# Download Contract Theory in Continuous-Time Models by Jakša Cvitanic, Jianfeng Zhang PDF

By Jakša Cvitanic, Jianfeng Zhang

In fresh years there was an important elevate of curiosity in continuous-time Principal-Agent types, or agreement thought, and their functions. Continuous-time types supply a robust and stylish framework for fixing stochastic optimization difficulties of discovering the optimum contracts among events, lower than quite a few assumptions at the info they've got entry to, and the influence they've got at the underlying "profit/loss" values. This monograph surveys fresh result of the speculation in a scientific approach, utilizing the procedure of the so-called Stochastic greatest precept, in versions pushed through Brownian Motion.

Optimal contracts are characterised through a process of Forward-Backward Stochastic Differential Equations. In a couple of attention-grabbing precise circumstances those should be solved explicitly, allowing derivation of many qualitative fiscal conclusions.

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**Additional resources for Contract Theory in Continuous-Time Models**

**Example text**

3. 2 Consider now this generalization of the previous example, where we allow volatility v to affect the drift in a nonlinear way: dXt = rt Xt + b(t, ut , vt ) dt + vt dBt where b is a deterministic function of (t, u, v), and rt is a given deterministic process. Assume exponential utility functions, Ui (x) = − R1i exp{−Ri x}. Also assume, with a slight abuse of notation g, that g(t, Xt , ut , vt ) = μt Xt + g(t, ut , vt ) where g is a deterministic function of (t, u, v), and μt is a given deterministic process.

In the notation of Sect. 1, we have σ = v, b = uv, g = UA (CT ). h = uA (t, ct , ut ), Define fA (t, c, z, u) := uA (t, c, u) + uz, fA∗ (t, c, z) := sup fA (t, c, z, u). 1, ∂z fA = u is bounded. 11) and the following BSDE have Lipschitz continuous generators: WtA = UA (CT ) + T t fA∗ t, ct , ZtA dt − T t ZtA dBt . 2 The agent’s admissible set U is a set of FB -adapted processes u taking values in U . 4, we first get the sufficient conditions. 2 hold, and T 2 E UA (CT ) + 0 uA (t, ct , 0) + fA∗ (t, ct , 0) dt 2 < ∞.

Thus, in the first best solution, the agent will choose volatility v such that XT = zλ Z¯ T 1 γA −1 1 + (zZ¯ T ) γP −1 . In particular, if the risk aversions are the same, γA = γP , we get 1 XT γ −1 Z¯ T A = (zλ ) 1 γA −1 1 + z γA −1 and CT = XT − IP (zZ¯ T ) = zλ XT 1 γA −1 (zλ ) 1 γA −1 1 + z γA −1 . 1 Linear Dynamics and Control of Volatility 23 That is, if the principal and agent have the same power utility, and they both behave optimally, the payoff turns out to be linear at time T . 3, hence the linear contract implements the first best.