diff --git a/src/asgi_lifespan/_concurrency/asyncio.py b/src/asgi_lifespan/_concurrency/asyncio.py
index 2096303..1ae5370 100644
--- a/src/asgi_lifespan/_concurrency/asyncio.py
+++ b/src/asgi_lifespan/_concurrency/asyncio.py
@@ -2,6 +2,7 @@
 import contextlib
 import types
 import typing
+from typing import Optional
 
 from .base import BaseEvent, BaseQueue, ConcurrencyBackend
 
@@ -68,8 +69,8 @@ async def run_and_silence_cancelled() -> None:
     async def __aexit__(
         self,
         exc_type: typing.Optional[typing.Type[BaseException]] = None,
-        exc_value: BaseException = None,
-        traceback: types.TracebackType = None,
+        exc_value: Optional[BaseException] = None,
+        traceback: Optional[types.TracebackType] = None,
     ) -> None:
         assert self.task is not None
 
diff --git a/src/asgi_lifespan/_concurrency/trio.py b/src/asgi_lifespan/_concurrency/trio.py
index 3599a18..042fcde 100644
--- a/src/asgi_lifespan/_concurrency/trio.py
+++ b/src/asgi_lifespan/_concurrency/trio.py
@@ -1,5 +1,6 @@
 import types
 import typing
+from typing import Optional
 
 import trio
 
@@ -66,7 +67,7 @@ async def __aenter__(self) -> None:
     async def __aexit__(
         self,
         exc_type: typing.Optional[typing.Type[BaseException]] = None,
-        exc_value: BaseException = None,
-        traceback: types.TracebackType = None,
+        exc_value: Optional[BaseException] = None,
+        traceback: Optional[types.TracebackType] = None,
     ) -> None:
         await self._exit_stack.__aexit__(exc_type, exc_value, traceback)
diff --git a/src/asgi_lifespan/_manager.py b/src/asgi_lifespan/_manager.py
index d73cfc4..a2748a9 100644
--- a/src/asgi_lifespan/_manager.py
+++ b/src/asgi_lifespan/_manager.py
@@ -1,5 +1,6 @@
 import typing
 from types import TracebackType
+from typing import Optional
 
 from ._compat import AsyncExitStack
 from ._concurrency import detect_concurrency_backend
@@ -94,9 +95,9 @@ async def __aenter__(self) -> None:
 
     async def __aexit__(
         self,
-        exc_type: typing.Type[BaseException] = None,
-        exc_value: BaseException = None,
-        traceback: TracebackType = None,
+        exc_type: Optional[typing.Type[BaseException]] = None,
+        exc_value: Optional[BaseException] = None,
+        traceback: Optional[TracebackType] = None,
     ) -> typing.Optional[bool]:
         if exc_type is None:
             self._exit_stack.push_async_callback(self.shutdown)