019) also show that for strong intra-personal equilibrium, which will be introduced momentarily, the choice of $\mathbf{D}$ is relevant.

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